diff --git "a/3NAzT4oBgHgl3EQffPyj/content/tmp_files/2301.01450v1.pdf.txt" "b/3NAzT4oBgHgl3EQffPyj/content/tmp_files/2301.01450v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/3NAzT4oBgHgl3EQffPyj/content/tmp_files/2301.01450v1.pdf.txt" @@ -0,0 +1,3973 @@ +arXiv:2301.01450v1 [math.AG] 4 Jan 2023 +KUMMER SURFACES AND QUADRIC LINE COMPLEXES IN +CHARACTERISTIC TWO +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +ABSTRACT. We give an analogue in characteristic 2 of the classical theory of quadric line +complexes and Kummer surfaces. +1. INTRODUCTION +Let C be a complex Riemann surface of genus 2 and J(C) its Jacobian. Let ι be the +inversion automorphism of J(C) and Θ the theta divisor on J(C). The complete linear +system |2Θ| gives a double covering from J(C) to a quartic surface S in P3 with 16 nodes +defined by +x4 + y4 + z4 + t4 + A +� +x2y2 + z2t2� ++ B +� +x2z2 + y2t2� ++ C +� +x2t2 + y2z2� ++ Dxyzt = 0, +where A, B, C, D are constants satisfying a certain explicit equation (e.g. Hudson [7, +p.81]). The quartic surface S is isomorphic to the quotient surface J(C)/⟨ι⟩ and is called a +Kummer quartic surface. It contains sixteen tropes (= double conics) which are the images +of Θ and its translations by 2-torsions. The incidence relation between sixteen nodes and +tropes is called a (166)-configuration, that is, each node is contained in six tropes and each +trope contains six nodes. Let Σ be the minimal resolution of S which is a K3 surface. The +surface Σ is realized as a complete intersection of three quadrics in P5 which is the image +of the rational map from J(C) defined by the linear system |4Θ − � pi| where {pi} are +sixteen 2-torsion points of A (cf. Griffiths-Harris [6, p.786]). Then Σ contains 32 lines +forming a (166)-configuration, that is, there are two sets of disjoint 16 lines on Σ such that +each member in one set meets exactly six members in another set. The 32 lines consist of +the proper transforms of sixteen tropes and sixteen exceptional curves over sixteen nodes. +We remark that by contracting the proper transforms of sixteen tropes we obtain another +quartic surface S∨ with 16 nodes which is the projective dual of S. +In the paper [12], Klein discovered a relationship between Kummer quartic surfaces and +quadric line complexes. A quadric line complex is a 3-dimensional family of lines in P3 +which is defined as the intersection of the Grassmannian G = G(2, 4) ⊂ P5 with a quadric +Research of the first author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) +No.20K03530 and the second author by JSPS Grant-in-Aid for Scientific Research (A) No.20H00112. +1 + +2 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Q in P5. In this paper we assume that G ∩ Q is non-singular. Then we can diagonalize +simultaneously G and Q as +(1.1) +G = +� +6 +� +i=1 +X2 +i = 0 +� +, +Q = +� +6 +� +i=1 +aiX2 +i = 0 +� +, +where (X1, · · · , X6) are homogeneous coordinates of P5 and a1, . . . , a6 ∈ C, ai ̸= aj(i ̸= +j). Recall that a non-singular quadric in P5 contains two irreducible families of planes and +a singular quadric of rank 5 contains an irreducible family of planes. Since the pencil of +quadrics in P5 defined by G and Q contains exactly six singular quadrics of rank 5, we +thus obtain a non-singular curve C of genus 2 defined by +(1.2) +y2 = +6 +� +i=1 +(x − ai) +parametrizing the irreducible families of planes contained in members of the pencil. The +classical theory claims that the surface Σ is isomorphic to the locus of special lines, called +singular lines, and S is the set of foci of these singular lines (see §2.1). The surface Σ is +given by the intersection of three quadrics: +(1.3) +Σ = +� +6 +� +i=1 +X2 +i = +6 +� +i=1 +aiX2 +i = +6 +� +i=1 +a2 +i X2 +i = 0 +� +. +Moreover the variety of lines in G ∩ Q is an abelian surface which is isomorphic to the Ja- +cobian J(C) of C. This is an outline of the classical theory. In the last century, Narasimhan +and Ramanan [20] reproved this in connection with the theory of vector bundles over C. +For modern treatment of this theory, we refer the reader to Griffiths and Harris [6], Cassels +and Flynn [2] and for a history of Kummer surfaces to Dolgachev [4]. The above theory +holds in any characteristic different from 2. +Now assume that the ground field is an algebraically closed field of characteristic 2. +Then the situation is entirely different. There are mainly two differences between the +case of characteristic p = 2 and the case of p ̸= 2. First of all, a quadratic form in +characteristic 2 is not determined by the associated alternating bilinear form. Secondly +the moduli space of curves of genus two and that of their Jacobians are stratified in terms +of 2-rank. There are three types, that is, ordinary (2-rank 2), 2-rank 1 and supersingular +(2-rank 0). Shioda [23] and the first author [10] found that J(C)/⟨ι⟩ has, instead of sixteen +nodes, four rational double points of type D4 for J(C) being ordinary, two rational double +points of type D8 for J(C) with 2-rank 1, an elliptic singularity of type 4⃝1 +0,1 in the sense +of Wagreich [24] for J(C) being supersingular. See the following Table 1 (see Figure 1 in +Subsection 2.4 for elliptic singularities). +In characteristic 2, Bhosle [1] studied pencils of quadrics in P2g+1. She presented a +canonical form of a pencil of quadrics in P2g+1, associated a hyperelliptic curve C of + +KUMMER SURFACES +3 +char p +p ̸= 2 +p = 2 +p = 2 +p = 2 +2-rank of J(C) +– +2 (ordinary) +1 +0 (supersingular) +# of branches of C → P1 +6 +3 +2 +1 +# of 2-torsion points of J(C) +16 +4 +2 +1 +Singularities of J(C)/⟨ι⟩ +16 A1 +4 D4 +2 D8 +4⃝1 +0,1 +TABLE 1 +genus g to the pencil in general case (that is, ordinary case for g = 2) and showed that +the Jacobian of C is isomorphic to the variety of (g − 1)-dimensional subspaces on the +base locus of the pencil. Also Laszlo and Pauly [14], [15] studied the linear system |2Θ| +and gave the equation of Kummer quartic surface in ordinary case, and Duquesne [5] in +arbitrary case. +The main purpose of this paper is to present an analogue of the theory of Kummer +surfaces and quadric line complexes in characteristic 2. We show that the stratification of +the moduli space of curves of genus 2 by the 2-rank bijectively corresponds to the one by +the canonical forms of pencils {λG + µQ}(λ,µ)∈P1 of quadratic forms. Let A, B be the +associated alternating forms of G, Q, respectively. Then there are three possibilities of +the associated alternating forms +� +0 +λA + µB +t(λA + µB) +0 +� +, where A = + + +0 +0 +1 +0 +1 +0 +1 +0 +0 + + +and B is as in the following Table 2. +2-rank of J(C) +2 +1 +0 +B + + +0 +0 +a1 +0 +a2 +0 +a3 +0 +0 + + + + +0 +0 +a1 +0 +a2 +0 +a2 +1 +0 + + + + +0 +0 +a1 +0 +a1 +1 +a1 +1 +0 + + +TABLE 2 +Here ai ̸= aj for i ̸= j (see Proposition 2.8 and the equations (2.12), (2.13), (2.14) +in Proposition 2.10). For each canonical form of a pencil of quadrics, we associate to +quartic surfaces S and S∨, an intersection of three quadrics Σ, a curve C of genus 2 and +its Jacobian J(C) in terms of line geometry. We remark that in characteristic p ̸= 2 a +quadratic form is determined by the associated bilinear form. Under the condition G ∩ Q +being non-singular, the case of non-diagonalizable pairs G, Q is excluded. On the other +hand, in the case p = 2 a quadratic form is not determined by the associated alternating +form. This difference allows the possibility of the above three cases of alternating forms, +and hence the exsistence of three types of curves of genus 2. + +4 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +The plan of this paper is as follows. In Section 2, we recall the classical theory of quadric +line complexes and Kummer surfaces, the theory of quadratic forms in characteristic 2 and +their pencils, and the theory of curves of genus 2, abelian surfaces and Kummer surfaces in +characteristic 2. We study quadric line complexes in Section 3, Kummer quartic surfaces +associated with quadric line complexes in Section 4, the intersections of three quadrics in +P5 in Section 5 and the curves of genus 2 in Section 6. Finally in Section 7 we discuss the +canonical map |4Θ − 2 �4 +i=1 pi| from the Jacobian of an ordinary curve of genus 2 to the +intersection of three quadrics, where p1, . . . , p4 are 2-torsion points on the Jacobian. This +is also an analogue of the map |4Θ − �16 +i=1 pi| mentioned above. +2. PRELIMINARIES +2.1. Classical theory of Quadric line complexes and Kummer surfaces. Let k be an +algebraically closed field of characteristic char(k) = p ≥ 0 (when we assume p ̸= 2, we +will mention it). In the following we recall the classical theory of quadric line complexes +over the base field k. The main references are Griffiths and Harris [6, Chapter 6], Cassels +and Flynn [2, Chapter 17]. +Let G = G(2, 4) be the Grassmannian variety of lines in P3 which can be embedded +into P5 as a non-singular quadric hypersurface (Pl¨ucker embedding). For a point p and a +hyperplane h in P3, let +σ(p) = {ℓ ∈ G : p ∈ ℓ}, σ(h) = {ℓ ∈ G : ℓ ⊂ h}, σ(p, h) = {ℓ ∈ G : p ∈ ℓ ⊂ h} +be the Schubert varieties. Both σ(p) and σ(h) are planes and σ(p, h) is a line in P5. Con- +versely any plane in G is of the form of either σ(p) or σ(h) for some p, h and any line in +G is of the form σ(p, h). Any non-singular quadric in P5 contains two 3-dimensional +irreducible families of planes and in case of G they are nothing but {σ(p)}p∈P3 and +{σ(h)}h⊂P3 (see Proposition 2.7 for p = 2). +Let Q be a quadric hypersurface in P5 such that Q ∩ G is non-singular. The variety +X = Q ∩ G is called a quadric line complex which parametrizes a 3-dimensional family +of lines in P3. The condition X being non-singular implies that X does not contain any +plane (see Lemma 2.9) and hence the intersection σ(p) ∩ Q is a conic in the plane σ(p). +Define +(2.1) +S = {p ∈ P3 : σ(p) ∩ Q is a singular conic}, +(2.2) +R = {p ∈ S : σ(p) ∩ Q is a double line}. +Similarly we define the dual version: +(2.3) +S∨ = {h ∈ (P3)∨ : σ(h) ∩ Q is a singular conic}, +(2.4) +R∨ = {h ∈ S∨ : σ(h) ∩ Q is a double line}. + +KUMMER SURFACES +5 +For x ∈ X , we denote by ℓx the line in P3 corresponding to x. A line ℓx is called singular +if x is a singular point of the conic σ(p) ∩ Q (resp. σ(h) ∩ Q) for some p (resp. h). The +point p (resp. the plane h) is uniquely determined by x and is called the focus (resp. the +plane) of ℓx. Let +(2.5) +Σ = {x ∈ X : ℓx is a singular line}. +Proposition 2.1. (Griffiths and Harris [6, p.767], Cassels and Flynn [2, Lemma 17.2.1]) +Let x ∈ X . Then x ∈ Σ if and only if the tangent space Tx(Q) of Q at x is tangent to G at +some point. +There exist canonical morphisms +π : Σ → S, +π∨ : Σ → S∨ +by sending x to the focus or the plane of ℓx. Note that when σ(p) ∩ Q is a double line, it +sends to the point p. +As mentioned above, a non-singular quadric in P5 contains two irreducible families of +planes and a singular quadric of rank 5 contains an irreducible family of planes for p ̸= 2. +The pencil of quadrics {t0G+t1Q}(t0:t1)∈P1 defines a curve C, which is a double covering +of P1, parametrizing irreducible components of families of planes contained in t0G + t1Q +((t0 : t1) ∈ P1). For p = 2, see Proposition 2.7. +Remark 2.2. In case p ̸= 2, by the assumption X = G ∩ Q being non-singular, we can +diagonalize G and Q simultaneously as in the equation (1.1) (cf. Klingenberg [13, Satz 1]). +Moreover it is known that S is a quartic surface with sixteen nodes and is called a Kummer +quartic surface and S∨ is isomorphic to S. The surface Σ is non-singular and hence is a +K3 surface. The both morphisms π and π∨ are the minimal resolutions of singularities. In +this case the curve C is given by the equation (1.2). +Let A be the variety of lines in X . For each L ∈ A, there exist a point pL and a plane +hL with L = σ(pL, hL). Thus we have morphisms +ϕ : A → S, +ϕ∨ : A → S∨ +defined by ϕ(L) = pL, ϕ∨(L) = hL when L = σ(pL, hL). Note