diff --git "a/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt" "b/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt" @@ -0,0 +1,622 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf,len=621 +page_content='FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS OF SURFACES C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Dedicated to Lorenzo, Sabrina, and Fabrizio Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth, complex Fano 4-fold, and ρX its Picard num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that if ρX > 12, then X is a product of del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The proof relies on a careful study of divisorial elementary contractions f : X → Y such that dim f(Exc(f)) = 2, together with the author’s previous work on Fano 4-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, given f : X → Y as above, under suitable assumptions we show that S := f(Exc(f)) is a smooth del Pezzo surface with −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Introduction Smooth, complex Fano varieties have been classically intensively studied, and have attracted a lot of attention also in the last decades, due to their role in the framework of the Minimal Model Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The Fano condition is a natural positivity condition of the tangent bundle, and it ensures a rich geometry, from both the points of view of birational geometry and of families of rational curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It has been known since the 90’s that Fano varieties form a bounded family in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Del Pezzo surfaces are known classically, and the classification of Fano 3-folds have been in achieved in the 80’s, there are 105 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Starting from dimension 4, there are probably too many families to get a com- plete classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' still we aim to better understand and describe the behavior and properties of these varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In this paper we focus on Fano 4-folds X with “large” Picard number ρX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' let us recall that since X is Fano, ρX is equal to the second Betti number b2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with ρX > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then X ∼= S1 ×S2, where Si are del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To the author’s knowledge, all known examples of Fano 4-folds which are not products of surfaces have ρ ≤ 9, so that we do not know whether the condition ρ > 12 in Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We refer the reader to [Cas22b, §6] for an overview of known Fano 4-folds with ρ ≥ 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' there are few examples and it is an interesting problem to construct new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' As ρS1×S2 = ρS1 + ρS2, and del Pezzo surfaces have ρ ≤ 9, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 14J45,14J35,14E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11953v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='AG] 27 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Let us note that Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 and Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 generalize to dimension 4 the analogous result for Fano 3-folds, established by Mori and Mukai in the 80’s: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 ([MM86], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 3-fold with ρX > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then X ∼= S × P1 where S is a del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular ρX ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 relies on a careful study of elementary contractions of X of type (3, 2), together with the author’s previous work on Fano 4-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To explain this, let us introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A contraction is a surjective morphism f : X → Y , with connected fibers, where Y is normal and projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' f is elementary if ρX−ρY = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' As usual, an elementary contraction can be of fiber type, divisorial, or small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We say that an elementary contraction f : X → Y is of type (3, 2) if it is divisorial with dim S = 2, where E := Exc(f) and S := f(E) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Such f can have at most finitely many 2-dimensional fibers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' outside the images of these fibers, Y and S are smooth, and f is just the blow-up of the surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If y0 ∈ S is the image of a two-dimensional fiber, then either Y or S are singular at y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' these singularities have been described by Andreatta and Wi´sniewski, see Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In any case, Y has at most isolated locally factorial and terminal singularities, while S can be not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by N1(X) the real vector space of one-cycles with real coefficients, modulo numerical equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have dim N1(X) = ρX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For any closed subset Z ⊂ X, we set N1(Z, X) := ι∗(N1(Z)) ⊂ N1(X) where ι: Z �→ X is the inclusion, so that N1(Z, X) is the subspace of N1(X) spanned by classes of curves in Z, and dim N1(Z, X) ≤ ρZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We study an elementary contraction f : X → Y of type (3, 2) under the hy- pothesis that: dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular this implies that Y is Fano too (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We would like to compare (−KY )|S to −KS, but since S may be singular, we consider the minimal resolution of singularities µ: S′ → S and set L := µ∗((−KY )|S), a nef and big divisor class on S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that KS′+L is semiample (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then our strategy is to look for curves in S′ on which KS′ + L is trivial, using other elementary contractions of X of type (3, 2) whose exceptional divisor intersects E in a suitable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Hence let us assume that X has another elementary contraction g1 of type (3, 2) whose exceptional divisor E1 intersects E, and such that E ·Γ1 = 0 for a curve Γ1 contracted by g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set D := f(E1) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that an irreducible component C1 of D ∩ S is a (−1)-curve contained in the smooth locus Sreg, and such that −KY · C1 = 1 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If C′ 1 ⊂ S′ is the transform of C1, we have (KS′ + L) · C′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally let us assume that X has three elementary contractions g1, g2, g3, all of type (3, 2), satisfying the same assumptions as g1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also assume that FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 3 E1 · Γ2 > 0 and E1 · Γ3 > 0, where E1 = Exc(g1) and Γ2, Γ3 are curves contracted by g2, g3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then we show that S is a smooth del Pezzo surface with −KS = (−KY )|S (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' let us give an overview of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The previous construction yields three distinct (−1)-curves C′ 1, C′ 2, C′ 3 ⊂ S′ such that (KS′ + L) · C′ i = 0 and C′ 1 intersects both C′ 2 and C′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This shows that the contraction of S′ given by KS′ +L cannot be birational, namely KS′ +L is not big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also rule out the possibility of a contraction onto a curve, and conclude that KS′ + L ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we show that ωS ∼= OY (KY )|S, where ωS is the dualizing sheaf of S, and conclude that S is smooth and del Pezzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We believe that these results can be useful in the study of Fano 4-folds besides their use in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It would be interesting to generalize this technique to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us now explain how we use these results to prove Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We define the Lefschetz defect of X as: δX := max � codim N1(D, X) | D ⊂ X a prime divisor � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This invariant, introduced in [Cas12], measures the difference between the Picard number of X and that of its prime divisors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we refer the reader to [Cas22b] for a survey on δX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fano 4-folds with δX ≥ 3 are classified, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4 ([Cas12], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If δX ≥ 4, then X ∼= S1 × S2 where Si are del Pezzo surfaces, and δX = maxi ρSi − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5 ([CRS22], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Smooth Fano 4-folds with δX = 3 are clas- sified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' They have 5 ≤ ρX ≤ 8, and if ρX ∈ {7, 8} then X is a product of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore in our study of Fano 4-folds we can assume that δX ≤ 2, that is, codim N1(D, X) ≤ 2 for every prime divisor D ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To prove that ρX ≤ 12, we look for a prime divisor D ⊂ X with dim N1(D, X) ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To produce such a divisor, we look at contractions of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has an elementary contraction of fiber type, or a divisorial elementary contraction f : X → Y with dim f(Exc(f)) ≤ 1, it is not difficult to find a prime divisor D ⊂ X such that dim N1(D, X) ≤ 3, hence ρX ≤ 5 (Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The case where X has a small elementary contraction is much harder and is treated in [Cas22a], where the following result is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 ([Cas22a], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has a small elementary contraction, then ρX ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We are left with the case where every elementary contraction f : X → Y is of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In this case we show (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1) that, if ρX ≥ 8, we can apply our previous study of elementary contractions of type (3, 2), so that if E := Exc(f) and S := f(E) ⊂ Y , then S is a smooth del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This implies that dim N1(S, Y ) ≤ ρS ≤ 9, dim N1(E, X) = dim N1(S, Y ) + 1 ≤ 10, and finally that