that the conic σ(pL) ∩ Q +splits into two lines, that is, one is σ(pL, hL) and another is σ(pL, h′) for some plane h′ +containing pL. Similarly σ(hL)∩Q = σ(pL, hL)+σ(p′, hL) for some point p′ ∈ hL. Thus +both morphisms ϕ, ϕ∨ have degree 2 branched at each point of R and R∨. +We will show that A is isomorphic to the Jacobian J(C) of C and the maps ϕ, ϕ∨ are the +quotient map by inversions of A. The following argument is given by Cassels and Flynn +[2, Chap. 17, §1]. We denote by a = (t0, t1, µ) a point of C over a point (t0 : t1) ∈ P1, +and by ¯a another point of C over (t0 : t1) ∈ P1, where µ denotes an irreducible family of +planes in the quadric t0G + t1Q. For a ∈ C and L ∈ A, we define a line Υ(a)L ∈ A as +follows. There exists a unique plane Π in the family of planes in t0G + t1Q corresponding + +6 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +to µ which contains L (Lemma 2.9). Then the conic Π∩X in Π splits into two lines L+M. +We now define Υ(a)L = M. Thus, if we denote by AL the set of lines {Υ(a)L}a∈C, then +AL is the set of lines meeting with L and is bitational to C. +Now, for an effective divisor a1 + a2 on C and L ∈ A, define +L′ = Υ(¯a1)Υ(a2)L. +We remark that if a1 + a2 is general, then Υ(a1)L ∩ Υ(a2)L = ∅, L′ ∩ L = ∅ and +L′ is the unique line meeting with Υ(a1)L and Υ(a2)L (Cassels and Flynn [2, Lemma +17.1.5]). Moreover, if we denote by Λ the 3-dimensional linear space spanned by L, +Υ(a1)L, Υ(a2)L, then +(2.6) +Λ ∩ X = L + Υ(a1)L + Υ(a2)L + L′. +Finally note that for ¯a + a we have Υ(a)Υ(a)L = L. Thus A has a natural structure of +principal homogeneous space over J(C). +Proposition 2.3. Let k be an algebraically closed field in characteristic p ≥ 0. Assume +that C is a non-singular curve. Then A is isomorphic to the Jacobian J(C) of C. +We remark that, by using the theory of vector bundles, Bhosle studied pencils of quadrics +in P2g+1 in characteristic 2. She presented a canonical form of a pencil of quadrics in +P2g+1, associated a hyperelliptic curve C of genus g to a pencil for generic case and +showed that the Jacobian of C is isomorphic to the variety of (g − 1)-dimensional sub- +spaces on the base locus of the pencil (Bhosle [1, Theorem 4]). +Let ι (resp. ι∨) be the covering transformation of the morphism ϕ : A → S (resp. +ϕ∨ : A → S∨). Then ι (resp. ι∨) has fixed points over R (resp. R∨). The following +Proposition is well known over the complex numbers (Griffiths and Harris [6, p.780]). +Proposition 2.4. Let k be an algebraically closed field in characteristic p ≥ 0. Assume +that C is a non-singular curve. The morphisms ϕ : A → S and ϕ∨ : A → S∨ are the +quotient maps by the inversion. In particular, S and S∨ are isomorphic. +Proof. Let ι be the covering transformation of ϕ. To prove that ι is an inversion of A, it is +enough to show that ι has an isolated fixed point and no fixed curves (Katsura [11, Lemma +3.5]). Assume that ι has a fixed curve G. Then, by definition of the map π : Σ → S, +π−1(ϕ(p)) ⊂ Σ is the line corresponding to p for any p ∈ G. Thus π−1(ϕ(G)) has +dimension 2. This implies that Σ is not irreducible and has singularities along a curve. On +the other hand, Σ is non-singular in charcteristic p ̸= 2. In case p = 2, we will show that +Σ is singular, but only isolated singularities (Theorems 5.3, 5.7, 5.11). Thus ι has at most +isolated fixed points. Assume now that ι has no fixed points. Then S is non-singular. On +the other hand, in characteristic p ̸= 2, S has 16 nodes. In case p = 2, we will show that +S has always singulatiries (Theorems 4.2, 4.4, 4.6). +□ +It is known that S∨ is the projective dual of S (cf. Cassels and Flynn [2, p.181]). + +KUMMER SURFACES +7 +Remark 2.5. Let Θ be the theta divisor of A. Then, over the complex numbers, it is well +known that the morphism ϕ : A → S is defined by the complete linear system |2Θ|, and +the linear system |4Θ − �16 +i=1 pi| also defines a rational map +ψ : A → Σ +of degree 2, where {pi} is the set of 2-torsion points on A (Griffiths and Harris [6, p.786]). +In case char(k) = 2, Laszlo and Pauly [14, Proposition 4.1] studied the map defined +by |2Θ| and found the equation of the Kummer quartic surface for ordinary case and +Duquesne [5] for arbitrary case. +2.2. Quadratic forms in characteristic 2. We recall fundamental facts on quadratic +forms in characteristic 2 in Dieudonn´e [3] and Bhosle [1]. +In case that k is a field of characteristic ̸= 2, a quadratic form on a vector space M over +k is a function q : M → k defined by q(x) = f(x, x) where f is a symmetric bilinear form +f : M × M → k. It is easy to see that q satisfies that +q(ax + by) = a2q(x) + b2q(y) + 2abf(x, y), +a, b ∈ k. +In particular f is given by f(x, y) = 1 +2(q(x + y) − q(x) − q(y)). +Now, let E be a vector space over an algebraically closed field k in characteristic 2. A +quadratic form q on E is a function q : E → k satisfying +(2.7) +q(ax + by) = a2q(x) + b2q(y) + abA(x, y), +a, b ∈ k, +where A is a bilinear form on E. Note that since char(k) = 2, A(x, x) = 0 for all x ∈ E +and hence A is alternating and symmetric. +A subspace V of E is called totally singular if q(x) = 0 for all x ∈ V . A totally singular +subspace is totally isotropic, that is, A(x, y) = 0 for all x, y ∈ E. The converse is not true. +The index ν of q is defined as the dimension of a maximal totally singular subspace. Two +totally singular subspaces of the same dimension are equivalent under the action of the +orthogonal group of E. +A quadratic form is called non-defective if the alternating form A is non-degenerate and +defective if A is degenerate. For a defective quadratic form q we define the null space N +of A by N = {x ∈ E : A(x, y) = 0 for all y ∈ E}. The dimension of N is called the +defect of q. +Proposition 2.6. (Bhosle [1, Lemma 2.5]) Let q be a quadratic form on k2m with the +associated alternating form A. +(1) Assume that q is non-defective. Let W be a maximal totally singular subspace of +k2m which is of dimension m and let e1, . . . , em be a basis of W. Then there exists a basis +e1, . . . , em, f1, . . . , fm of k2m such that f1, . . . , fm span a totally singular subspace for q +and A(ei, fj) = δij for i, j = 1, . . . , m. +(2) Assume that the defect of q is two and the null space N contains a unique singular +subspace N0 of dimension one. Let W be a maximal totally singular subspace of k2m. + +8 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Then there exists a basis e1, . . . , em, f1, . . . , fm of k2m such that e1, . . . , em is a basis of +W and the span of f2, . . . , fm is totally singular, A(ei, fj) = δij for i, j = 2, . . . , m, e1 +spans N0 and e1, f1 is a basis of N. +Proposition 2.7. (Bhosle [1, Lemma 2.6]) Let q be a quadratic form on k2m. +(1) Assume that q is non-defective. Then the space of maximal totally singular sub- +spaces for q has two connected components each of which is a non-singular variety of +dimension m(m − 1)/2. +(2) Assume that the defect of q is two and the null space N contains a unique singular +subspace N0 of dimension one. Then the space of maximal totally singular subspaces for +q is a non-singular variety of dimension m(m − 1)/2. +2.3. Pencils of quadratics in P5. Let q1, q2 be quadratic forms on k2m and consider the +pencil {xq1 +yq2}(x:y)∈P1 of quadratic forms generated by q1, q2. We may write the pencil +as {q1 + xq2}x∈k. We have a pencil of alternating forms {A1 + xA2} associated with +{q1 + xq2}. By choosing a basis of k2m, we denote the corresponding alternating matrix +by the same symbol A1 + xA2. The Pfaffian of the pencil is defined as the square root of +det(A1 + xA2). +Proposition 2.8. (Klingenberg [13, Satz 2], Bhosle [1, Proposition 2.8]) Let A1, A2 be +alternating forms with A1 non-degenerate. Let � +i(t − ai)2mi be the characteristic poly- +nomial of A−1 +1 A2. Then the pencil can be written in the following form: +A1 = +m +� +i=1 +mi +� +j=1 +XijYij, +A2 = +� +i +� +ai +� mi +� +j=1 +XijYij +� ++ +mi +� +j=2 +XijYi(j−1) +� +, +where XijYij denote the alternating form +A((xij, yij), (x′ +ij, y′ +ij)) = xijy′ +ij − x′ +ijyij. +In [13, Satz 2], the author assumed the charcteristic p ̸= 2, however as Bhosle pointed +out [1, Proposition 2.8], the proof of Satz 2 works well in characteristic 2. +Let (X1, X2, X3, Y1, Y2, Y3) be homogeneous coordinates of P5. We consider a pencil P +of quadrics generated by Q1, Q2 with the associated alternating forms A1, A2. We assume +that Q1 is non-defective and hence det(A1) ̸= 0. For simplicity we use the same symbols +Q1, Q2 for the hypersurfaces defined by them. First we recall the following fact (e.g. +Griffiths and Harris [6, p.762, Lemma]). +Lemma 2.9. Assume that Q1∩Q2 is non-singular. Then Q1∩Q2 does not contain a plane. +Proof. Assume that Q1 ∩ Q2 contains a plane Π. By changing of coordinates (Proposition +2.6 (1)), we may assume that Π is defined by X1 = X2 = X3 = 0, and Q1, Q2 are given +by +Q1 : +� +i +XiYi + ciX2 +i = 0, +Q2 : +� +i,j +aijXiYj + +� +i,j +bijXiXj + +� +i +diX2 +i = 0, + +KUMMER SURFACES +9 +respectively. By restricting the Jacobian of Q1 ∩ Q2 to Π, we obtain the three conics +Y1 +3 +� +j=1 +a2jYj + Y2 +3 +� +j=1 +a1jYj = 0, +Y2 +3 +� +j=1 +a3jYj + Y3 +3 +� +j=1 +a2jYj = 0, +Y1 +3 +� +j=1 +a3jYj + Y3 +3 +� +j=1 +a1jYj = 0 +which have a common zero. +□ +Let �3 +i=1(t−ai)2 be the characteristic polynomial of A−1 +1 A2. The following three cases +occur. +(a) a1, a2, a3 are different; +(b) a1 ̸= a2 = a3; +(c) a = a1 = a2 = a3. +Now by using Proposition 2.8, P is given as follows: +(2.8) +(a) : + + + +Q1 : �3 +i=1(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = 0, +Q2 : �3 +i=1(aiXiYi + r2 +i X2 +i + s2 +i Y 2 +i ) = 0. +(2.9) +(b) : + + + +Q1 : �3 +i=1(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = 0, +Q2 : �3 +i=1(aiXiYi + r2 +i X2 +i + s2 +i Y 2 +i ) + X3Y2 = 0. +(2.10) +(c) : + + + +Q1 : �3 +i=1(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = 0, +Q2 : �3 +i=1(aXiYi + r2 +i X2 +i + s2 +i Y 2 +i ) + X2Y1 + X3Y2 = 0. +As in the proof of Bhosle [1, Corollary 2.10], by changing the coordinates +(2.11) + + + +xi = αiXi + βiYi, +yi = γiXi + δiYi +with αiδi+βiγi = 1, αiγi = p2 +i , βiδi = t2 +i , we may assume that P is given by the equations +in Proposition 2.10. +Proposition 2.10. +(2.12) +(a) : + + + +Q1 : �3 +i=1 XiYi = 0, +Q2 : �3 +i=1 aiXiYi + ciX2 +i + diY 2 +i = 0. + +10 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +(2.13) +(b) : + + + +Q1 : �3 +i=1 XiYi = 0, +Q2 : �3 +i=1 (aiXiYi + ciX2 +i + diY 2 +i ) + B = 0 +where B = b1X2Y3 + b2X3Y2 + b3X2X3 + b4Y2Y3 and b1 = β3γ2, b2 = α2δ3, b3 = +γ2δ3, b4 = α2β3. +(2.14) +(c) : + + + +Q1 : �3 +i=1 XiYi = 0, +Q2 : �3 +i=1 (aXiYi + ciX2 +i + diY 2 +i ) + C = 0 +where C = b1X2Y3 +b2X3Y2 +b3X2X3 +b4Y2Y3 +b5X1Y2 +b6X2Y1 +b7X1X2 +b8Y1Y2, +b1 = β3γ2, b2 = α2δ3, b3 = γ2δ3, b4 = α2β3, b5 = β2γ1, b6 = α1δ2, b7 = γ1δ2, b8 = +α1β2. +We use (2.12), (2.13), (2.14) in Proposition 2.10 as canonical forms of pencil of quadrics. +Note that sQ1+tQ2 is non-defective for (s, t) ̸= (ai, 1) and aiQ1+Q2 has the defect 2 and +the null space contains a unique singular subspace of dimension one as stated in Proposi- +tion 2.7 (2). +Remark 2.11. Note that in cases (b), (c), b1b2 = b3b4, b5b6 = b7b8. Since αiδi + βiγi = 1, +(b1, b2, b3, b4) ̸= (0, 0, 0, 0), and (b5, b6, b7, b8) ̸= (0, 0, 0, 0). Also by the relations b1b2 = +b3b4 and b5b6 = b7b8, we have a relation +(2.15) +(b1b5 + b4b7)(b2b6 + b3b8) = (b1b8 + b4b6)(b2b7 + b3b5). +Similarly we have +(2.16) +b2(b1b8 + b4b6) = b4(b2b6 + b3b8), +b3(b1b8 + b4b6) = b1(b2b6 + b3b8), +b1(b2b7 + b3b5) = b3(b1b5 + b4b7), +b4(b2b7 + b3b5) = b2(b1b5 + b4b7), +b5(b1b8 + b4b6) = b8(b1b5 + b4b7), +b7(b1b8 + b4b6) = b6(b1b5 + b4b7), +b6(b2b7 + b3b5) = b7(b2b6 + b3b8), +b8(b2b7 + b3b5) = b5(b2b6 + b3b8). +We use the following Lemma 2.12 and Remark 2.13 to construct some involutions of +Kummer quartic surfaces (Propositions 4.5, 5.8 and Lemma 5.10). +Lemma 2.12. Assume that b1b2b3b4 ̸= 0 and b5b6b7b8 ̸= 0. +(1) In the equation (2.13), after changing the coordinates, we may assume that b2b4c2 = +b1b3d2. +(2) In the equation (2.14), after changing the coordinates, we may assume that b6b8c1 = +b5b7d1. +Proof. (1) Recall that we start the pencil of quadrics given in (2.9), and by changing of co- +ordinates (2.11), we have the equation (2.13). Here b1 = β3γ2, b2 = α2δ3, b3 = γ2δ3, b4 = +α2β3 and ci = aiγiδi + r2 +i δ2 +i + s2 +i γ2 +i , di = aiαiβi + r2 +i β2 +i + s2 +i α2 +i . Since +b2b4c2 = β3δ3(a2α2 +2γ2δ2+r2 +2α2 +2δ2 +2+s2 +2α2 +2γ2 +2), b1b3d2 = β3δ3(a2α2β2γ2 +2+r2 +2β2 +2γ2 +2+s2 +2α2 +2γ2 +2), + +KUMMER SURFACES +11 +and α2δ2 + β2γ2 = 1, the condition c2b2b4 = d2b1b3 is equivalent to a2α2γ2 + r2 +2 = 0, that +is, a2p2 +2 + r2 +2 = 0. Now first by applying the changing of coordinates +(2.17) +� +Y2 = ǫX′ +2 + Y ′ +2 +(ǫ ∈ k) +Y1 = Y ′ +1, Y3 = Y ′ +3, Xi = X′ +i +(i = 1, 2, 3) +to the equation (2.9) and then by applying the changing of coordinates (2.11), the condition +a2p2 +2 + r2 +2 = 0 is replaced by a2p2 +2 + r2 +2 + ε(a2t2 +2 + s2 +2) + ε2(a2t2 +2 + s2 +2) = 0. Thus +if a2t2 +2 + s2 +2 ̸= 0, then we may assume a2p2 +2 + r2 +2 = 0 and hence c2b2b4 = d2b1b3. If +a2t2 +2 + s2 +2 = 0, then first apply the changing of coordinates +(2.18) +� +X2 = X′ +2 + εY ′ +2 +(ε ∈ k), +X1 = X′ +1, X3 = X′ +3, Yi = Y ′ +i +(i = 1, 2, 3) +to the equation (2.9). Then the condition a2t2 +2+s2 +2 = 0 is replaced by a2t2 +2 +s2 +2+ǫ2(a2p2 +2+ +r2 +2) = 0, and hence we may assume a2t2 +2 + s2 +2 ̸= 0. Thus we have proved the assertion (1). +(2) The proof is the same as in the case (1). +□ +Remark 2.13. In case b1b2b3b4 = 0, Lemma 2.12 holds after a modification. For example, +if b1 = b4 = 0 and b2b3 ̸= 0, then instead of b2b4c2 = b1b3d2, we may assume b2 +2c2 = b2 +3d2. +Also, by the same way, we may assume b1b4c3 = b2b3d3. We remark that the changing +of coordinates (2.18) does not change {bi}, but (2.17) may change {bi}. Therefore we can +not assume both b2b4c2 = b1b3d2 and b1b4c3 = b2b3d3 at the same time. +2.4. Curves of genus 2, abelian surfaces and Kummer surfaces in characteristic 2. +Let k be an algebraically closed field of characteristic 2, and we recall fundamental results +of curves of genus 2, abelian surfaces and Kummer surfaces in characteristic 2. Let A be +an abelian surface over k and let ι be the inversion. Then, for the singularities of A/⟨ι⟩, +we have the following theorem (cf. Katsura [10]). +Theorem 2.14. +(i) 4 rational double points of type D4 if A is ordinary. +(ii) 2 rational double points of type D8 if the 2-rank of A is one. +(iii) An elliptic double point of type 19 +⃝0 if A is superspecial. +(iv) An elliptic double point of type 4⃝1 +0,1 if A is supersingular and not superspecial. +Here a supersingular abelian surface is called superspecial if it is isomorphic to the +product of two elliptic curves, and elliptic double points of type 19 +⃝0, 14 +⃝1 +0,1 are in the sense +of Wagreich [24]. The dual graphs of the minimal resolutions of these singularities are +given in Figure 1. Each component is a non-singular rational curve, −3 or −4 is the +self-intersection number of the component and other components are (−2)-curves. The +central component has multiplicity 2. In Figure 1, an elliptic singularity of type 14 +⃝(0),(1) +0,0,1 +is also in the sense of Wagreich whose minimal resolution is obtained from the one of a +singularity of type 14 +⃝1 +0,1 by blowing up a point on the (−3)-curve. This singularity appears +in Theorem 5.11. + +12 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +r +r +r +r +r +r +▲ +▲ +▲ +▲ +▲ +▲ +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +−3 +2 +19 +⃝0 : +−3 +2 +4⃝1 +0,1 : +−4 +2 +14 +⃝(0),(1) +0,0,1 +: +FIGURE 1. Dual graphs of minimal resolutions of singularities +Note that in Cases (i) and (ii) the non-singular model of A/⟨ι⟩ is a K3 surface and that +in Cases (iii) and (iv) A/⟨ι⟩ is a rational surface (cf. Shioda [23] and Katsura [10]). +For a non-singular curve C of genus 2, we denote by J(C) the Jacobian variety of C. +As for a normal form of C, by Igusa [9] we have the following result. +(2.19) +y2 + y = + + + +αx + βx−1 + γ(x − 1)−1 +if J(C) is ordinary, +x3 + αx + βx−1 +if J(C) is of 2-rank 1, +x5 + αx3 +if J(C) is supersingular , +where α, β, γ ∈ k and α, β, γ in the first case and β in the second case are different from +0. Note that in characteristic 2 there exists no curves C of genus 2 such that J(C) is +superspecial (cf. Ibukiyama-Katsura-Oort [8]), and hence only an elliptic singularity of +type 4⃝1 +0,1 appears in the quotient surface J(C)/⟨ι⟩. +3. QUADRIC LINE COMPLEXES AND KUMMER SURFACES IN CHARACTERISTIC 2: +THE INTERSECTION OF TWO QUADRICS +We use the same notation as in Subsection 2.3. In this section we discuss the intersection +of two quadrics in P5 associated with a quadric line complex. Let P, P1 or P0 be the pencil +of quadrics in P5 given by (2.12), (2.13) or (2.14) in Proposition 2.10, respectively. We +identify Q1 with the grassmannian G. Let X , X1 or X0 be the intersection of Q1 and Q2 +for P, P1 or P0, respectively. +Lemma 3.1. (a) X is singular if and only if � +i cidi = 0. +(b) X1 is singular if and only if c1d1 = 0 or b2b4c2 + b1b4c3 + b1b3d2 + b2b3d3 = 0. +(c) X0 is singular if and only if c1(b1b8 +b4b6)(b2b6+b3b8)+c3(b1b5 +b4b7)(b1b8+b4b6) ++d1(b1b5 + b4b7)(b2b7 + b3b5) + d3(b2b6 + b3b8)(b2b7 + b3b5) = 0. +Proof. Let J be the Jacobian matrix of X , X1 or X0 with respect to the homogeneous +coordinates (X1, Y1, X2, Y2, X3, Y3). We denote by ∆ij the determinant of the matrix con- +sisting i-th and j-th columns. Assume rank(J) ≤ 1. +(a) In this case, J is given by +� +Y1 +X1 +Y2 +X2 +Y3 +X3 +a1Y1 +a1X1 +a2Y2 +a2X2 +a3Y3 +a3X3 +� +. + +KUMMER SURFACES +13 +By the relation � XiYi = 0, 5 columns of J vanish. Hence, if Xi ̸= 0, then ci = 0 by the +equation Q2 = 0. Conversely if c1 = 0, then (X1, Y1, X2, Y2, X3, Y3) = (1, 0, 0, 0, 0, 0) is +a singular point. +(b) In this case we use the equations of the pencil given in (2.9) instead of (2.13). +Moreover we use the pair Q1 and Q2 + a2Q1. Then J is given by +� +Y1 +X1 +Y2 +X2 +Y3 +X3 +(a1 + a2)Y1 +(a1 + a2)X1 +0 +X3 +Y2 +0 +� +. +Obviously X3 = Y2 = 0 and then J is given by +� +0 +0 +0 +X2 +Y3 +0 +(a1 + a2)Y1 +(a1 + a2)X1 +0 +0 +0 +0 +� +. +Thus rank(J) ≤ 1 if and only if X2 = Y2 = X3 = Y3 = 0 or X1 = Y1 = X3 = Y2 = 0. +First assume that X2 = Y2 = X3 = Y3 = 0. Then, by changing the coordinates (2.11), +the new coordinates should satisfy x1y1 = 0 and a1x1y1 + c1x2 +1 + d1y2 +1 = 0, that is, +c1d1 = 0. +Next assume that X1 = Y1 = X3 = Y2 = 0. Then, by changing the coordinates (2.11), +we have +α2γ2X2 +2 + β3δ3Y 2 +3 = 0, +(c2α2 +2 + d2γ2 +2)X2 +2 + (c3β2 +3 + d3δ2 +3)Y 2 +3 = 0. +Thus X1 has a singularity if and only if +α2γ2(c3β2 +3 + d3δ2 +3) + β3δ3(c2α2 +2 + d2γ2 +2) = 0, +by using the relation between {bi} and {αi, βi, γi, δi}, which is equivalent to +b2b4c2 + b1b4c3 + b1b3d2 + b2b3d3 = 0. +(c) In this case we also use the equations of the pencil given in (2.10) instead of (2.14). +Moreover we use the pair Q1 and Q2 + aQ1. Then J is given by +� +Y1 +X1 +Y2 +X2 +Y3 +X3 +0 +X2 +Y1 +X3 +Y2 +0 +� +. +Obviously X3 = Y1 = 0 and hence J is +� +0 +X1 +Y2 +X2 +Y3 +0 +0 +X2 +0 +0 +Y2 +0 +� +. +Thus rank(J) ≤ 1 if and only if X2 = X3 = Y1 = Y2 = 0. By changing the coordinates +(2.11), we have +α1γ1X2 +1 + β3δ3Y 2 +3 = 0, +(c1α2 +1 + d1γ2 +1)X2 +1 + (c3β2 +3 + d3δ2 +3)Y 2 +3 = 0. +Thus X0 has a singularity if and only if +α1γ1(c3β2 +3 + d3δ2 +3) + β3δ3(c1α2 +1 + d1γ2 +1) = 0. + +14 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +By using the relation between {bi} and {αi, βi, γi, δi}, we have obtained the assertion. +□ +Assumption: In the following of this paper we assume that X , X1, X0 are non-singular. +Remark 3.2. Under the assumption, the cases b1 = b2 = 0 and b3 = b4 = 0 do not occur in +case (b) by the condition b3(b1d2 +b2d3)+b4(b2c2 +b1c3) ̸= 0. Only the case b1 = b3 = 0, +b1 = b4 = 0, b2 = b3 = 0 or b2 = b4 = 0 is possible. In case (c), +(b1b5 + b4b7, b2b6 + b3b8) ̸= (0, 0), +(b1b8 + b4b6, b2b7 + b3b5) ̸= (0, 0). +The following Lemma 3.3 will be used to study singularities of Kummer quartic surfaces +and their blowing-ups in Theorems 4.6, 5.11. +Lemma 3.3. Let ξ1 = √b2b4 + b5b8, ξ2 = √b1b3 + b6b7. Then (ξ1, ξ2) ̸= (0, 0). +Proof. Assume that ξ2 +1 = α2 +2β3δ3 + β2 +2α1γ1 = 0 and ξ2 +2 = γ2 +2β3δ3 + δ2 +2α1γ1 = 0. Then it +follows that α2 +2δ2 +2α1γ1β3δ3 = γ2 +2β2 +2α1γ1β3δ3, that is, α1γ1β3δ3(α2δ2 + β2γ2)2 = 0. Since +α2δ2 + β2γ2 = 1, we have α1γ1β3δ3 = 0. Again by αiδi + βiγi = 1, we can easily +see that α1 = β3 = 0, α1 = δ3 = 0, γ1 = β3 = 0 or γ1 = δ3 = 0. These imply +that b1 = b4 = b6 = b8 = 0, b2 = b3 = b6 = b8 = 0, b1 = b4 = b5 = b7 = 0 or +b2 = b3 = b5 = b7 = 0. In any case, this contradicts to the condition (c) in Lemma +3.1. +□ +4. QUADRIC LINE COMPLEXES AND KUMMER SURFACES IN CHARACTERISTIC 2: +THE KUMMER QUARTIC SURFACES +In this section we discuss the Kummer quartic surfaces associated with quadratic line +complexes. Let S, S1 or S0 be the Kummer quartic surface associated with P, P1 or P0, +respectively. +Theorem 4.1. (a) The Kummer quartic surface S is given by the equation +(a1 + a2)2(c3x2y2 + d3z2t2) + (a1 + a3)2(c2x2z2 + d2y2t2) ++(a2 + a3)2(c1x2t2 + d1y2z2) + (a1 + a2)(a2 + a3)(a3 + a1)xyzt = 0. +(b) The Kummer quartic surface S1 is given by the equation +b2 +3c1x4 + b2 +2d1y4 + b2 +1d1z4 + b2 +4c1t4 ++(b2 +3d2 + b2 +2c2 + (a1 + a2)2c3 + (a1 + a2)b2b3)x2y2 ++(b2 +3d3 + b2 +1c3 + (a1 + a2)2c2 + (a1 + a2)b1b3)x2z2 ++(b2 +2d3 + b2 +4c3 + (a1 + a2)2d2 + (a1 + a2)b2b4)y2t2 ++(b2 +1d2 + b2 +4c2 + (a1 + a2)2d3 + (a1 + a2)b1b4)z2t2 ++(a1 + a2)2(b3x2yz + b2xy2t + b1xz2t + b4yzt2) = 0. + +KUMMER SURFACES +15 +(c) The Kummer quartic surface S0 is given by the equation +(b2 +3c1 + b2 +7c3)x4 + (b2 +2d1 + b2 +8c3)y4 + (b2 +1d1 + b2 +6d3)z4 + (b2 +4c1 + b2 +5d3)t4 ++b5(b1b5 + b4b7)xt3 + b7(b2b7 + b3b5)x3t + b2(b2b6 + b3b8)xy3 + b8(b2b6 + b3b8)y3z ++b3(b2b7 + b3b5)x3y + b4(b1b5 + b4b7)zt3 + b6(b1b8 + b4b6)yz3 + b1(b1b8 + b4b6)z3t ++(b2 +2c2 + b2 +3d2)x2y2 + (b2 +1c3 + b2 +3d3 + b2 +6c1 + b2 +7d1)x2z2 + (b2 +5c2 + b2 +7d2)x2t2 ++(b2 +6d2 + b2 +8c2)y2z2 + (b2 +2d3 + b2 +4c3 + b2 +5d1 + b2 +8c1)y2t2 + (b2 +1d2 + b2 +4c2)z2t2 ++b7(b2b6 + b3b8)x2yz + b3(b1b5 + b4b7)x2zt + b8(b2b7 + b3b5)xy2t + b2(b1b8 + b4b6)y2zt ++b1(b2b6+b3b8)xyz2+b6(b1b5+b4b7)xz2t+b4(b2b7+b3b5)xyt2+b5(b1b8+b4b6)yzt2 = 0. +Proof. We consider the pencil P0 of quadrics, but not assuming a1 = a2 = a3. We identify +Q1 and the grassmannian G = G(2, 4). Consider three points +(X1, X2, X3, Y1, Y2, Y3) = (x, 0, 0, 0, y, z), (0, x, 0, y, 0, t), (0, 0, x, z, t, 0) +in P5 and the plane +Π = {(αx, βx, γx, βy + γz, αy + γt, αz + βt) : (α, β, γ) ∈ P2} +generated by these points, where (x, y, z, t) ∈ P3. The plane Π is a generic member of an +irreducible family of planes on G. The conic Π ∩ Q2 on Π is given by +(4.1) +(c1x2 + d2y2 + d3z2 + b4yz + b5xy)α2 ++(c2x2 + d1y2 + d3t2 + b1xt + b6xy)β2 ++(c3x2 + d1z2 + d2t2 + b2xt + b8zt)γ2 ++((a1 + a2)xy + b1xz + b4yt + b7x2 + b8y2)αβ ++((a2 + a3)xt + b6xz + b8yt + b3x2 + b4t2)βγ ++((a1 + a3)xz + b2xy + b4zt + b5xt + b8yz)γα = 0. +Then this conic has a singularity if and only if +α = (a2 + a3)xt + b6xz + b8yt + b3x2 + b4t2, +β = (a1 + a3)xz + b2xy + b4zt + b5xt + b8yz, +γ = (a1 + a2)xy + b1xz + b4yt + b7x2 + b8y2. + +16 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +By combining this and (4.1), we obtain the sextic equation. The terms of degree ≤ 1 +in x of this sextic are identically 0, and hence by dividing by x2 we have the following +equation: +(b2 +3c1 + b2 +7c3)x4 + (b2 +2d1 + b2 +8c3)y4 + (b2 +1d1 + b2 +6d3)z4 + (b2 +4c1 + b2 +5d3)t4 ++b5(b1b5 + b4b7)xt3 + b7(b2b7 + b3b5)x3t + b2(b2b6 + b3b8)xy3 + b8(b2b6 + b3b8)y3z ++b3(b2b7 + b3b5)x3y + b4(b1b5 + b4b7)zt3 + b6(b1b8 + b4b6)yz3 + b1(b1b8 + b4b6)z3t ++(a1 + a3)(b3b7x3z + b1b6xz3 + b2b8y3t + b4b5yt3) ++(b2 +2c2+b2 +3d2+c3(a1+a2)2+(a1+a2)b2b3)x2y2+(b2 +5c2+b2 +7d2+c1(a2+a3)2+(a2+a3)b5b7)x2t2 ++(b2 +1c3 + b2 +3d3 + b2 +6c1 + b2 +7d1 + c2(a1 + a3)2 + (a1 + a3)(b1b3 + b6b7))x2z2 ++(b2 +6d2+b2 +8c2+d1(a2+a3)2+(a2+a3)b6b8)y2z2+(b2 +1d2+b2 +4c2+d3(a1+a2)2+(a1+a2)b1b4)z2t2 ++(b2 +2d3 + b2 +4c3 + b2 +5d1 + b2 +8c1 + d2(a1 + a3)2 + (a1 + a2)b2b4 + (a2 + a3)b5b8)y2t2 ++(b7(b2b6+b3b8)+b3(a1+a2)(a1+a3))x2yz+(b3(b1b5+b4b7)+b7(a1+a3)(a2+a3))x2zt ++(b8(b2b7+b3b5)+b2(a1+a2)(a1+a3))xy2t+(b2(b1b8+b4b6)+b8(a1+a3)(a2+a3))y2zt ++(b1(b2b6+b3b8)+b6(a1+a3)(a2+a3))xyz2+(b6(b1b5+b4b7)+b1(a1+a2)(a1+a3))xz2t ++(b4(b2b7+b3b5)+b5(a1+a3)(a2+a3))xyt2+(b5(b1b8+b4b6)+b4(a1+a2)(a1+a3))yzt2 ++(b2b7(a2 + a3) + b3b5(a1 + a2))x2yt + (b2b6(a1 + a2) + b3b8(a2 + a3))xy2z ++(b1b5(a2 + a3) + b4b7(a1 + a2))xzt2 + (b1b8(a1 + a2) + b4b6(a2 + a3))yz2t ++((a1 +a2)(a2 +a3)(a3 +a1)+(a1 +a2)(b5b6 +b7b8)+(a2 +a3)(b1b2 +b3b4))xyzt = 0. +Now by putting a1 = a2 = a3 we have the equation (c) of the Kummer quartic surface. +The case (a) (resp. the case (b)) is obtained by putting b1 = · · · = b8 = 0 (resp. a2 = a3 +and b5 = · · · = b8 = 0). +□ +Next we study the Kummer quartic surfaces individually. First we consider the case (a). +Theorem 4.2. The quartic surface S has exactly four rational double points +P1 = (1, 0, 0, 0), P1 = (0, 1, 0, 0), P3 = (0, 0, 1, 0), P4 = (0, 0, 0, 1) +of type D4 and contains four tropes: +Θ1 : x = (a1 + a2) +� +d3zt + (a1 + a3) +� +d2yt + (a2 + a3) +� +d1yz = 0, +Θ2 : y = (a1 + a2) +� +d3zt + (a1 + a3)√c2xz + (a2 + a3)√c1xt = 0, +Θ3 : z = (a1 + a2)√c3xy + (a1 + a3) +� +d2yt + (a2 + a3)√c1xt = 0, +Θ4 : t = (a1 + a2)√c3xy + (a1 + a3)√c2xz + (a2 + a3) +� +d1yz = 0. +The trope Θi passes through three points Pj (j ̸= i) among the four points. + +KUMMER SURFACES +17 +Proof. It follows from the Jacobian criterion that S has exactly four singular points P1, . . . , +P4. By blowing up the singular points, we can see that Pi is a rational double point of +type D4 (or it follows from Theorem 6.1, Remark 6.2, Proposition 2.4 and Theorem 2.14 +that the singularities are of type D4). Each trope is defined by a hyperplane section, e.g. +2Θ1 = {x = 0}. The last assertion is straightforward. +□ +The Cremona transformation +(4.2) +cr : (x, y, z, t) → +�� +d1d2d3/x, +� +c1c2d3/y, +� +c1d2c3/z, +� +d1c2c3/t +� +preserves the equation of the Kummer quartic surface and interchanges Pi and Θi (1 ≤ +i ≤ 4). Also there are the following involutions of S which generate (Z/2Z)2: +ϕ1 : (x, y, z, t) → +�� +d1d2y, √c1c2x, +� +c1d2t, +� +d1c2z +� +ϕ2 : (x, y, z, t) → +�� +d1d3z, +� +c1d3t, √c1c3x, +� +d1c3y +� +. +Remark 4.3. Laszlo and Pauly [14, Proposition 4.1] gave the equation of the Kummer +quartic surface associated with the Jacobian J(C) of an ordinary curve C of genus 2 as +follows: +λ2 +10(x2 +00x2 +10 + x2 +01x2 +11) + λ2 +01(x2 +00x2 +01 + x2 +10x2 +11) + λ2 +11(x2 +00x2 +11 + x2 +01x2 +10) ++λ−1 +00 λ10λ01λ11x00x01x10x11 = 0. +Here {xij} is a basis of H0(J(C), 2Θ). In the equation in Theorem 4.1 (a), by putting +X = x 4√c1c2c3, Y = y +4� +d1d2c3, Z = z +4� +d1c2d3, T = t +4� +c1d2d3 +and then dividing by (a1+a2)(a2+a3)(a3+a1) +√c1c2c3d1d2d3 +, we obtain +(4.3) + + + + + +√c3d3(a1+a2) +(a1+a3)(a2+a3)(X2Y 2 + Z2T 2) + +√c2d2(a1+a3) +(a1+a2)(a2+a3)(X2Z2 + Y 2T 2) ++ +√c1d1(a2+a3) +(a1+a2)(a1+a3)(X2T 2 + Y 2Z2) + XY ZT = 0 +which is the same equation as Laszlo and Pauly’s one above. +Next we consider the Kummer quartic surface in the case (b). +Theorem 4.4. The surface S1 has exactly two singular points +P1 = +� +0, +� +b1, +� +b2, 0 +� +, P2 = +�� +b4, 0, 0, +� +b3 +� +of type D8 and contains two tropes Θ1 and Θ2 both of which are double conics cutting by +the hyperplane section √b2y + √b1z = 0 and √b3x + √b4t = 0, respectively. Two tropes +Θ1 and Θ2 meet at P1 and P2. + +18 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Proof. First note that (b1, b2) ̸= (0, 0), (b3, b4) ̸= (0, 0) (Remark 3.2). By the Jacobian +criterion, we can see that S1 has two singular points. We will show that the Jacobian +surface has 2-rank 1 (Theorem 6.3, Remark 6.4). It follows from Proposition 2.4 and +Theorem 2.14 that the singularities are of type D8. For the second assertion, for example, +by putting the equation √b2y + √b1z = 0 in the last term of the equation in Theorem 4.4, +we have a decomposition +b3x2yz + b2xy2t + b1xz2t + b4yzt2 = +� +b1/b2z2 �� +b3x + +� +b4t +�2 +by using b1b2 = b3b4. This implies the assertion. +□ +Recall that we may assume that b2b4c2 = b1b3d2 for b1b2b3b4 ̸= 0 and b2 +2c2 = b2 +3d2 for +b1 = b4 = 0 (Lemma 2.12, Remark 2.13). +Proposition 4.5. S1 has an involution ι defined by +ι : (x, y, z, t) → +� +4� +d1b2 +2/c1b2 +3y, +4� +c1b2 +3/d1b2 +2x, +4� +c1b2 +4/d1b2 +1t, +4� +d1b2 +1/c1b2 +4z +� +for b1b2b3b4 ̸= 0 and +ι : (x, y, z, t) → +� +4� +d1b2 +2/c1b2 +3y, +4� +c1b2 +3/d1b2 +2x, +4� +c1b2 +2/d1b2 +3t, +4� +d1b2 +3/c1b2 +2z +� +for b1 = b4 = 0 which satisfies ι(¯Θ1) = ¯Θ2 and ι(P1) = P2. +Proof. The proof is straightforward. +□ +We do not know an exact formula of a Cremona transformation interchanging {¯Θ1, ¯Θ2} +and {P1, P2} as the one given in (4.2) for ordinary case. +Finally we consider the case (c). Let ξ1 = √b2b4 + b5b8, ξ2 = √b1b3 + b6b7. Recall +that (b1b5 + b4b7, b2b6 + b3b8) ̸= (0, 0), (b1b8 + b4b6, b2b7 + b3b5) ̸= (0, 0) (Remark 3.2) +and (ξ1, ξ2) ̸= (0, 0) (Lemma 3.3). Let +x0 = +� +(b1b8 + b4b6)ξ1, +y0 = +� +(b1b5 + b4b7)ξ2, +z0 = +� +(b2b7 + b3b5)ξ1, +t0 = +� +(b2b6 + b3b8)ξ2, +x′ +0 = +� +(b1b8 + b4b6)ξ2, +y′ +0 = +� +(b1b5 + b4b7)ξ1, +z′ +0 = +� +(b2b7 + b3b5)ξ2, +t′ +0 = +� +(b2b6 + b3b8)ξ1. +Theorem 4.6. (1) S0 has a unique singular point P0 = (x0, y0, z0, t0) of type 4⃝1 +0,1. +(2) S0 contains a trope ¯Θ by cutting by the hyperplane +z′ +0x + t′ +0y + x′ +0z + y′ +0t = 0. + +KUMMER SURFACES +19 +Proof. (1) By the Jacobian criterion, we can check that P0 is a singular point of the surface +by elementary but long calculation. Later we show that S0 is the quotient of the supersin- +gular abelian surface (Theorem 6.5, Remark 6.6). It follows from Proposition 2.4 and +Theorem 2.14 that S0 has a unique singularity of type 4⃝1 +0,1. We remark that the point P0 +is nothing but the base point of the linear system defined by six quadrics +X1 = b3x2 + b4t2 + b6xz + b8yt, Y1 = b2y2 + b1z2 + b7xz + b5yt, +X2 = b2xy + b5xt + b8yz + b4zt, Y2 = b3xy + b7xt + b6yz + b1zt, +X3 = b7x2 + b8y2 + b1xz + b4yt, Y3 = b6z2 + b5t2 + b3xz + b2yt. +Here X2 +i , Y 2 +i are coefficients of ci, di when we consider the quartic equation of S0 as a +linear form of ci, di. +(2) Consider a hyperplane t = αx + βy + γz (α, β, γ ∈ k). By putting this into the +equation of S0, we have the equation +f(x, y, z)2 + xyL1(x, y, z)2 + yzL2(x, y, z)2 + zxL3(x, y, z)2 = 0 +where f is a quadric and L1, L2, L3 are linear forms. We can check that L1, L2, L3 coincide +up to constant by elementary but long calculation. Then we choose α, β, γ satisfying +L1 ≡ 0 which gives the desired hyperplane. Now a direct calculation shows that the +desired hyperplane is given in (2). +□ +We do not know an exact formula of a Cremona transformation interchanging ¯Θ and P0 +as the one given in (4.2) for the ordinary case. +5. QUADRIC LINE COMPLEXES AND KUMMER SURFACES IN CHARACTERISTIC 2: +THE INTERSECTION OF THREE QUADRICS +Denote by Σ, Σ1 or Σ0 the set of singular lines (see the equation (2.5)) of the quadric +line complex defined by the pencil P, P1 or P0, respectively. +Theorem 5.1. (a) The surface Σ is given by the equations +Σ = +� +3 +� +i=1 +XiYi = +3 +� +i=1 +aiXiYi + ciX2 +i + diY 2 +i = +3 +� +i=1 +a2 +i XiYi = 0 +� +. +(b) Σ1 is given by the equations +3 +� +i=1 +XiYi = +3 +� +i=1 +(aiXiYi + ciX2 +i + diY 2 +i ) + b1X2Y3 + b2X3Y2 + b3X2X3 + b4Y2Y3 += +3 +� +i=1 +a2 +i XiYi + b1b3X2 +2 + b2b3X2 +3 + b2b4Y 2 +2 + b1b4Y 2 +3 = 0 +where a2 = a3 and b1b2 = b3b4. + +20 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +(c) Σ0 is given by the equations +3 +� +i=1 +XiYi = +3 +� +i=1 +(aXiYi + ciX2 +i + diY 2 +i ) ++b1X2Y3 + b2X3Y2 + b3X2X3 + b4Y2Y3 + b5X1Y2 + b6X2Y1 + b7X1X2 + b8Y1Y2 += a2 +3 +� +i=1 +XiYi +b5b7X2 +1 +(b1b3 +b6b7)X2 +2 +b2b3X2 +3 +b6b8Y 2 +1 +(b2b4 +b5b8)Y 2 +2 +b1b4Y 2 +3 ++(b1b5 + b4b7)X1Y3 + (b2b6 + b3b8)X3Y1 + (b2b7 + b3b5)X1X3 + (b1b8 + b4b6)Y1Y3 = 0, +where b1b2 = b3b4, b5b6 = b7b8. +Proof. We use Proposition 2.1. Consider the case (c) without assuming a1 = a2 = a3. For +x = (u1, u2, u3, v1, v2, v3) ∈ G, x′ = (u′ +1, u′ +2, u′ +3, v′ +1, v′ +2, v′ +3) ∈ G ∩ Q2, +the tangent space of G at x is given by +3 +� +i=1 +(viXi + uiYi) = 0, +and that of Q2 at x′ is given by +(a1v′ +1 + b7u′ +2 + b5v′ +2)X1 + (a1u′ +1 + b6u′ +2 + b8v′ +2)Y1 + (a2v′ +2 + b7u′ +1 + b3u′ +3 + b6v′ +1 + b1v′ +3)X2 ++(a2u′ +2+b5u′ +1+b2u′ +3+b8v′ +1+b4v′ +3)Y2+(a3v′ +3+b3u′ +2+b2v′ +2)X3+(a3u′ +3+b1u′ +2+b4v′ +2)Y3 = 0. +The condition that these two tangent spaces coincide implies that the coefficients of Xi, Yi +coincide, and by putting these into � +i uivi = 0, we obtain the equation +(a2 +1 + b5b6 + b7b8)u′ +1v′ +1 + (a2 +2 + b1b2 + b3b4 + b5b6 + b7b8)u′ +2v′ +2 + (a2 +3 + b1b2 + b3b4)u′ +3v′ +3 ++b5b7u′2 +1 + b6b8v′2 +1 + (b1b3 + b6b7)u′2 +2 + (b2b4 + b5b8)v′2 +2 + b2b3u′2 +3 + b1b4v′2 +3 ++(b1b5 + b4b7)u′ +1v′ +3 + (b2b6 + b3b8)u′ +3v′ +1 + (b2b7 + b3b5)u′ +1u′ +3 + (b1b8 + b4b6)v′ +1v′ +3 ++(a1 + a2)b7u′ +1u′ +2 + (a1 + a2)b5u′ +1v′ +2 + (a1 + a2)b6u′ +2v′ +1 + (a1 + a2)b8v′ +1v′ +2 ++(a2 + a3)b3u′ +2u′ +3 + (a2 + a3)b1u′ +2v′ +3 + (a2 + a3)b2u′ +3v′ +2 + (a2 + a3)b4v′ +2v′ +3. +Now by using the equations a1 = a2 = a3 and b1b2 = b3b4, b5b6 = b7b8, we obtain the +equation (c). By putting b1 = · · · = b8 = 0, we have the equation (a), and by putting +a2 = a3, b1b2 = b3b4 and b5 = · · · = b8 = 0, we obtain the equation (b). +□ + +KUMMER SURFACES +21 +Remark 5.2. Using (2.8), (2.9) and (2.10), we see, in a similar way to the proof of Theorem +5.1, that the surfaces Σ, Σ1 and Σ0 are given as follows: +(a) The surface Σ is isomorphic to the surface defined by +3 +� +i=1 +(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = +3 +� +i=1 +(aiXiYi + r2 +i X2 +i + s2 +i Y 2 +i ). += +3 +� +i=1 +(a2 +i XiYi + a2 +i p2 +i X2 +i + a2 +i t2 +i Y 2 +i ) = 0. +(b) The surface Σ1 is isomorphic to the surface defined by +3 +� +i=1 +(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = +3 +� +i=1 +(aiXiYi + r2 +i X2 +i + s2 +i Y 2 +i ) + X3Y2. += +3 +� +i=1 +(a2 +i XiYi + a2 +i p2 +i X2 +i + a2 +i t2 +i Y 2 +i ) + p2 +2X2 +3 + t2 +3Y 2 +2 = 0 +(c) The surface Σ0 is isomorphic to the surface defined by +3 +� +i=1 +(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = +3 +� +i=1 +(aXiYi + r2 +i X2 +i + s2 +i Y 2 +i ) + X2Y1 + X3Y2. += +3 +� +i=1 +(a2XiYi + a2p2 +i X2 +i + a2t2 +i Y 2 +i ) + p2 +1X2 +2 + p2 +2X2 +3 + t2 +2Y 2 +1 + t2 +3Y 2 +2 + X3Y1 = 0 +These equations are sometimes useful to examine the properties of Σ, Σ1 and Σ0, for +example, to calculate the singularities. +Next we study the intersection of three quadrics in Theorem 5.1 individually. First we +consider the case (a). +Theorem 5.3. (1) The lines on Σ are exactly the following eight ones: +�Θ1 : X1 = X2 = X3 = � +i +√diYi = 0, +�Θ2 : Y1 = Y2 = X3 = √c1X1 + √c2X2 + √d3Y3 = 0, +�Θ3 : Y1 = X2 = Y3 = √c1X1 + √d2Y2 + √c3X3 = 0, +�Θ4 : X1 = Y2 = Y3 = √d1Y1 + √c2X2 + √c3X3 = 0, +E1 : Y1 = Y2 = Y3 = � +i +√ciXi = 0, +E2 : X1 = X2 = Y3 = √d1Y1 + √d2Y2 + √c3X3 = 0, +E3 : X1 = Y2 = X3 = √d1Y1 + √c2X2 + √d3Y3 = 0, +E4 : Y1 = X2 = X3 = √c1X1 + √d2Y2 + √d3Y3 = 0. + +22 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +(2) The surface Σ has exactly twelve nodes at the twelve intersection points of �Θi and +Ej. In particular Σ is a K3 surface with rational double points. +Proof. Since any line on Σ is of the form σ(p, h) cutting by planes σ(p) and σ(h), it +corresponds to a singularity of the Kummer quartic surface or a trope. Hence the number +of lines is eight. The assertion (2) follows from the Jacobian criterion and the resolution +of singularities. +□ +The configuration of {�Θi, Ej} is given as in Figure 2: +E1 +E2 +E3 +E4 +�Θ1 +�Θ2 +�Θ3 +�Θ4 +FIGURE 2. Eight