ρX ≤ 12, proving Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In §2 we gather some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then in §3 we develop our study of elementary contractions of type (3, 2), while in §4 we prove Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Notation We work over the field of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by N1(X) (respectively, N 1(X)) the real vector space of one-cycles (respectively, Cartier divisors) with real coefficients, modulo numerical equiva- lence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' dim N1(X) = dim N 1(X) = ρX is the Picard number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let C be a one-cycle of X, and D a Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by [C] (respec- tively, [D]) the numerical equivalence class in N1(X) (respectively, N 1(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also denote by D⊥ ⊂ N1(X) the orthogonal hyperplane to the class [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The symbol ≡ stands for numerical equivalence (for both one-cycles and divi- sors), and ∼ stands for linear equivalence of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' NE(X) ⊂ N1(X) is the convex cone generated by classes of effective curves, and NE(X) is its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' An extremal ray R is a one-dimensional face of NE(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If D is a Cartier divisor in X, we write D·R > 0, D·R = 0, and so on, if D·γ > 0, D · γ = 0, and so on, for a non-zero class γ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We say that R is K-negative if KX · R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has terminal and locally factorial singularities, and is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then NE(X) is a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Given a contraction f : X → Y , we denote by NE(f) the convex subcone of NE(X) generated by classes of curves contracted by f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we recall that there is a bijection between contractions of X and faces of NE(X), given by f �→ NE(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover dim NE(f) = ρX − ρY , in particular f is elementary if and only if NE(f) is an extremal ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' When dim X = 4, we say that an extremal ray R is of type (3, 2) if the as- sociated elementary contraction f is of type (3, 2), namely if f is divisorial with dim f(Exc(f)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also set ER := Exc(f) and denote by CR ⊂ ER a general fiber of f|ER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' note that ER · CR = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We will also consider the cones Eff(X) ⊂ N 1(X) of classes of effective divisors, and mov(X) ⊂ N1(X) of classes of curves moving in a family covering X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since X is Fano, both cones are polyhedral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have the duality relation Eff(X) = mov(X)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Preliminaries In this section we gather some preliminary results that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Andreatta and Wi´sniewski have classified the possible 2-dimensional fibers of an elementary contraction of type (3, 2) of a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In doing this, they also describe precisely the singularities both of the target, and of the image of the exceptional divisor, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 ([AW98], Theorem on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set S := f(Exc(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 5 Then f can have at most finitely many 2-dimensional fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Outside the images of these fibers, Y and S are smooth, and f is the blow-up of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let y0 ∈ S ⊂ Y be the image of a 2-dimensional fiber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' then one of the following holds: (i) S is smooth at y0, while Y has an ordinary double point at y0, locally factorial and terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' (ii) Y is smooth at y0, while S is singular at y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' More precisely either S is not normal at y0, or it has a singularity of type 1 3(1, 1) at y0 (as the cone over a twisted cubic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular the singularities of Y are at most isolated, locally factorial, and terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Now we give some simple preliminary results on extremal rays of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' set E := Exc(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim N1(E, X) ≥ 4, then E · R ≥ 0 for every extremal ray R of X different from NE(f), and Y is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It follows from [Cas17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='16 and Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='17] that NE(f) is the unique extremal ray of X having negative intersection with E, −KX + E = f ∗(−KY ) is nef, and (−KX + E)⊥ ∩ NE(X) = NE(f), so that −KY is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and R1, R2 extremal rays of X of type (3, 2) such that dim N1(ER1, X) ≥ 4 and ER1 · R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ER2 ·R1 = 0 and R1+R2 is a face of NE(X) whose associated contraction is birational, with exceptional