lines on Σ +Let �Σ be the minimal resolution of Σ. Then �Σ is a K3 surface and contains twenty +(−2)-curves (i.e. non-singular rational curves) which are the proper transforms of �Θi, Ej +and the twelve exceptional curves over the twelve nodes. We denote the proper transforms +by the same symbols �Θi, Ej. Then the dual graph of twenty (−2)-curves is given as in +Figure 3: +✤ +✤ +✤ +✤ +✤ +✤ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❴ +❴ +❴ +❴ +❴ +❴ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +• +E1 +�Θ3 +E3 +�Θ2 +�Θ1 +E4 +�Θ4 +E2 +FIGURE 3. The dual graph of 20 (−2)-curves on �Σ + +KUMMER SURFACES +23 +Remark 5.4. Over the complex numbers, Peters and Stienstra [21] studied a 1-dimensional +family of K3 surfaces containing 20 (−2)-curves forming the dual graph in Figure 3 and +Mukai and Ohashi [16] also studied quartic surfaces given in Remark 4.3. +The projective transformations +ι1 : (X1, X2, X3, Y1, Y2, Y3) → +�� +d1/c1Y1, X2, X3, +� +c1/d1X1, Y2, Y3 +� +, +ι2 : (X1, X2, X3, Y1, Y2, Y3) → +� +X1, +� +d2/c2Y2, X3, Y1, +� +c2/d2X2, Y3 +� +, +ι3 : (X1, X2, X3, Y1, Y2, Y3) → +� +X1, X2, +� +d3/c3Y3, Y1, Y2 : +� +c3/d3X3 +� +act on Σ and hence induce an automorphism group of �Σ isomorphic to (Z/2Z)3. The +subgroup (Z/2Z)2 generated by ι1◦ι2, ι1◦ι3 preserves {�Θi} and the remaining involutions +interchange {�Θi} and {Ej}. +Proposition 5.5. The linear system defined by six quadrics +X1 = (a2 + a3)xt, X2 = (a1 + a3)xz, X3 = (a1 + a2)xy, +Y1 = (a2 + a3)yz, Y2 = (a1 + a3)yt, Y3 = (a1 + a2)zt +gives a birational map µ from the Kummer quartic surface S to Σ. +Proof. Note that X2 +i , Y 2 +i are coefficients of ci, di when we consider the quartic equa- +tion of S as a linear form of ci, di. By definition of µ, its image satisfies the equations +�3 +i=1 XiYi = �3 +i=1 a2 +i XiYi = 0, and the relation �3 +i=1(aiXiYi+ciX2 +i +diY 2 +i ) = 0 follows +from the quartic equation of S. The base locus of the linear system consists of four singu- +lar points of S. Let ˜S be the blowing-up of the four singular points of S. Then µ induces a +proper morphism from ˜S to Σ. Thus if we show that the inverse image µ−1(P) of a point P +of Σ consists of one point, then the assertion follows. Let P be a general point of the line +˜Θ1. Then we can write P = (0, 0, 0, α, β, γ) satisfying α, β, γ ∈ k∗ and √d1α + √d2β + +√d3γ = 0. Then the inverse image µ−1(P) consists of only one point by solving the +equation (xt, xz, xy, yz, yt, zt) = (0, 0, 0, α/(a2 + a3), β/(a1 + a3), γ/(a1 + a2)) . +□ +Remark 5.6. In Figure 3, by contracting sixteen (−2)-curves except �Θ1, . . . , �Θ4, we obtain +the Kummer quartic surface S. If we contract sixteen (−2)-curves except E1, . . . , E4, then +we obtain the dual Kummer surface S∨. +Next we consider the case (b). +Theorem 5.7. (1) The lines on Σ1 are exactly the following four ones: +�Θ1 : Y1 = +� +b1X2 + +� +b2X3 = +� +b2Y2 + +� +b1Y3 = 0, +�Θ2 : X1 = +� +b3X2 + +� +b4Y3 = +� +b3X3 + +� +b4Y2 = 0, + +24 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +E1 : X1 = +� +b1X2 + +� +b2X3 = +� +b2Y2 + +� +b1Y3 = 0, +E2 : Y1 = +� +b3X2 + +� +b4Y3 = +� +b3X3 + +� +b4Y2 = 0. +Here we give only the equation of a plane Π such that Π ∩ Σ1 is a double line. The dual +graph of the four lines is a square, that is, �Θ1 · �Θ2 = E1 · E2 = 0 and �Θi · Ej = 1, 1 ≤ +i, j ≤ 2. +(2) The surface Σ1 has exactly four singular points which are the intersection points of +the four lines. +Proof. The proof of (1) is the same as that of Theorem 5.3. We use the condition of Lemma +3.1(b) for the intersection multiplicities. The assertion (2) follows from the Jacobian cri- +terion and resolutions of singularities. +□ +Consider the involutions ι1, ι2 of P5: +ι1 : (X1, X2, X3, Y1, Y2, Y3) → +�� +d1/c1Y1, X2, X3, +� +c1/d1X1, Y2, Y3 +� +; +In case b1b2b3b4 ̸= 0, +ι2 : (X1, X2, X3, Y1, Y2, Y3) → +�� +d1/c1Y1, +� +b2b4/b1b3Y2, X3, +� +c1/d1X1, +� +b1b3/b2b4X2, Y3 +� +, +and in case b1b2b3b4 = 0, for example, b1 = b4 = 0, +ι2 : (X1, X2, X3, Y1, Y2, Y3) → +�� +d1/c1Y1, (b2/b3)Y2, X3, +� +c1/d1X1, (b3/b2)X2, Y3 +� +. +Obviously ι1 preserves the equations of Σ1 and hence induces an involution of Σ1 and +ι1(�Θ1) = E1 and ι1(�Θ2) = E2. Note that ι2 preserves Σ1 if and only if b2b4c2 = b1b3d2 +for b1b2b3b4 ̸= 0 and b2 +2c2 = b2 +3d2 for b1 = b4 = 0. +Proposition 5.8. After changing the coordinates, we may assume that b2b4c2 = b1b3d2 for +b1b2b3b4 ̸= 0 and b2 +2c2 = b2 +3d2 for b1 = b4 = 0, and hence the projective transformation ι2 +preserves Σ1 and ι2(�Θ1) = �Θ2 and ι2(E1) = E2. +Proof. The assertion follows from Lemma 2.12 and Remark 2.13. +□ +Proposition 5.9. The linear system defined by six quadrics +X1 = b3x2 + b4t2, X2 = b2xy + b4zt + (a1 + a2)xz, X3 = b1xz + b4yt + (a1 + a2)xy, +Y1 = b2y2 + b1z2, Y2 = b3xy + b1zt + (a1 + a2)yt, Y3 = b3xz + b2yt + (a1 + a2)zt +gives a birational map µ1 from the Kummer quartic surface S1 to Σ1. +Proof. We remark that X2 +i , Y 2 +i are the coefficients of ci, di when we consider the quartic +equation of S1 as a linear form of ci, di. The proof of the assertion is similar to the one of +Proposition 5.5. In this case consider a general point (α, √b2, √b1, 0, √b1, √b2) of the line +˜Θ1 (α ∈ k). Then we can easily check that (x, y, z, t) is uniquely determined by α. +□ + +KUMMER SURFACES +25 +The involution ι2 of Σ1 corresponds to the involution ι of S1 defined in Proposition 4.5 +under the birational map µ1. +Finally we consider the case (c). Let ι be the projective transformation of P5 defined by +ι(X1) = +� +b6b8/b5b7Y1, ι(Y1) = +� +b5b7/b6b8X1, ι(Xi) = Xi, ι(Yi) = Yi (i = 2, 3) +if b5b6b7b8 ̸= 0, +ι(X1) = (b6/b7)Y1, ι(Y1) = (b7/b6)X1, ι(Xi) = Xi, ι(Yi) = Yi (i = 2, 3) +if, for example, b5 = b8 = 0. By Lemma 2.12, Remark 2.13, we may assume that b6b8c1 = +b5b7d1 in the first case and b2 +6c1 = b2 +7d1 in the second case. +Lemma 5.10. The involution ι acts on Σ0 as an automorphism. +Proof. This is straightforward. +□ +Theorem 5.11. (1) The surface Σ0 has only one singular point P, which is the intersection +point of the two conics defined by b7X1+b6Y1 = 0 and b2X3+b4Y3 = 0 or by b5X1+b8Y1 = +0 and b3X3 + b1Y3 = 0. The singular point P is an elliptic double point of type 14 +⃝(0),(1) +0,0,1 +(Figure 1). Let ξ1 = √b2b4 + b5b8 and ξ2 = √b1b3 + b6b7. Then, the singular point +P = (X1, X2, X3, Y1, Y2, Y3) is concretely given by +X1 = b8(ξ1 +√b1b3c2 + ξ2 +√b1b3d2 + b1 +√ξ1ξ2c3 + b3 +√ξ1ξ2d3), +X2 = ξ1(b8 +√b1b3c1 + b5 +√b1b3d1 + b1 +√b5b8c3 + b3 +√b5b8d3), +X3 = b1(b8 +√ξ1ξ2c1 + b5 +√ξ1ξ2d1 + ξ1 +√b5b8c2 + ξ2 +√b5b8d2), +Y1 = b5(ξ1 +√b1b3c2 + ξ2 +√b1b3d2 + b1 +√ξ1ξ2c3 + b3 +√ξ1ξ2d3), +Y2 = ξ2(b8 +√b1b3c1 + b5 +√b1b3d1 + b1 +√b5b8c3 + b3 +√b5b8d3), +Y3 = b3(b8 +√ξ1ξ2c1 + b5 +√ξ1ξ2d1 + ξ1 +√b5b8c2 + ξ2 +√b5b8d2). +(2) Let x0, y0, z0, t0, x′ +0, y′ +0, z′ +0, t′ +0 be as in Theorem 4.6. The lines on Σ0 are exactly the +following two: +�E : y0X1 + t0X3 + x0Y2 = y0X2 + z0X3 + x0Y1 = z0X1 + t0X2 + x0Y3 += +�√c1x0 + +� +d2y0 + +� +d3z0 + +� +b4y0z0 + b5x0y0 +� +X1 ++ +�√c2x0 + +� +d1y0 + +� +d3t0 + +� +b1x0t0 + b6x0y0 +� +X2 ++ +�√c3x0 + +� +d1z0 + +� +d2t0 + +� +b2x0t0 + b8z0t0 +� +X3 = 0 +and +�Θ : x′ +0Y1 + z′ +0X3 + y′ +0Y2 = x′ +0X2 + t′ +0X3 + y′ +0X1 = z′ +0Y1 + z′ +0X2 + y′ +0Y3 += +�� +d1y′ +0 + +� +d2x′ +0 + +� +d3t′ +0 + +� +b4x′ +0z′ +0 + b8x′ +0y′ +0 +� +Y1 + +26 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O ++ +�√c2y′ +0 + √c1x′ +0 + +� +d3z′ +0 + +� +b1y′ +0z′ +0 + b7x′ +0y′ +0 +� +X2 ++ +�√c3y′ +0 + √c1t′ +0 + +� +d2z′ +0 + +� +b2y′ +0z′ +0 + b5y′ +0t′ +0 +� +X3 = 0, +respectively. Moreover �Θ = ι( �E), and �E and �Θ meet at P. +Proof. Note that by Lemma 3.3 we have (ξ1, ξ2) ̸= (0, 0). To calculate the singularities of +Σ0, we use the equations in Remark 5.2 (c). Subtracting suitable constant multiples of the +first equation from the second and the third equations, the surface Σ0 is isomorphic to the +surface defined by the following three equations: +(1) �3 +i=1(XiYi + p2 +i X2 +i + t2 +i Y 2 +i ) = 0, +(2) �3 +i=1{(r2 +i + ap2 +i )X2 +i + (s2 +i + at2 +i )Y 2 +i } + X2Y1 + X3Y2 = 0, +(3) p2 +1X2 +2 + p2 +2X2 +3 + t2 +2Y 2 +1 + t2 +3Y 2 +2 + X3Y1 = 0 +The Jacobian matrix with respect to the coordinates (X1, X2, X3, Y1, Y2, Y3) is given by + + +Y1 +Y2 +Y3 +X1 +X2 +X3 +0 +Y1 +Y2 +X2 +X3 +0 +0 +0 +Y1 +X3 +0 +0 + + . +In this matrix, if Y1 ̸= 0, then the first, the second and the third rows are linearly indepen- +dent. If X3 ̸= 0, then the 4th, the 5th and the 6th rows are linearly independent. Therefore, +singular points must satisfy the equation Y1 = X3 = 0. In the change of coordinates +(2.11), we have +Y1 = γ1x1 + α1y1, X3 = δ3x3 + β3y3. +Therefore, for the equations which define Σ0 in Theorem 5.1, the singularities are defined +by the following two equations: +(5.1) +γ1X1 + α1Y1 = 0, +(5.2) +δ3X3 + β3Y3 = 0. +Note that either α2δ2 ̸= 0 or β2γ2 ̸= 0 holds by α2δ2 + β2γ2 = 1. If α2δ2 ̸= 0, then +multiplying (5.1) by δ2 and (5.2) by α2, we have b7X1 + b6Y1 = 0 and b2X3 + b4Y3 = 0. +If β2γ2 ̸= 0, multiplying (5.1) by β2 and (5.2) by γ2, we have b5X1 + b8Y1 = 0 and +b3X3 + b1Y3 = 0. Since we are in charactersitic 2, using these equations, we have 3 linear +equations from the defining equations of Σ0. Solving the equations, we get the concrete +coordinates of the only one singular point P. +Let Π be the plane defined by +y0X1 + t0X3 + x0Y2 = y0X2 + z0X3 + x0Y1 = z0X1 + t0X2 + x0Y3 = 0. +In Theorem 4.6, we showed that the Kummer quartic surface S0 has a singular point P0 = +(x0, y0, z0, t0). The plane Π is nothing but the one consisting of lines in P3 passing through + +KUMMER SURFACES +27 +P0. We can also check that Π cuts Σ0 along the double line �E, �E contains the singular +point P and �Θ = ι( �E) by direct but long calculation. Since P is the unique singular point, +it is fixed by ι and hence ι( �E) also contains P. Therefore �E and �Θ meet at P. +Recall that there exists a canonical morphism π : Σ0 → S0 by sending x to the focus of +ℓx (see §2.1) which is nothing but the one sending ˜E to P0. It is easy to see that π is the +blowing-up of P0 ∈ S0. On the other hand, it follows from the canonical resolution given +in Katsura [10, §6] that the minimal resolution of a singularity of type 14 +⃝(0),(1) +0,0,1 +is obtained +from the one of a singularity of type 4⃝1 +0,1 by blowing-up a point on the (−3)-curve (Figure +1). Since P0 is of type 4⃝1 +0,1 (Theorem 4.6), P is of type 14 +⃝(0),(1) +0,0,1 . +□ +Remark 5.12. In Theorem 5.11 (2), if x0 = 0, then the equations +y0X1 + t0X3 + x0Y2 = y0X2 + z0X3 + x0Y1 = z0X1 + t0X2 + x0Y3 = 0 +define a 3-dimensional subspace instead of a plane. However, together with the relation +� XiYi = 0, it defines a unique line. The same thing holds if y′ +0 = 0. +Proposition 5.13. The linear system defined by six quadrics +X1 = b3x2 + b4t2 + b6xz + b8yt, Y1 = b2y2 + b1z2 + b7xz + b5yt, +X2 = b2xy + b5xt + b8yz + b4zt, Y2 = b3xy + b7xt + b6yz + b1zt, +X3 = b7x2 + b8y2 + b1xz + b4yt, Y3 = b6z2 + b5t2 + b3xz + b2yt +gives a birational map µ0 from the Kummer quartic surface S0 to Σ0. +Proof. Note that X2 +i , Y 2 +i are coefficients of ci, di when we consider the quartic equation of +S0 as a linear form of ci, di. The proof is similar to the one of Proposition 5.5. By using +the relations b1b2 = b3b4, b5b6 = b7b8, we have +� +b2b6 + b3b8y + +� +b1b5 + b4b7t = +� +b7X1 + b6Y1 + b3X3 + b1Y3, +� +b2b7 + b3b5x + +� +b1b8 + b4b6z = +� +b5X1 + b8Y1 + b2X3 + b4Y3, +(b2b6 + b3b8)xy + (b1b8 + b4b6)zt = b6X2 + b8Y2. +It follows that (x, y, z, t) is uniquely determined by (X1, X2, X3, Y1, Y2, Y3). +□ +6. QUADRIC LINE COMPLEXES AND KUMMER SURFACES IN CHARACTERISTIC 2: +CURVES OF GENUS 2 +6.1. General theory. Let fi(t) and gi(t) (i = 1, 2, 3) be non-zero rational functions, and +let ai (i = 1, 2, 3) be elements of k. Let +(6.1) +Q(t) += f1(t)X2 +1 + (t + a1)X1Y1 + g1(t)Y 2 +1 ++f2(t)X2 +2 + (t + a2)X2Y2 + g2(t)Y 2 +2 ++f3(t)X2 +3 + (t + a3)X3Y3 + g3(t)Y 2 +3 = 0 + +28 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +be a pencil of quadrics. Let G(3, 6) be the Grassmannian of 3-dimensional subspaces of +k6 and let Z be the correspondence variety in P1 × G(3, 6) defined by +Z = {(t, W) | Wis totally singular for Q(t)}. +For a general point t ∈ A1 ⊂ P1, the fiber of the morphism f : Z −→ P1 has two +connected components. We set C = Spec(f∗OZ). Then, C is a double cover of P1 (cf. +Bhosle [1, Proposition 2.16]). We calculate the concrete equation of C. +For this purpose, we factorize polynomials fi(t)X2 +i +(t+ai)XiYi+gi(t)Y 2 +i (i = 1, 2, 3) +over the algebraic closure of k(t) as +fi(t)X2 +i + (t + ai)XiYi + gi(t)Y 2 +i = fi(t)(Xi + αiYi)(Xi + α′ +iYi). +Here, we have +αi + α′ +i = t + ai +fi(t) and αiα′ +i = gi(t) +fi(t) +(i = 1, 2, 3). +We have 3 vectors defined by the equations + + + +X1 + α1Y1 = 1, X1 + α′ +1Y1 = 0, +X2 + α2Y2 = 0, X2 + α′ +2Y2 = a, +X3 + α3Y3 = 0, X3 + α′ +3Y3 = b, + + + +X1 + α1Y1 = 0, X1 + α′ +1Y1 = a, +X2 + α2Y2 = 1, X2 + α′ +2Y2 = 0, +X3 + α3Y3 = 0, X3 + α′ +3Y3 = c, + + + +X1 + α1Y1 = 0, X1 + α′ +1Y1 = b, +X2 + α2Y2 = 0, X2 + α′ +2Y2 = c, +X3 + α3Y3 = 1, X3 + α′ +3Y3 = 0, +which are a basis of a 3-dimensional vector space V corresponding with a point of Z over +a general point t ∈ A1 ⊂ P1. Here, a, b and c are arbitrary elements of k. Solving each +system of equations, we have 3 vectors u1, u2 and u3 whose coordinates are arranged as +(X1, X2, X3, Y1, Y2, Y3): +u1 = +� +f1(t)α′ +1 +t+a1 , f2(t)aα2 +t+a2 , f3(t)bα3 +t+a3 , f1(t) +t+a1 , f2(t)a +t+a2 , f3(t)b +t+a3 +� +, +u2 = +� +f1(t)aα1 +t+a1 , f2(t)α′ +2 +t+a2 , f3(t)cα3 +t+a3 , f1(t)a +t+a1 , f2(t) +t+a2 , f3(t)c +t+a3 +� +, +u3 = +� +f1(t)bα1 +t+a1 , f2(t)cα2 +t+a2 , f3(t)α′ +3 +t+a3 , f1(t)b +t+a1 , f2(t)c +t+a2 , f3(t) +t+a3 +� +. +We set +v1 = t+a1 +f1(t)u1, +v2 = u2 + u1, +v3 = u3 + u1. + +KUMMER SURFACES +29 +Again we set +w1 = v1 + t+a1 +t+a2 +f2(t)α2 +f1(t) v2 + t+a1 +t+a3 +f3(t)α3 +f1(t) v3, +w2 = v2, +w3 = v3. +Then, {w1, w2, w3} is also a basis of V and the matrix + + +w1 +w2 +w3 + + +is the homogeneous coordinates for the point in G(3, 6) which corresponds with V . Putting +a = b = c = 1 in the matrix, we have a multi-section of f : Z −→ P1. The homogeneous +coordinates are given by + + +α′ +1 + t+a1 +t+a2 +f2(t)α2 +f1(t) + t+a1 +t+a3 +f3(t)α3 +f1(t) +0 +0 +1 +t+a1 +t+a2 +f2(t) +f1(t) +t+a1 +t+a3 +f3(t) +f1(t) +1 +1 +0 +0 +0 +0 +1 +0 +1 +0 +0 +0 + + , +and certain affine coordinates for the point in G(3, 6) which corresponds with V is given +by + + +α′ +1 + t+a1 +t+a2 +f2(t)α2 +f1(t) + t+a1 +t+a3 +f3(t)α3 +f1(t) +t+a1 +t+a2 +f2(t) +f1(t) +t+a1 +t+a3 +f3(t) +f1(t) +1 +0 +0 +1 +0 +0 + + , +Using (1, 1) component of this matrix, we set +z = (t + a2)(t + a3)f1(t) +� +α′ +1 + t + a1 +t + a2 +f2(t)α2 +f1(t) ++ t + a1 +t + a3 +f3(t)α3 +f1(t) +� +. +Then, we know +z = (t + a2)(t + a3)f1(t)α′ +1 + (t + a1)(t + a3)f2(t)α2 + (t + a1)(t + a2)f3(t)α3 +and we have +(6.2) +z2 + (t + a1)(t + a2)(t + a3)z += (t + a2)2(t + a3)2f1(t){f1(t)α′ +1 +2 + (t + a1)α′ +1} ++(t + a1)2(t + a3)2f2(t){f2(t)α2 +2 + (t + a2)α2} ++(t + a1)2(t + a2)2f3(t){f3(t)α2 +3 + (t + a3)α3} += (t + a2)2(t + a3)2f1(t)g1(t) + (t + a1)2(t + a3)2f2(t)g2(t) ++(t + a1)2(t + a2)2f3(t)g3(t). +We denote by C the curve defined by this equation (6.2). + +30 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +6.2. Cases (a), (b) and (c). Now, we consider 3 cases (a), (b) and (c) in Proposition 2.10. +Case (a). In this case the pencil P of quadrics is given by +3 +� +i=1 +� +ciX2 +i + (t + ai)XiYi + diY 2 +i +� += 0. +Therefore, we set fi(t) = ci and gi(t) = di. Thus we have proved the following theorem. +Theorem 6.1. The curve C associated with P is given by +(6.3) +z2 + (t + a1)(t + a2)(t + a3)z += c1d1(t + a2)2(t + a3)2 + c2d2(t + a1)2(t + a3)2 + c3d3(t + a1)2(t + a2)2. +where � +i cidi ̸= 0. If � +i cidi = 0, then the curve has a singularity. +Remark 6.2. The equation of Igusa’s canonical model of C is given by +y2 + y = +√c1d1(a2 + a3) +(a1 + a2)(a1 + a3)x + +√c2d2(a1 + a3) +(a1 + a2)(a2 + a3)x + +√c3d3(a1 + a2) +(a1 + a3)(a2 + a3)(x + 1). +The Jacobian variety J(C) is ordinay, that is, of 2-rank 2. Recall that the coefficients +of Igusa’s canonical model are different from 0, that is, � cidi ̸= 0 (see (2.19)) which +coincides with the condition of the smoothness of X (Lemma 3.1(a)). Combining this +equation and Laszlo and Pauly [15, Proposition 3.1], we obtain the equation (4.3) of the +Kummer quartic surface. +Case (b). In this case the pencil of quadrics is given by +3 +� +i=1 +� +ciX2 +i + (t + ai)XiYi + diY 2 +i +� ++ B = 0, +B = b1X2Y3 + b2X3Y2 + b3X2X3 + b4Y2Y3, with b1b2 = b3b4 +as in (2.13). By the condition b2b4c2 + b1b4c3 + b1b3d2 + b2b3d3 ̸= 0 for the smoothness of +X1 (Lemma 3.1), we may assume b4 ̸= 0. Since B = b4 +� +Y2 + b1 +b4X2 +� � +Y3 + b2 +b4X3 +� +, we +set +Z2 = Y2 + b1 +b4 +X2, Z3 = Y3 + b2 +b4 +X3. +Moreover, we set +W3 = X3 + +b4 +t + a3 +Z2. +Then, our pencil becomes +(6.4) +c1X2 +1 + (t + a1)X1Y1 + d1Y 2 +1 ++ +� +c2 + b1(t+a2) +b4 ++ b2 +1d2 +b2 +4 +� +X2 +2 + (t + a2)X2Z2 + +� +b2 +2d3+b2 +4c3 +(t+a3)2 + b2b4 +t+a3 + d2 +� +Z2 +2 ++ +� +c3 + b2(t+a3) +b4 ++ b2 +2d3 +b2 +4 +� +W 2 +3 + (t + a3)W3Z3 + d3Z2 +3 = 0. + +KUMMER SURFACES +31 +Comparing this equation with (6.1), we see that the curve C1 is given by the equation +z2 + (t + a1)(t + a2)(t + a3)z += (t + a2)2(t + a3)2c1d1 ++(t + a1)2(t + a3)2 � +c2 + b1(t+a2) +b4 ++ b2 +1d2 +b2 +4 +� � +b2 +2d3+b2 +4c3 +(t+a3)2 + b2b4 +t+a3 + d2 +� ++(t + a1)2(t + a2)2 � +c3 + b2(t+a3) +b4 ++ b2 +2d3 +b2 +4 +� +d3 +Using a2 = a3 and b1b2 = b3b4, we have +z2 + (t + a1)(t + a2)2z += (t + a2)4c1d1 + (t + a1)2(t + a2)2 � +b1b2 + c2d2 + c3d3 + ( b1d2+b2d3 +b4 +)2� ++(t + a1)2(t + a2)3( b1d2+b2d3 +b4 +) + (t + a1)2(t + a2){b3(b1d2 + b2d3) + b4(b2c2 + b1c3)} ++(t + a1)2{b2 +1c3d2 + b2 +2c2d3 + b2 +3d2d3 + b2 +4c2c3} +Setting +z = v+(t+a1)(b1 +� +c3d2+b2 +� +c2d3+b3 +� +d2d3+b4 +√c2c3)+(t+a1)(t+a2)b1d2 + b2d3 +b4 +, +we have proved the following theorem. +Theorem 6.3. The curve C1 associated with P1 is given by +v2 + (t + a1)(t + a2)2v = (t + a2)4c1d1 ++(t + a1)2(t + a2)2 � +b1b2 + c2d2 + c3d3 + b1 +√c3d2 + b2 +√c2d3 + b3 +√d2d3 + b4√c2c3 +� ++(t + a1)2(t + a2){b3(b1d2 + b2d3) + b4(b2c2 + b1c3)}, +where c1d1 ̸= 0, b3(b1d2 + b2d3) + b4(b2c2 + b1c3) ̸= 0. If c1d1 = 0 or b3(b1d2 + b2d3) + +b4(b2c2 + b1c3) = 0, then C1 is singular. +Remark 6.4. The equation of Igusa’s canonical model of C1 is given by +y2 + y = x3 + αx + βx−1 +with +α = +6√ +b4(b2c2+b1c3)+b3(b1d2+b2d3) +√a1+√a2 ++ +√ +c2d2+c3d3+b1b2+b4√c2c3+b2 +√c2d3+b1 +√c3d2+b3 +√d2d3 +3√ +b4(b2c2+b1c3)+b3(b1d2+b2d3) ++ +3√ +{b4(b2c2+b1c3)+b3(b1d2+b2d3)}2 +a2 +1+a2 +2 +β = +√c1d1 3√ +b4(b2c2+b1c3)+b3(b1d2+b2d3) +a2 +1+a2 +2 +. +The Jacobian variety J(C1) is of 2-rank 1. Recall that β ̸= 0 (see (2.19)), that is, c1d1 ̸= 0 +and b4(b2c2 + b1c3) + b3(b1d2 + b2d3) ̸= 0 which coincides with the condition of the +smoothness of X1 (Lemma 3.1(b)). + +32 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Case (c). In this case the pencil of quadrics is given by +3 +� +i=1 +� +ciX2 +i + (t + ai)XiYi + diY 2 +i +� ++ C = 0, +C = b1X2Y3 + b2X3Y2 + b3X2X3 + b4Y2Y3 + b5X1Y2 + b6X2Y1 + b7X1X2 + b8Y1Y2, +with b1b2 = b3b4 and b5b6 = b7b8 as in (2.14). By the condition for the smoothness of +X0 (Lemma 3.1), the cases b1 = b2 = b3 = b4 = 0 and b5 = b6 = b7 = b8 = 0 do +not occur. So we may assume that b4 ̸= 0 and b8 ̸= 0. In this case b1b8 + b4b6 ̸= 0. +In fact if b1b8 + b4b6 = 0, then we have b2b6 + b3b8 = b1b5 + b4b7 = 0 by the relations +b2(b1b8+b4b6) = b4(b2b6+b3b8) and b5(b1b8+b4b6) = b8(b1b5+b4b7), which contradicts the +smoothness of X0 (Lemma 3.1). Thus in the following we may assume that b4 ̸= 0, b8 ̸= 0 +and b1b8 + b4b6 ̸= 0. +Since b1b2 = b3b4 and b5b6 = b7b8, we have +C = b4 +�b1 +b4 +X2 + Y2 +� �b2 +b4 +X3 + Y3 +� ++ b8 +�b5 +b8 +X1 + Y1 +� �b6 +b8 +X2 + Y2 +� +. +We set +Z2 = b1 +b4X2 + Y2, Z3 = b2 +b4X3 + Y3, +Z1 = b5 +b8X1 + Y1, W2 = b6 +b8X2 + Y2. +Then, we have +X2 = +b4b8 +b1b8+b4b6(Z2 + W2), +Y2 = +b4b6 +b1b8+b4b6Z2 + +b1b8 +b1b8+b4b6W2, +and our pencil becomes +� +c1 + b2 +5d1 +b2 +8 + (t + a1) b5 +b8 +� +X2 +1 + (t + a1)X1Z1 + d1Z2 +1 ++ b2 +4b2 +8c2+b2 +1b2 +8d2+(t+a2)b1b4b2 +8 +(b1b8+b4b6)2 +W 2 +2 + (t + a2) +b4b8 +b1b8+b4b6W2Z2 + b2 +4b2 +8c2+b2 +4b2 +6d2+(t+a2)b2 +4b6b8 +(b1b8+b4b6)2 +Z2 +2 ++ +� +c3 + b2 +2d3 +b2 +4 + (t + a3) b2 +b4 +� +X2 +3 + (t + a3)X3Z3 + d3Z2 +3 + b4Z2Z3 + b8Z1W2 = 0. +We set +W1 = X1 + +b8 +t + a1 +W2, W3 = X3 + +b4 +t + a3 +Z2. +Then, we have +� +c1 + b2 +5d1 +b2 +8 + (t + a1) b5 +b8 +� +W 2 +1 + (t + a1)W1Z1 + d1Z2 +1 ++ +� +b2 +8c1+b2 +5d1+(t+a1)b5b8 +(t+a1)2 ++ b2 +4b2 +8c2+b2 +1b2 +8d2+(t+a2)b1b4b2 +8 +(b1b8+b4b6)2 +� +W 2 +2 ++(t + a2) +b4b8 +b1b8+b4b6W2Z2 + +� +b2 +4c3+b2 +2d3+(t+a3)b2b4 +(t+a3)2 ++ b2 +4b2 +8c2+b2 +4b2 +6d2+(t+a2)b2 +4b6b8 +(b1b8+b4b6)2 +� +Z2 +2 ++ +� +c3 + b2 +2d3 +b2 +4 + (t + a3) b2 +b4 +� +W 2 +3 + (t + a3)W3Z3 + d3Z2 +3 = 0 + +KUMMER SURFACES +33 +Setting +˜W2 = +√b4b8 +√b1b8 + b4b6 +W2, ˜Z2 = +√b4b8 +√b1b8 + b4b6 +Z2, +we have +� +c1 + b2 +5d1 +b2 +8 + (t + a1) b5 +b8 +� +W 2 +1 + (t + a1)W1Z1 + d1Z2 +1 ++ +� +(b2 +8c1+b2 +5d1+(t+a1)b5b8)(b1b8+b4b6) +(t+a1)2b4b8 ++ b2 +4b8c2+b2 +1b8d2+(t+a2)b1b4b8 +(b1b8+b4b6)b4 +� +˜W 2 +2 ++(t + a2) ˜W2 ˜Z2 + +� +(b2 +4c3+b2 +2d3+(t+a3)b2b4)(b1b8+b4b6) +(t+a3)2b4b8 ++ b4b2 +8c2+b4b2 +6d2+(t+a2)b4b6b8 +(b1b8+b4b6)b8 +� +˜Z2 +2 ++ +� +c3 + b2 +2d3 +b2 +4 + (t + a3) b2 +b4 +� +W 2 +3 + (t + a3)W3Z3 + d3Z2 +3 = 0. +Comparing this equation with (6.1), we see that the curve C0 is given by the equation +z2 + (t + a1)(t + a2)(t + a3)z += (t + a2)2(t + a3)2 � +c1 + b2 +5d1 +b2 +8 + (t + a1) b5 +b8 +� +d1 ++(t + a1)2(t + a3)2 � +(b2 +8c1+b2 +5d1+(t+a1)b5b8)(b1b8+b4b6) +(t+a1)2b4b8 ++ b2 +4b8c2+b2 +1b8d2+(t+a2)b1b4b8 +(b1b8+b4b6)b4 +� +× +� +(b2 +4c3+b2 +2d3+(t+a3)b2b4)(b1b8+b4b6) +(t+a3)2b4b8 ++ b4b2 +8c2+b4b2 +6d2+(t+a2)b4b6b8 +(b1b8+b4b6)b8 +� ++(t + a1)2(t + a2)2 � +c3 + b2 +2d3 +b2 +4 + (t + a3) b2 +b4 +� +d3 +Using a1 = a2 = a3 = a, we have +z2 + (t + a)3z += (t + a)6 +b1b4b6b8 +(b1b8+b4b6)2 + (t + a)5 � +b5 +b8d1 + b2 +b4d3 + b4b8c2+b1b6d2 +b1b8+b4b6 +� ++(t + a)4 �� +c1 + b2 +5d1 +b2 +8 +� +d1 + +� +c3 + b2 +2d3 +b2 +4 +� +d3 + (b2 +4c2+b2 +1d2)(b2 +8c2+b2 +6d2) +(b1b8+b4b6)2 ++ b1b2 + b5b6 +� ++(t + a)3 � +(b2 +8c1+b2 +5d1)b6+(b2 +8c2+b2 +6d2)b5 +b8 ++ (b2 +4c3+b2 +2d3)b1+(b2 +4c2+b2 +1d2)b2 +b4 +� ++(t + a)2 � +(b2 +8c1+b2 +5d1)(b2 +8c2+b2 +6d2) +b2 +8 ++ (b2 +4c3+b2 +2d3)(b2 +4c2+b2 +1d2) +b2 +4 ++ (b1b8+b4b6)2b2b5 +b4b8 +� ++(t + a)(b1b8 + b4b6)2 � +(b2 +8c1+b2 +5d1)b2 +b4b2 +8 ++ (b2 +4c3+b2 +2d3)b5 +b2 +4b8 +� ++ (b1b8+b4b6)2(b2 +8c1+b2 +5d1)(b2 +4c3+b2 +2d3) +b2 +4b2 +8 +. +Let α be a root of the equation x2 + x + +b1b4b5b6 +(b1b8+b4b6)2 = 0. Replacing z by +z + (t + a)3α + (t + a)2 +�b5 +b8 +d1 + b2 +b4 +d3 + b4b8c2 + b1b6d2 +b1b8 + b4b6 +� +, +we have proved the following theorem. + +34 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Theorem 6.5. The curve C0 associated with P0 is given by +z2 + (t + a)3z += (t + a)4(c1d1 + c2d2 + c3d3 + b1b2 + b5b6) ++(t + a)3{b6b8c1 + (b2b4 + b5b8)c2 + b1b4c3 + b5b7d1 + (b1b3 + b6b7)d2 + b2b3d3} ++(t + a)2{b2 +8c1c2 + b2 +6c1d2 + b2 +5c2d1 + b2 +7d1d2 ++b2 +4c2c3 + b2 +2c2d3 + b2 +1c3d2 + b2 +3d2d3 + b1b3b5b8 + b2b4b6b7} ++(t + a){(b1b3b2 +8 + b2b4b2 +6)c1 + (b1b3b2 +5 + b2b4b2 +7)d1 ++(b2 +1b5b8 + b2 +4b6b7)c3 + (b2 +3b5b8 + b2 +2b6b7)d3} ++(b1b8 + b4b6)2c1c3 + (b2b6 + b3b8)2c1d3 + (b1b5 + b4b7)2c3d1 + (b2b7 + b3b5)2d1d3. +Remark 6.6. The equation of Igusa’s canonical model of C0 is given by +y2 + y = x5 + αx3 +with +α = +α1 +5√ +α3 +2, +α1 = b6b8c1 + (b5b8 + b2b4)c2 + b1b4c3 + b5b7d1 + (b1b3 + b6b7)d2 + b2b3d3 ++(b1b8 + b4b6)√c1c3 + (b3b8 + b2b6)√c1d3 + (b1b5 + b4b7)√c3d1 + (b3b5 + b2b7)√d1d3, +α2 = (b1b3b2 +8 + b2b4b2 +6)c1 + (b1b3b2 +5 + b2b4b2 +7)d1 + (b2 +1b5b8 + b2 +4b6b7)c3 + (b2 +3b5b8 + b2 +2b6b7)d3. +The Jacobian variety J(C0) is supersingular, that is, of 2-rank 0. The necessary and suffi- +cient condition for the curve C0 to be non-singular is α2 ̸= 0, that is, (b1b3b2 +8 + b2b4b2 +6)c1 + +(b1b3b2 +5 + b2b4b2 +7)d1 + (b2 +1b5b8 + b2 +4b6b7)c3 + (b2 +3b5b8 + b2 +2b6b7)d3 ̸= 0, which coincides with +the condition of the smoothness of X0 (Lemma 3.1(c)) by b1b2 = b3b4 and b5b6 = b7b8. +7. MULTI-LINEAR SYSTEMS +In this section, let k be an algebraically closed field of characteristic 2, C a curve of +genus 2 over k, and J(C) the Jacobian variety of C. The curve C gives a principal polar- +ization of J(C) which we call the theta divisor. We may assume C contains the zero point +O of J(C) and is symmetric with respect to the inversion ι, that is, ι∗C = C. Through- +out this section, we assume that J(C) is ordinary. Namely, the 2-rank of J(C) is equal +to two and denoting by J(C)[2] the group scheme of 2-torsion points of J(C), we have +J(C)[2]red ∼= Z/2Z ⊕ Z/2Z. We denote by ai (i = 1, 2, 3) the 2-torsion points of J(C). +We may assume that a1 and a2 are contained in C and that the points O, a1 and a2 give the +ramification points of the double covering π : C −→ P1. In this case, we have a3 ̸∈ C. +We sometimes set a0 = O. +For a point x ∈ J(C), we denote by Tx the translation by the point x and denote by +ˆJ(C) the dual abelian surface of J(C). For a divisor D, we have a homomorphism +ΦD : +J(C) +−→ +ˆJ(C) +x +�→ +T ∗ +x(D) − D + +KUMMER SURFACES +35 +TABLE 3. The elements of J(C)[2]red which are contained in the curve +curve +elements of J(C)[2]red +C +O, a1, a2 +T ∗ +a1C +a1, O, a3 +T ∗ +a2C +a2, a3, O +T ∗ +a3C +a3, a2, a1 +If D is ample, then ΦD is an isogeny, and if D is a principal polarization, then ΦD is an +isomorphism (cf. Mumford [17]) +7.1. Points on the theta divisor C. We examine intersection points of some divisors. +Lemma 7.1. The theta divisor C contains no 4-torsion point of J(C). +Proof. Suppose there exists an element x ∈ C of order 4. Then, 2x = a is a 2-torsion point +of J(C). Assume a ∈ C. Then, since C = ι∗C ∋ −x, T ∗ +xC contains O, −2x = 2x = a +and O − x = −x. Therefore, C ∩ T ∗ +xC contains three different points O, a and −x. If +T ∗ +xC ̸= C, then we have 3 ≤ (T ∗ +xC · C) = C2 = 2, a contradiction. Therefore, we have +T ∗ +xC = C in this case. Therefore, we have x ∈ Ker ΦC. However, since C is a principal +polarization, we have Ker ΦC = {O}, a contradiction. Hence we have 2x = a = a3 ̸∈ C. +In this case, T ∗ +xC contains x − x = O, a1 − x and −x − x = −2x = −a3 = a3. T ∗ +a1C +contains a1 − a1 = O, −x − a1 = −x + a1 and a2 − a1 = a3. If T ∗ +xC ̸= T ∗ +a1C, then we +have 3 ≤ (T ∗ +xC · T ∗ +a1C) = C2 = 2, a contradiction. Therefore, we have T ∗ +xC = T ∗ +a1C. +Therefore, we have T ∗ +x−a1C = C. This means the non-zero element x − a1 is contained +in Ker ΦC. However, since C is a principal polarization, we have Ker ΦC = {O}, and we +have a contradiction again. Hence, C contains no 4-torsion point of J(C). +□ +Corollary 7.2. Let x be a 4-torsion point of J(C). Then, T ∗ +xC contains no point of +J(C)[2]. +Proof. This corollary follows from Lemma 7.1. +□ +Lemma 7.3. Let x be a general point of J(C). Then, T ∗ +xC contains no point of J(C)[2]. +Proof. There exists a point x ∈ J(C) which is not contained in ∪3 +i=0T ∗ +aiC. Then, T ∗ +xC +contains no point of J(C)[2]. +□ +The following lemma is well-known. +Lemma 7.4. Let xi (i = 1, 2, . . . , n) be points of J(C). Then, +n +� +i=1 +T ∗ +xiC − nC ∼ 0 +if and only if +n +� +i=1 +xi = O. + +36 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Proof. Let O ˆJ(C) be the zero point of ˆJ(C). Since ΦC(�n +i=1 xi) = �n +i=1 ΦC(xi) = +�n +i=1 T ∗ +xiC − nC, we see �n +i=1 T ∗ +xiC − nC ∼ 0 if and only if ΦC(�n +i=1 xi) = O ˆJ(C). +Since C is a principal polarization, we see ΦC(�n +i=1 xi) = O ˆJ(C) if and only if �n +i=1 xi = +O. +□ +Remark 7.5. If i ̸= j, T ∗ +aiC ∩ T ∗ +ajC consists of different two 2-torsion points. Since +(T ∗ +aiC · T ∗ +ajC) = C2 = 2, T ∗ +aiC intersects T ∗ +ajC at each intersection point transversely. +Therefore, the defining equations for T ∗ +aiC and T ∗ +ajC make the system of parameters at +each intersection point. +7.2. The linear system |2C|. In this subsection, we examine the linear system |2C|. Let +L(2C) be the vector space given by the divisor 2C. By the Riemann-Roch theorem, we +have dim L(2C) = 4. By Lemma 7.4, we have +2(T ∗ +ai(C) − C) ∼ 0 +(i = 0, 1, 2, 3). +Therefore, there exists rational functions fi (i = 0, 1, 2, 3) such that +2(T ∗ +ai(C) − C) = (f3−i). +We take a natural isomorphism +(7.1) +ρ : L(2C) ∼= H0(J(C), OJ(C)(2C)) +and we set ˜fi = ρ(fi). +The following Lemmas 7.6 and 7.8 are known (see Laszlo and Pauly [14]). But we give +their proofs for the sake of completeness. +Lemma 7.6. fi (i = 0, 1, 2, 3) gives a basis of L(2C). +Proof. Since dim L(2C) = 4 and (7.1), it suffices to show that ˜fi (i = 0, 1, 2, 3) are +linearly independent over k. We consider a linear relation +α0 ˜f0 + α1 ˜f1 + α2 ˜f2 + α3 ˜f3 = 0 +(a0, a1, a2, a3 ∈ k). +Note that ˜fi has 3 zero points in J(C)[2] and ˜fi(ai) ̸= 0. Therefore, we have αi ˜fi(ai) = 0 +and ˜fi(ai) ̸= 0. Therefore, we have αi = 0. +□ +Remark 7.7. Let F : J(C) −→ J(C)(2) be the relative Frobenius morphism and we denote +by Θ the descent divisor of 2C, i.e., ρ∗(Θ) ∼ 2C. We may assume Θ is an effective divisor. +Since dim H0(J(C)(2), OJ(C)(2)(Θ)) is one-dimensional, we take a basis θ. Then, we may +assume F ∗(θ) = ˜f3, and we may assume ˜f3−i = T ∗ +ai ˜f3. Then, fi (i = 0, 1, 2, 3) is our +basis and ˜fi (i = 0, 1, 2, 3) gives the canonical basis in Laszlo and Pauly [14]. From here +on, we take our basis fi (i = 0, 1, 2, 3) of L(2C) like this. +Lemma 7.8. The inversion ι acts on L(2C) identically. + +KUMMER SURFACES +37 +Proof. Since C is symmetric, we have +(ι∗f3−i) = 2ι∗(T ∗ +aiC − C) = 2(T ∗ +aiC − C) = (f3−i). +Therefore, there exists α ∈ k, α ̸= 0 such that ι∗f3−i = αf3−i. Since ι∗ is of order 2 +and the characteristic p = 2, we have α = 1. Therefore, by Lemma 7.6 we complete our +proof. +□ +Remark 7.9. We consider the map +ϕ|2C| : +J(C) +−→ +P3 +P +�→ +(f3(P), f2(P), f1(P), f0(P)) +The divisor 2C is base point free (cf. Mumford [17], for instance). Therefore, ϕ|2C| is a +morphism. We set x00 = f3, x10 = f2, x01 = f1 and x11 = f0. Then, by Laszlo and Pauly +[14] the image ϕ|2C|(J(C)) is given by the equation +λ2 +10(x2 +00x2 +10 + x2 +01x2 +11) + λ2 +01(x2 +00x2 +01 + x2 +10x2 +11) + λ2 +11(x2 +00x2 +11 + x2 +01x2 +10) ++ λ10λ01λ11 +λ00 +x00x10x01x11 = 0 +with certain constants λij (λ10λ01λ11λ00 ̸= 0), and the image ϕ|2C|(J(C)) is isomorphic +to the Kummer quartic surface J(C)/⟨ι⟩. +7.3. The linear system |4C|. We denote by V the subspace of L(4C) generated by fifj +(i, j = 0, 1, 2, 3): +V = ⟨fifj (i, j = 0, 1, 2, 3)⟩, +and we set ˜V = ρ(V ). +Lemma 7.10. dim V = dim ˜V = 10. +Proof. By Remark 7.9, fi (i = 0, 1, 2, 3) has a quartic relation and there exist no quadric +relations. Therefore, fifj ∈ L(4C) (i, j = 0, 1, 2, 3) are linearly independent over k. +Therefore, we have dim V = dim ˜V = 10. +□ +Corollary 7.11. +dim ˜V ∩ H0(J(C), OJ(C)(4C − 2 +3 +� +i=0 +ai)) = 6. +In particular, +dim H0(J(C), OJ(C)(4C − 2 +3 +� +i=0 +ai)) ≥ 6. +Proof. ˜fi ˜fj (i, j = 0, 1, 2, 3; i ̸= j) are contained in H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)). +Therefore, we have dim ˜V ∩ H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)) ≥ 6. On the other hand, +˜f 2 +i (i = 0, 1, 2, 3) has 3 zeros in J(C)[2] and ˜f 2 +i (ai) ̸= 0. Therefore, we have ⟨ ˜f 2 +i (i = +0, 1, 2, 3)⟩∩H0(J(C), OJ(C)(4C −2 �3 +i=0 ai)) = {0}, from which the result follows. +□ + +38 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +We denote by L(4C)⟨ι∗⟩ the ι∗-invariant subspace of L(4C). +Corollary 7.12. V ⊂ L(4C)⟨ι∗⟩. In particular, dim L(4C)⟨ι∗⟩ ≥ 10. +Proof. This follows from Lemmas 7.8 and 7.10. +□ +We take a 4-torsion point x ∈ J(C)[4] \ J(C)[2]. Then, by Lemma 7.4, there exists a +rational function ϕx such that 4(T ∗ +xC − C) = (ϕx). Considering the action of ι, we have +(ι∗ϕx) = 4(T ∗ +−xC − C) = (ϕ−x). +Considering the constant multiple, we can choose ϕ−x as ϕ−x = ι∗ϕx. For an element x ∈ +J(C)[2], we have already 2(T ∗ +xC − C) = (fx). Therefore, we have 4(T ∗ +xC − C) = (f 2 +x). +In this case we choose ϕx as ϕx = f 2 +x. We denote by U the subspace of L(4C) generated +by 16 functions ϕx. Since we work in characteristic 2, the theta group G(4C) acts on U +(cf. Mumford [17]). The action of the closed points is given by +(x, ϕx) ◦ ϕy = ϕx ◦ T ∗ +xϕy = constant · ϕx+y +(cf. Mumford [17], [19]). By Mumford [18] (also see Sekiguchi [22]) the representation +of G(4C) on L(4C) is irreducible. Therefore, we have U = L(4C). Hence, we have the +following lemma. +Lemma 7.13. ϕx (x ∈ J(C)[4]) are a basis of L(4C). +Proposition 7.14. dim L(4C)⟨ι∗⟩ = 10. In particular, V = L(4C)⟨ι∗⟩. +Proof. We consider the representation of ι∗ with respect to the basis {ϕx}. We arrange ϕx +(x ∈ J(C)[2]) first, and then for a 4-torsion point x we arrange ϕ−x next to ϕx. Then, the +representation matrix is given by + + + + + + + + + + + + + + +1 +0 +1 +1 +1 +0 +1 +1 +0 +... +0 +1 +0 +1 +0 + + + + + + + + + + + + + + +Therefore, we have dim L(4C)⟨ι∗⟩ = 10. The latter part follows from Corollary 7.12. +□ +Lemma 7.15. dim H0(J(C), OJ(C)(4C − �3 +i=0 ai)) = 12. + +KUMMER SURFACES +39 +Proof. By the Riemann-Roch theorem, we have dim H0(J(C), OJ(C)(4C)) = 16. We +have ˜f 2 +i (ai) ̸= 0 (i = 0, 1, 2, 3) and ˜f 2 +i (aj) = 0 for j ̸= i. Subtracting a suitable linear +combination of these four functions, any element of H0(J(C), OJ(C)(4C)) becomes zero +at all points of J(C)[2]. Therefore, we have dim H0(J(C), OJ(C)(4C − �3 +i=0 ai)) = +12. +□ +Let Oai be the local ring at the point ai and mai the maximal ideal of Oai. The cotangent +space at the point ai of J(C) is isomorphic to mai/m2 +ai and it is 2-dimensional. We have +a natural exact sequence +0 −→ OJ(C)(−2 +3 +� +i=0 +ai) −→ OJ(C)(− +3 +� +i=0 +ai) −→ ⊕3 +i=0mai/m2 +ai −→ 0. +Tensoring OJ(C)(4C), we have an exact sequence +0 −→ OJ(C)(4C − 2 +3 +� +i=0 +ai) −→ OJ(C)(4C − +3 +� +i=0 +ai) −→ ⊕3 +i=0mai/m2 +ai −→ 0. +Therefore, we have a long exact sequence +(7.2) +0 −→ H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)) −→ H0(J(C), OJ(C)(4C − �3 +i=0 ai)) +ψ +−→ ⊕3 +i=0mai/m2 +ai. +To calculate the dimension of Im ψ, we construct some elements of H0(J(C), OJ(C)(4C− +�3 +i=0 ai)). For this purpose, we choose different two elements ai, aj in J(C)[2]. Then, +there exist an element xk of order 4 in J(C) such that 2xk = ai + aj. By Lemma 7.4, we +see +2T ∗ +xkC + T ∗ +aiC + T ∗ +ajC − 4C ∼ 0. +Therefore, there exists a rational function ϕij such that +2T ∗ +xkC + T ∗ +aiC + T ∗ +ajC − 4C = (ϕij). +We set ˜ϕij = ρ(ϕij). Note that T ∗ +xkC contains no element of J(C)[2] by Lemma 7.2. By +Table 3 the situation of ˜ϕij at the points of J(C)[2] is listed as in Table 4. +Now, let z be a general point of C. Since a1 + a2 + a3 = a0 = O, by Lemma 7.4 there +exist rational functions g0 and h0 such that +T ∗ +z+a1C + T ∗ +−zC + T ∗ +a2C + T ∗ +a3C − 4C = (g0), +T ∗ +z+a2C + T ∗ +−zC + T ∗ +a1C + T ∗ +a3C − 4C = (h0). +Note that T ∗ +−zC contains no point in J(C)[2] and that T ∗ +z+a1C contains only a1 among +the points of J(C)[2], and that T ∗ +z+a2C contains only a2 among the points of J(C)[2]. +Therefore, the situation of ˜g0 = ρ(g0) and ˜h0 = ρ(h0) at the points of J(C)[2] is as +follows. + +40 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +TABLE 4. Zero points of ˜ϕij +simple zero point in J(C)[2] +double zero point in J(C)[2] +˜ϕ01 +a2, a3 +a0, a1 +˜ϕ02 +a1, a3 +a0, a2 +˜ϕ03 +a0, a3 +a1, a2 +˜ϕ12 +a1, a2 +a0, a3 +˜ϕ13 +a0, a2 +a1, a3 +˜ϕ23 +a0, a1 +a2, a3 +TABLE 5. Zero points of ˜g0, ˜h0 +simple zero point in J(C)[2] +double zero point in J(C)[2] +˜g0 +a0 +a1, a2, a3 +˜h0 +a0 +a1, a2, a3 +Lemma 7.16. ˜g0 and ˜h0 make a basis of the cotangent space ma0/m2 +a0 at the point O = a0. +Proof. We have a0 = O ∈ T ∗ +a1C ∩ T ∗ +a2C. Therefore, this lemma follows from Remark +7.5. +□ +By Table 4, ˜ϕij gives non-zero vectors at cotangent spaces of two points of J(C)[2] and +becomes 0-vector at cotangent spaces of the other two points of J(C)[2]. We denote by +v(k) +ij the non-zero vector given by ˜ϕij at the cotangent spaces of the point ak. By Table 3 and +Lemma 7.16, ˜g0 and ˜h0 make a basis, say ⟨v1, v2⟩, of the cotangent space ma0/m2 +a0 at a0, +and are 0-vectors of the cotangent spaces at the 2-torsion points. In Table 5, we summarize +the situation of these functions at the cotangent spaces of the points of J(C)[2]. +TABLE 6. Cotangent vectors +ma0/m2 +a0 +ma1/m2 +a1 +ma2/m2 +a2 +ma3/m2 +a3 +˜ϕ01 +0 +0 +v(2) +01 +v(3) +01 +˜ϕ02 +0 +v(1) +02 +0 +v(3) +02 +˜ϕ03 +v(0) +03 +0 +0 +v(3) +03 +˜ϕ12 +0 +v(1) +12 +v(2) +12 +0 +˜g0 +v1 +0 +0 +0 +˜h0 +v2 +0 +0 +0 +Lemma 7.17. dim Im ψ ≥ 6. + +KUMMER SURFACES +41 +Proof. By Remark 7.5, T ∗ +aiC intersects T ∗ +ajC (j ̸= i) at different two points and the inter- +section is transversal. Therefore, the defining equations of T ∗ +aiC and T ∗ +ajC give a basis at +the cotangent spaces of the two points. Therefore, in Table 6, two or two of three non-zero +vectors at the cotangent spaces are linearly independent over k. +The 6 functions in Table 6 are contained in H0(J(C), OJ(C)(4C − �3 +i=0 ai)). Suppose +that the images of 6 functions in Table 6 by ψ are linearly dependent. Then, there exist +bi ∈ k (i = 1, 2, . . . , 6) such that +b1ψ( ˜ϕ01) + b2ψ( ˜ϕ02) + b3ψ( ˜ϕ03) + b4ψ( ˜ϕ12) + b5ψ(˜g0) + b6ψ(˜h0) = 0 +Considering the component of the cotangent space ma1/m2 +a1, we have b2v(1) +02 + b4v(1) +12 = 0. +Since v(1) +02 and v(1) +12 linearly independent over k as we explained above, we have b2 = +b4 = 0. Considering the components of the cotangent spaces ma3/m2 +a3 and ma0/m2 +a0 +successively, by similar arguments we have bi = 0 for all i = 1, 2, · · · , 6. +□ +Proposition 7.18. dim H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)) = 6 and dim Im ψ = 6. +Proof. By Lemmas 7.15, 7.17 and (7.2), we have dim H0(J(C), OJ(C)(4C−2 �3 +i=0 ai)) ≤ +6. The former part follows from Corollary 7.11. The latter part follows from the exact +sequence (7.2). +□ +We denote by ˜W the subspace of ˜V which is generated by ˜fi ˜fj (i, j = 0, 1, 2, 3; i ̸= j). +Theorem 7.19. ˜W = H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)). +Proof. We have ˜W ⊂ H0(J(C), OJ(C)(4C−2 �3 +i=0 ai)). Since both sides are 6-dimensional, +we get our result. +□ +Using H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)), we consider the following rational map. +f : +J(C) +−→ +P5 +P +�→ +(X0, X1, X2, X3, X4, X5) = (f3f2, f3f1, f3f0, f2f1, f2f0, f1f0). +Then, by Remark 7.9 we have the relation +(7.3) +λ2 +10(X2 +0+X2 +5)+λ2 +01(X2 +1+X2 +4)+λ2 +11(X2 +2+X2 +3)+λ10λ01λ11 +λ00 +X0X5 = 0 +(λ10λ01λ11λ00 ̸= 0) +We also have two trivial equations +(7.4) +X0X5 + X1X4 = 0, +X0X5 + X2X3 = 0. +Theorem 7.20. Let S be a surface in P5 defined by the equations (7.3), (7.4). Then, S is +a K3 surface with 12 A1-rational double points. + +42 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Proof. By a direct calculation, we have 12 singularities at such as a point (λ01, λ10, 0, 0, 0, 0). +By a blowing-up, the singular points can be resolved and the types are A1-rational double. +The dualizing sheaf of S is isomorphic to OS(−6+2+2+2) ∼= OS. Since the singularities +are rational, the minimal resolution of S is a K3 surface. +□ +Remark 7.21. The 12 singular points of S are given as follows: +P34 = (0, 0, 0, λ01, λ11, 0), P24 = (0, 0, λ01, 0, λ11, 0), +P13 = (0, λ11, 0, λ01, 0, 0), P12 = (0, λ11, λ01, 0, 0, 0), +P35 = (0, 0, 0, λ10, 0, λ11), P25 = (0, 0, λ10, 0, 0, λ11), +P03 = (λ11, 0, 0, λ10, 0, 0), P02 = (λ11, 0, λ10, 0, 0, 0), +P45 = (0, 0, 0, 0, λ10, λ01), P15 = (0, λ10, 0, 0, 0, λ01), +P04 = (λ01, 0, 0, 0, λ10, 0), P01 = (λ01, λ10, 0, 0, 0, 0). +We have a commutative diagram of rational maps. +J(C) +ϕ|2C| +−→ +P3 ∋ (f3, f2, f1, f0) +f ց +↓ ϕ +P5 ∋ (f3f2, f3f1, f3f0, f2f1, f2f0, f1f0). +In this diagram, ϕ|2C| is a morphism and Im ϕ|2C| is isomorphic to J(C)/⟨ι⟩. +Theorem 7.22. f is a rational map whose base points consist of the points in J(C)[2]. ϕ +is a biratinal map whose base points consist of the singular points J(C)/⟨ι⟩. The inverse +map ϕ−1 is a blow-down morphism and S has 4 exceptional curves with respect to ϕ−1. +Each exceptional curve contains 4 singular points of S. +Proof. Since ˜fi ˜fj ∈ H0(J(C), OJ(C)(4C − 2 �3 +i=0 ai)) (i ̸= j), the points in J(C)[2] +are base points of f. Since some ˜fi ˜fj is not zero outside of J(C)[2], we have the first +statement. By a general theory of Abelian variety, |2C| is base point free (cf. Mumford +[17]). Therefore, we get the second statement. +The inverse map ϕ−1 is given by +(X0, X1, X2, X3, X4, X5) �→ (X0X1, X0X3, X1X3, X0X5). +If only one component of coordinates of a point P on P5 is not zero, then P is not a point +on S. Therefore, for a point P, there exist at least two non-zero components of coordinates +of P, say i, j. Since we can express ϕ−1 as a map which includes XiXj as a coordinate, +ϕ−1 is a morphism. For example, if i = 3 and j = 5, then we can express ϕ−1 as +(X0, X1, X2, X3, X4, X5) �→ (X0X5, X3X4, X3X5, X4X5), +which coincides with the original one by X0X5 = X1X4 = X2X3 (cf. (7.4)). Therefore, +ϕ−1 is a morphism. We can show the other statements by direct calculations. +□ + +KUMMER SURFACES +43 +Remark 7.23. We list up the 4 exceptional curves ℓi (i = 1, 2, 3, 4) for ϕ−1 and the singular +points on them. +ℓ1 : +X0 = X2 = X4 = 0, λ10X5 + λ01X1 + λ11X3 = 0, +the singular points on ℓ1 : +P12, P15, P35, +ℓ2 : +X1 = X2 = X5 = 0, λ10X0 + λ01X4 + λ11X3 = 0, +the singular points on ℓ2 : +P04, P03, P34, +ℓ3 : +X3 = X4 = X5 = 0, λ10X0 + λ01X1 + λ11X2 = 0, +the singular points on ℓ3 : +P01, P02, P12, +ℓ4 : +X0 = X1 = X3 = 0, λ10X5 + λ01X4 + λ11X2 = 0, +the singular points on ℓ4 : +P24, P25, P45. +7.4. Relations to a paper by Duquesne. In his paper [5], Duquesne examined the linear +system |2C| in characteristic 2 and showed that the image of the rational map ϕ|2C| is +isomorphic to the Kummer surface. In this subsection, we examine the relation between +our theory and his results. +Let C be a non-singular curve of genus 2 and we consider the symmetric product +Sym2(C) of two C’s. Then, as is well-known, we have morphisms +˜φ : +C × C +−→ +Sym2(C) +ν +−→ +J(C) +(P1, P2) +�→ +P1 + P2 +�→ +P1 + P2 − KC. +Here, ν is the blowing-up at the zero point O, and KC is a canonical divisor of C. We have +an inclusion morphism +φ : C × {∞} ֒→ C × C −→ Sym2(C) +ν +−→ J(C). +Here, ∞ is a point of a ramification point of the hyperelliptic structure C −→ P1. We +denote by C∞ the image of this inclusion morphism. Then, C∞ gives the principal polar- +ization of J(C). +(a) The ordinary case. +Igusa’s normal form of the curve C of genus 2 such that the Jacobian variety J(C) is +ordinary is given by (2.19). It is easy to see that this curve is isomorphic to the curve given +by +y2 + (x2 + x)y = αx5 + (α + β + γ)x3 + γx2 + βx. +We denote this affine curve by Caff and the point at infinity by ∞. Then, we have +C = Caff ∪ {∞}. +We consider points P1 = (1, 0) and P2 = (0, 0), and set +a0 = φ((∞, ∞)), a1 = φ((P1, ∞)), a2 = φ((P2, ∞)), a3 = ˜φ((P1, P2)). +Then, a0 is the zero point of J(C), and ai (i = 1, 2, 3) are the 2-torsion points of J(C). +Note a0, a1, a2 ∈ C∞ and a3 ̸∈ C∞. + +44 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Using the notation in Duquesne [5], we consider symmetric functions +k1 = 1, k2 = x1 + x2, k3 = x1x2, +k4 = (x1+x2)(αx2 +1x2 +2+(α+β+γ)x1x2+β)+(x2 +2+x2)y1+(x2 +1+x1)y2 +(x1+x2)2 +. +Then, these four functions give a basis of L(2C∞) and we have a morphism +ϕ|2C∞| : +J(C) +−→ +P3 +P +�→ +(k1(P), k2(P), k3(P), k4(P)). +The image of ϕ|2C∞| is given by the equation +(7.5) +k2 +2k2 +4 + k2 +1k3k4 + k1k2 +3k4 + k1k2k3k4 + β2k4 +1βk3 +1k3 + βk2 +1k2k3 ++(α2 + β2 + γ2 + α + β)k2 +1k2 +3 + αk1k2k2 +3 + αk1k3 +3 + α2k4 +3 = 0 +(cf. Duquesne [5, Sections 2 and 3]) and ϕ|2C∞|(C∞) is a trope defined by k1 = 0. +The singularities of this surface are +ϕ|2C∞|(a0) = (0, 0, 0, 1), +ϕ|2C∞|(a1) = (0, 1, 1, α), +ϕ|2C∞|(a2) = (0, 1, 0, 0), +ϕ|2C∞|(a3) = (1, 1, 0, β). +They are rational double points of type D4. We consider the change of coodinates +X1 = k1, X2 = k2 + k1 + k3, X3 = k3, X4 = k4 + βk1 + αk3. +Using the notation of Subsection 7.2, we set C = C∞ and ˜Xi = ρ(Xi) (i = 1, 2, 3, 4). +Then, the zero points and the non-zero points of ˜Xi in J(C)[2] are listed as in Table 7. +TABLE 7. Zero points of ˜Xi +zero points in J(C)[2] +non-zero points in J(C)[2] +˜X1 +a0, a1, a2 +a3 +˜X2 +a0, a1, a3 +a2 +˜X3 +a0, a2, a3 +a1 +˜X4 +a1, a2 , a3 +a0 +Proposition 7.24. Xi coincides with f4−i given in Lemma 7.6 up to constant. +Proof. Since f0, f1, f2 and f3 make a basis of L(2C∞), there exist elements bj ∈ k +(j = 0, 1, 2, 3) such that Xi = b0f0 + b1f1 + b2f2 + b3f3. Therefore, we have +˜Xi = b0 ˜f0 + b1 ˜f1 + b2 ˜f2 + b3 ˜f3. +Taking values at the points a0, a1, a2 and a3, we get the result. +□ + +KUMMER SURFACES +45 +Using this change of coordinates, the quartic surface (7.5) is isomorphic to the surface +defined by +(X2 +1X2 +4 + α2X2 +2X2 +3) + (X2 +3X2 +4 + β2X2 +1X2 +2) + (X2 +2X2 +4 + γ2X2 +1X2 +3) + X1X2X3X4 = 0. +Using the theory in Theorem 7.19, we have a rational map +f : +J(C) +−→ +P5 +P +�→ +(Y0, Y1, Y2, Y3, Y4, Y5) = (X1X2, X1X3, X1X4, X2X3, X2X4, X3X4). +and the image is given by +Y0Y5 + Y1Y4 = Y0Y5 + Y2Y3 = Y 2 +2 + αY 2 +3 + Y 2 +5 + β2Y 2 +0 + Y 2 +4 + γ2Y 2 +1 + Y0Y5 = 0. +This surface is a K3 surface with 12 rational double points of type A1, as we already +examined. +(b) 2-rank 1 case. +Igusa’s normal form of the curve C of gneus 2 such that the the Jacobian variety J(C) +is of 2-rank 1 is given by (2.19). It is easy to see that this curve is isomorphic to the curve +defined by +y2 + xy = x5 + αx3 + βx. +Using the notation in Duquesne [5], we consider symmetric functions +k1 = 1, k2 = x1 + x2, k3 = x1x2, +k4 = (x1+x2)(x2 +1x2 +2+αx1x2+β)+x2y1+x1y2 +(x1+x2)2 +. +Then, these four functions give a basis of L(2C∞), and the image of ϕ|2C∞| is given by the +equation +k2 +2k2 +4 + k2 +1k3k4 + β2k4 +1 + α2k2 +1k2 +3 + k1k2k2 +3 + k4 +3 = 0 +(cf. Duquesne [5, Sections 2 and 3]). The singularities are +(0, 0, 0, 1), +(0, 1, 0, 0), +which are rational double points of type D8. We set +Y0 = k2k4, Y1 = k2 +1, Y2 = k2 +3, Y3 = k1k3, Y4 = k1k4, Y5 = k2k3. +Then we have a surface in P5 defined by +Y1Y2 + Y 2 +3 = Y0Y3 + Y4Y5 = Y 2 +0 + β2Y 2 +1 + Y 2 +2 + α2Y 2 +3 + Y3Y4 + Y3Y5 = 0. +This surface has 4 singular points. They are rational double points of type A3 at (0, 0, 0, 0, 0, 1) +and (0, 0, 0, 0, 1, 0), and rational double points of type D4 at (1, 0, 1, 0, 0, 0) and (β, 1, 0, 0, 0, 0). +(c) Supersingular case. +As we stated in (2.19) Igusa’s normal form of the curve C of genus 2 such that the +Jacobian variety J(C) is supersingular is given by +y2 + y = x5 + αx3. + +46 +TOSHIYUKI KATSURA AND SHIGEYUKI KOND ¯O +Using the notation in Duquesne [5], we consider symmetric functions +k1 = 1, k2 = x1 + x2, k3 = x1x2, +k4 = (x1+x2)(x2 +1x2 +2+αx1x2)+y1+y2 +(x1+x2)2 +. +Then, these four functions give a basis of L(2C∞), and the image of ϕ|2C∞| is given by the +equation +k2 +2k2 +4 + k3 +1k4 + αk3 +1k2 + k2 +1k2k3 + α2k2 +1k2 +3 + k1k3 +2 + k4 +3 = 0 +(cf. Duquesne [5, Sections 2 and 3]). This surface has only one singular point at (0, 0, 0, 1), +which is an elliptic double point of type 4⃝1 +0,1. We don’t know how to connect this surface +with a (2, 2, 2)-surface in P5. +REFERENCES +[1] U. 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Japan 29 (1977), 707–727. +[23] T. Shioda, Kummer surfaces in characteristic 2, Proc. Japan Acad., 50 (1974), 718–722. +[24] P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math., 92 (1970), 419–454. +GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO, MEGURO-KU, +TOKYO 153-8914, JAPAN +Email address: tkatsura@ms.u-tokyo.ac.jp +GRADUATE SCHOOL OF MATHEMATICS, NAGOYA UNIVERSITY, NAGOYA, 464-8602, JAPAN +Email address: kondo@math.nagoya-u.ac.jp +