locus ER1 ∪ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let H be a nef divisor on X such that H⊥ ∩ NE(X) = R2, and set H′ := H + (H · CR1)ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then H′ · CR1 = H′ · CR2 = 0, and if R3 is an extremal ray of NE(X) different from R1 and R2, we have ER1 · R3 ≥ 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, hence H′·R3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore H′ is nef and (H′)⊥∩NE(X) = R1+R2 is a face of NE(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If Γ ⊂ X is an irreducible curve with [Γ] ∈ R1 + R2, then H′ · Γ = 0, so that either ER1 · Γ < 0 and Γ ⊂ ER1, or H · Γ = 0, [Γ] ∈ R2 and Γ ⊂ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This shows that the contraction of R1 + R2 is birational with exceptional locus ER1 ∪ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we have ER2 · R1 = 0 by [Cas13b, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2(b) and its proof].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and R1, R2 distinct extremal rays of X of type (3, 2) with dim N1(ERi, X) ≥ 4 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If there exists a birational contraction g: X → Z with R1, R2 ⊂ NE(g), then ER1 · R2 = ER2 · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We note first of all that ERi · Rj ≥ 0 for i ̸= j by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that ER1 · R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ER1 · (CR1 + CR2) = ER1 · CR2 − 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover ER2 · R1 > 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3, so that ER2 · (CR1 + CR2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand for every prime divisor D different from ER1, ER2 we have D · (CR1 + CR2) ≥ 0, therefore [CR1 + CR2] ∈ Eff(X)∨ = mov(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since [CR1 + CR2] ∈ NE(g), g should be of fiber type, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2, and g: X → Z a contraction of fiber type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρZ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This follows from [Cas12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' for the reader’s convenience we report the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim Z ≤ 1, then ρZ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If Z is a surface, take any prime divisor D ⊂ X such that g(D) ⊊ Z, so that N1(g(D), Z) = {0} if g(D) = {pt}, and N1(g(D), Z) = R[g(D)] if g(D) is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Consider the pushforward of one-cycles g∗ : N1(X) → N1(Z), and note that dim ker g∗ = ρX−ρZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have g∗(N1(D, X)) = N1(g(D), Z) and dim N1(g(D), Z) ≤ 1, thus codim N1(D, X) ≥ ρZ − 1, and δX ≤ 2 yields ρZ ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim Z = 3, then as in [Cas12, proof of Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6] one shows that there exists a prime divisor D ⊂ X such that dim N1(g(D), Z) ≤ 2, and reasoning as before we get ρZ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 ([Cas17], Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='17(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has a divisorial elementary contraction not of type (3, 2), then ρX ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Showing that S is a del Pezzo surface In this section we study elementary contractions of type (3, 2) of a Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We focus on the surface S which is the image of the exceptional divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' as explained in the Introduction, our goal is to show that under suitable assumptions, S is a smooth del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that S has isolated singularities by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then KS′ + L is semiample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Note that −KY is Cartier by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1, and ample by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, so that L is nef and big on S′, and for every irreducible curve Γ ⊂ S′, we have L · Γ = 0 if and only if Γ is µ-exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Consider the pushforward of one-cycles f∗ : N1(X) → N1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then f∗(N1(E, X)) = N1(S, Y ), therefore ρS′ ≥ ρS ≥ dim N1(S, Y ) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let R be a KS′-negative extremal ray of NE(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The contraction associated to R can be onto a point (if S′ ∼= P2), onto a curve (so that ρS′ = 2), or the blow-up of a smooth point (see for instance [Mat02, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1-4-8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since ρS′ > 2, R is generated by the class of a (−1)-curve Γ, that cannot be µ-exceptional, because µ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then L · Γ > 0 and (KS′ + L) · Γ = L · Γ − 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover, if γ ∈ NE(S′)KS′≥0, then (KS′ + L) · γ = KS′ · γ + L · γ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By the Cone Theorem, we conclude that KS′+L is nef on S′, and also semiample by the Base-Point-Free Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The varieties in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' g E X ER1 T1 Y S C1 D = f(ER1) f Z h(E) h(ER1) h dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has an extremal ray R1 of type (3, 2) such that: E · R1 = 0 and E ∩ ER1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set D := f(ER1) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then D|S = C1 + · · · + Cr where Ci are pairwise disjoint (−1)-curves contained in Sreg, ER1 = f ∗(D), and f∗(CR1) ≡Y Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover if C′ i ⊂ S′ is the transform of Ci, we have (KS′ + L) · C′ i = 0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 we have ER1·NE(f) = 0 and NE(f)+R1 is a face of NE(X), whose associated contraction h: X → Z is birational with Exc(h) = E ∪ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have a diagram (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4) X f � h � Y g � Z where g is an elementary, K-negative, divisorial contraction, with Exc(g) = D (recall that Y is is locally factorial by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1, and Fano by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since ER1·NE(f) = E·R1 = 0, both h(E) and h(ER1) are surfaces in Z, and the general fiber of h over these surfaces is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover h(E) ∩ h(ER1) is finite, and the connected components of E ∩ ER1 are 2-dimensional fibers of h over these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Using the classification of the possible 2-dimensional fibers of h in [AW98], as in [Cas22a, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='15] we see that every connected component Ti of E ∩ ER1 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE (which is non-empty by assumption) is isomorphic to P1 ×P1 with normal bundle O(−1, 0) ⊕ O(0, −1), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set Ci := f(Ti), so that D ∩ S = f(E ∩ ER1) = f(∪iTi) = ∪iCi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then Ci ∼= P1, Ci ∩ Cj = ∅ if i ̸= j, and f has fibers of dimension one over Ci, therefore Ci ⊂ Sreg and Ci ⊂ Yreg by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover g(D) = h(ER1) is a surface, namely g is of type (3, 2), and Ci is a one-dimensional fiber of g contained in Yreg, hence KY · Ci = D · Ci = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also have ER1 = f ∗(D) and f∗(CR1) ≡Y Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since Ci ⊂ Sreg, it is a Cartier divisor in S, and we can write D|S = m1C1 + · · + mrCr with mi ∈ Z>0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In S we have Ci · Cj = 0 for i ̸= j, hence for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r} we get −1 = D · Ci = (m1C1 + · · · + mrCr) · Ci = miC2 i and we conclude that mi = 1 and C2 i = −1, so that Ci is a (−1)-curve in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally −KS · Ci = −KY · Ci = 1, hence if C′ i ⊂ S′ is the transform of Ci, we have (KS′ + L) · C′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has an extremal ray R1 of type (3, 2) such that E · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then R′ 1 := f∗(R1) is an extremal ray of Y of type (3, 2), and ER1 = f ∗(ER′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If E∩ER1 ̸= ∅, we are in the setting of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' consider the elementary contraction g: Y → Z as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then NE(g) = f∗(R1) = R′ 1 is an extremal ray of Y of type (3, 2), and f ∗(ER′ 1) = ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If E ∩ ER1 = ∅, then we still have a diagram as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4), where g is locally isomorphic to the contraction of R1 in X, and the statement is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has two extremal rays R1, R2 of type (3, 2) such that: ER1 · R2 > 0 and E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then one of the following holds: (i) S is a smooth del Pezzo surface and −KS = (−KY )|S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' (ii) ER1 · CR2 = ER2 · CR1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 to f, R1 and to f, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Write f(ER1)|S = C1+· · ·+Cr, and let Γ2 be an irreducible component of f(ER2)|S, so that C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Cr, Γ2 are (−1)-curves contained in Sreg, and Γ2 ≡ f∗(CR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7) 0 < ER1 · CR2 = f ∗(f(ER1)) · CR2 = f(ER1) · Γ2 = (C1 + · · · + Cr) · Γ2, hence Ci · Γ2 > 0 for some i, say i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover let Γ′ 2 and C′ 1 in S′ be the transforms of Γ2 and C1 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 9 then Γ′ 2 and C′ 1 are disjoint from the µ-exceptional locus, are (−1)-curves in S′, (KS′ + L) · C′ 1 = (KS′ + L) · Γ′ 2 = 0, and C′ 1 · Γ′ 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that KS′ + L is semiample by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, the face (KS′ + L)⊥ ∩ NE(S′) contains the classes of two distinct (−1)-curves which meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This means that the associated contraction cannot be birational, and we have two possibilities: either KS′ + L ≡ 0, or the contraction associated to KS′ + L is onto a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that these two cases yield respectively (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose first that KS′ + L ≡ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' in particular −KS′ is nef and big, namely S′ is a weak del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set for simplicity F := OY (KY )|S, invertible sheaf on S, and let ωS be the dualizing sheaf of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have KS′ ≡ µ∗(F), and since S′ is rational, we also have OS′(KS′) ∼= µ∗(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By restricting to the open subset µ−1(Sreg), we conclude that (ωS)|Sreg ∼= F|Sreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Now we use the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let S be a reduced and irreducible projective surface with isolated singularities, and ωS its dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If there exists an invertible sheaf F on S such that (ωS)|Sreg ∼= F|Sreg, then S is normal and ωS ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This should be well-known to experts, we include a proof for lack of references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We postpone the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8 and carry on with the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8 we have that S is normal and ωS ∼= F, in particular ωS is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If y0 is a singular point of S, then by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 y0 is a singularity of type 1 3(1, 1), but this contradicts the fact that ωS is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that S is smooth, and finally that −KS = (−KY )|S is ample, so that S is a del Pezzo surface, and we have (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Assume now that KS′ +L yields a contraction g: S′ → B onto a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let F ⊂ S′ be a general fiber F of g, so that −KS′ · F = L · F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since F is not µ-exceptional, we have L · F > 0 and hence −KS′ · F > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus there is a non-empty open subset B0 ⊆ B such that (−KS′)|g−1(B0) is g-ample, therefore g|g−1(B0) : g−1(B0) → B0 is a conic bundle, F ∼= P1, and −KS′ · F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The curves C′ 1 and Γ′ 2 are components of the same fiber F0 of g, and −KS′ ·F0 = 2 = −KS′ · (C′ 1 + Γ′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For any irreducible curve C0 contained in F0 we have −KS′ · C0 = L · C0 ≥ 0, so that if C0 is different from C′ 1 and Γ′ 2, we must have −KS′ · C0 = L · C0 = 0 and C0 is µ-exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus C0 ∩ (C′ 1 ∪ Γ′ 2) = ∅, and since F0 is connected, we conclude that F0 = C′ 1 + Γ′ 2 and F0 ⊂ g−1(B0), hence F0 is isomorphic to a reducible conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This also shows that C′ i for i > 1 are contained in different fibers of g, so that C1 · Γ2 = Γ2 · C1 = 1 and Ci · Γ2 = 0 for every i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r, and finally using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7) ER1 · CR2 = (C1 + · · · + Cr) · Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Similarly we conclude that ER2 · CR1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In the setting of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6(i), we cannot conclude that Y is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A priori Y could have isolated singularities at some y0 ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' by [AW98] in this case f −1(y0) ∼= P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that S has isolated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The surface S is reduced, thus it satisfies condition (S1), namely depth OS,y ≥ 1 for every y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then by [Har07, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3] the dualizing sheaf ωS satisfies condition (S2): depth ωS,y ≥ 2 for every y ∈ S, where depth ωS,y is the depth of the stalk ωS,y as an OS,y-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then, for every open subset U ⊂ S such that S ∖ U is finite, we have ωS = j∗((ωS)|U), where j : U �→ S is the inclusion, see [Har07, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This is analogous to the properties of reflexive sheaves on normal varieties, see [Har80, Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6], and can be proved using local cohomology [Gro67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Hence we have ωS = j∗((ωS)|Sreg), where j : Sreg �→ S is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since F is locally free, we get ωS = j∗((ωS)|Sreg) ∼= j∗(F|Sreg) = F, in particular ωS is an invertible sheaf and for every y ∈ Y we have ωS,y ∼= OS,y as an OS,y-module, thus depth OS,y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore S has property (S2), and it is normal by Serre’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has three distinct extremal rays R1, R2, R3 of type (3, 2) such that: E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 to f, R1, R2 and to f, R1, R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us keep the same notation as in the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' moreover we denote by Γ3 an irreducible component of f(ER3)|S and Γ′ 3 ⊂ S′ its transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that KS′ + L ≡ 0, which yields the statement by the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Otherwise, KS′ + L yields a contraction g: S′ → B onto a curve, and F0 = C′ 1 + Γ′ 2 is a fiber of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand also Γ′ 3 is contained in a fiber of g, it is different from C′ 1 and Γ′ 2, and C′ 1 · Γ′ 3 > 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has four distinct extremal rays R0, R1, R2, R3 of type (3, 2) such that: ER0 · Ri = 0 for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then one of the following holds: (i) dim N1(ERi, X) ≤ 3 for some i ∈ {0, 1, 2, 3}, in particular ρX ≤ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 11 (ii) dim N1(ER0, X) ≤ 10, in particular ρX ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover if f : X → Y is the contraction of R0 and S := f(ER0), then S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We assume that dim N1(ERi, X) ≥ 4 for every i = 0, 1, 2, 3, and prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If ER0 ∩ ERi = ∅ for some i ∈ {1, 2, 3}, then for every curve C ⊂ ER0 we have ERi · C = 0, so that [C] ∈ (ERi)⊥, and N1(ER0, X) ⊂ (ERi)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since the classes [ER1], [ER2], [ER3] ∈ N 1(X) generate distinct one dimensional faces of Eff(X) (see [Cas13a, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='19]), they are linearly independent, hence in N1(X) we have codim � (ER1)⊥ ∩ (ER2)⊥ ∩ (ER3)⊥� = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand codim N1(ER0, X) ≤ δX ≤ 2, thus N1(ER0, X) cannot be contained in the above intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then N1(ER0, X) ̸⊂ (ERh)⊥ for some h ∈ {1, 2, 3}, hence ER0 ∩ ERh ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, since ER0 · Rh = 0, there exists an irreducible curve C ⊂ ER0 with [C] ∈ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For j = 2, 3 we have ER1 · Rj > 0, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 also ERj · R1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This implies that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For instance say h = 3: then ER1 · R3 > 0 yields ER1 ∩ C ̸= ∅, hence ER0 ∩ ER1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then there exists an irreducible curve C′ ⊂ ER0 with [C′] ∈ R1, and ER2 ·R1 > 0 yields ER0 ∩ER2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we apply Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10 to get that S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore dim N1(S, Y ) ≤ ρS ≤ 9 and dim N1(ER0, X) = dim N1(S, X) + 1 ≤ 10, so we get (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 In this section we show how to apply the results of §3 to bound ρX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' the following is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2 and ρX ≥ 8, and with no small elementary contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ δX + 10 ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover every elementary contraction f : X → Y is of type (3, 2), and S := f(Exc(f)) ⊂ Y is a smooth del Pezzo surface with −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In the proof we will use the following terminology: if R1, R2 are distinct one- dimensional faces of a convex polyhedral cone C, we say that R1 and R2 are adjacent if R1 + R2 is a face of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A facet of C is a face of codimension one, and RC is the linear span of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We will also need the following elementary fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 ([Ewa96], Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let C be a convex polyhedral cone not containing non-zero linear subspaces, and R0 a one-dimensional face of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Rm be the one-dimensional faces of C that are adjacent to R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then the linear span of R0, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Rm is RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let f : X → Y be an elementary contraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' note that ρY = ρX − 1 ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then f is not of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, and not small by assumption, so that f is divisorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover f is of type (3, 2) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E) ⊂ Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have dim N1(E, X) ≥ ρX − δX ≥ 6, and if R′ ̸= NE(f) is another extremal ray of X, we have E · R′ ≥ 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover, if R′ is adjacent to NE(f), then E ·R′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed the contraction g: X → Z of the face R′ + NE(f) cannot be of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, thus it is birational and we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We are going to show that there exists three extremal rays R′ 1, R′ 2, R′ 3 adjacent to NE(f) such that ER′ 1 · R′ j > 0 for j = 2, 3, and then apply Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us consider the cone NE(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It is a convex polyhedral cone whose extremal rays R are in bijection with the extremal rays R′ of X adjacent to NE(f), via R = f∗(R′), see [Cas08, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, R is still of type (3, 2), and f ∗(ER) = ER′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus for every pair R1, R2 of distinct extremal rays of Y , with Ri = f∗(R′ i) for i = 1, 2, we have ER1 · R2 = ER′ 1 · R′ 2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If R1 and R2 are adjacent, we show that ER1·R2 = ER2·R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed consider the contraction Y → Z of the face R1 + R2 and the composition g: X → Z, which contracts R′ 1 and R′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Again g cannot be of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, thus it is birational and we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4 to get ER′ 1 · R′ 2 = ER′ 2 · R′ 1 = 0, thus ER1 · R2 = ER2 · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fix an extremal ray R1 of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that there exist two distinct extremal rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed since ER1 is an effective divisor, there exists some curve C ⊂ Y with ER1 · C > 0, hence there exists some extremal ray R2 with ER1 · R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By contradiction, let us assume that ER1 · R = 0 for every extremal ray R of Y different from R1, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This means that the cone NE(Y ) has the extremal ray R1 in the halfspace N1(Y )ER1<0, the extremal ray R2 in the halfspace N1(Y )ER1>0, and all other extremal rays in the hyperplane (ER1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fix R ̸= R1, R2, and let τ be a facet of NE(Y ) containing R and not R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Note that Rτ ̸= (ER1)⊥, as ER1 and −ER1 are not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 the rays adjacent to R in τ cannot be all contained in (ER1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that R2 is adjacent to R, therefore ER2 · R = 0, namely R ⊂ (ER2)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Summing up, we have shown that every extremal ray R ̸= R1, R2 of Y is contained in both (ER1)⊥ and (ER2)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand these rays include all the rays adjacent to R1, so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 their linear span must be at least a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore (ER1)⊥ = (ER2)⊥ and the classes [ER1], [ER2] ∈ N 1(Y ) are proportional, which is impossible, because they generate distinct one dimensional faces of the cone Eff(Y ) (see [Cas13a, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that there exist two distinct extremal rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 13 For i = 1, 2, 3 we have Ri = f∗(R′ i) where R′ i is an extremal ray of X adjacent to NE(f), so that E · R′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover for j = 2, 3 we have ER′ 1 · R′ j = ER1 · Rj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11 to NE(f), R′ 1, R′ 2, R′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have already excluded (i), and (ii) yields the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ We can finally prove the following more detailed version of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold which is not a product of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ 12, and if ρX = 12, then there exist X ϕ ��� X′ g→ Z where ϕ is a finite sequence of flips, X′ is smooth, g is a contraction, and dim Z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since X is not a product of surfaces, we have δX ≤ 3 by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover δX = 3 yields ρX ≤ 6 by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, while δX ≤ 2 yields ρX ≤ 12 by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If ρX = 12, the statement follows from [Cas22a, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ References [AW98] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Andreatta and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Wi´sniewski, On contractions of smooth varieties, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 7 (1998), 253–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas08] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Casagrande, Quasi-elementary contractions of Fano manifolds, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 144 (2008), 1429–1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas12] , On the Picard number of divisors in Fano manifolds, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 45 (2012), 363–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas13a] , On the birational geometry of Fano 4-folds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 355 (2013), 585–628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas13b] , Numerical invariants of Fano 4-folds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 286 (2013), 1107–1113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas17] , Fano 4-folds, flips, and blow-ups of points, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Algebra 483 (2017), 362–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas22a] , Fano 4-folds with a small contraction, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 405 (2022), 1–55, paper no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 108492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas22b] , The Lefschetz defect of Fano varieties, Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Palermo (2), pub- lished online 19 December, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [CRS22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Casagrande, E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Mori and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Mukai, Classification of Fano 3-folds with b2 ≥ 2, I, Algebraic and Topological Theories – to the memory of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Takehiko Miyata (Kinosaki, 1984), Ki- nokuniya, Tokyo, 1986, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 496–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Universit`a di Torino, Dipartimento di Matematica, via Carlo Alberto 10, 10123 Torino - Italy Email address: cinzia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='casagrande@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'}