On the supersingular K3 surface in characteristic 5 with Artin invariant 1
We present three interesting projective models of the supersingular surface in characteristic with Artin invariant . For each projective model, we determine smooth rational curves on with the minimal degree and the projective automorphism group. Moreover, by using the superspecial abelian surface, we construct six sets of disjoint smooth rational curves on , and show that they form a beautiful configuration.
2000 Mathematics Subject Classification:14J28, 14G17
Let be a surface defined over an algebraically closed field , and the Picard number of . Then it is well-known that or . The last case occurs only when is of positive characteristic. A surface is called supersingular if its Picard number is . Let be a supersingular surface in characteristic . Let denote its Néron-Severi lattice and let be the dual of . Then Artin  proved that is a -elementary abelian group of rank , where is an integer such that . This integer is called the Artin invariant of . It is known that the isomorphism class of depends only on and (Rudakov and Shafarevich ). On the other hand, supersingular surfaces with Artin invariant form a -dimensional family and a supersingular surface with Artin invariant in characteristic is unique up to isomorphisms (Ogus [24, 25], Rudakov and Shafarevich ).
Supersingular surfaces in small characteristic with Artin invariant are especially interesting because big finite groups act on them by automorphisms. (See Dolgachev and Keum ). For example, the group in case or in case acts on the surface by automorphisms. Moreover these surfaces contain a finite set of smooth rational curves on which the above group acts as symmetries. For example, in case , there exist smooth rational curves which form a -configuration (see Dolgachev and Kondo , Katsura and Kondo ). In case , the Fermat quartic surface is a supersingular surface with Artin invariant , and it contains lines (e.g. Katsura and Kondo , Kondo and Shimada ).
In this paper we consider a similar problem for the supersingular surface in characteristic with Artin invariant . We work over an algebraically closed field of characteristic containing the finite field . Let be the Fermat sextic curve in defined by
Note that the left hand side of the equation (1.1) is a Hermitian form over and the projective unitary group acts on by automorphisms. Let be the double cover of branched along . Then is a supersingular surface in characteristic with Artin invariant , on which the finite group acts by automorphisms (e.g. Dolgachev and Keum ). Let be an -rational point of . Then the tangent line to at intersects at with multiplicity 6. Hence the pullback of on splits into two smooth rational curves meeting at one point with multiplicity . Since the number of -rational points of is , we obtain smooth rational curves on .
The main result of this paper is to exhibit three projective models of and determine smooth rational curves of minimal degree on with respect to the corresponding polarizations.
There exist three polarizations of degree on satisfying the following conditions:
The projective model is the double cover of branched along . Here is the class of the pull-back of a line on by the covering morphism . The projective automorphism group of is a central extension of by the cyclic group of order generated by the deck-transformation of over . The double plane contains exactly smooth rational curves of degree , on which acts transitively.
The projective automorphism group of is isomorphic to the alternating group . The minimal degree of curves on is , and contains exactly smooth rational curves of degree , on which acts transitively.
The projective automorphism group of is isomorphic to
of order . The minimal degree of curves on is , and contains exactly smooth rational curves of degree , which decompose into two orbits under the action of .
The set of the smooth rational curves in Theorem 1.1 (3) possesses the following remarkable property. Let and be two sets of disjoint smooth rational curves on a surface. We say that and form a -configuration if every member in one set intersects exactly members in the other set with multiplicity and is disjoint from the remaining members. For example, a -configuration appears in the theory of Kummer surfaces associated to the Jacobian of a smooth curve of genus two: sixteen 2-torsion points on the Jacobian, the theta divisor and its translations by 2-torsion points (Chapter 6 of Griffiths and Harris ).
There exist six sets
of disjoint smooth rational curves on with the following properties.
If , then and form a -configuration for and .
For , the sets and form a -configuration.
If , then and form a -configuration.
In fact, the set of the smooth rational curves of degree on decomposes into the disjoint union of six sets with the properties (a), (b), (c).
Since , however, it is difficult to present these curves explicitly. Instead, we construct the six sets with the properties (a), (b), (c) on the Kummer surface model of . Let be the elliptic curve defined by , and let be the product abelian surface . It is well-known that is isomorphic to the Kummer surface associated with . In Section 8, we construct these six sets explicitly on by giving the pull-back of rational curves by the rational map . As a corollary of this construction, we have the following result. Let be a projective line over with an affine parameter. We define four subsets of by the following:
They are mutually disjoint. See Remark 8.9 for the geometric characterization of the decomposition .
There exists a model of defined over , and a set of the rational curves defined over on that admits a decomposition into disjoint six subsets and satisfying of Theorem 1.2. Moreover, any intersection point of two curves in is an -rational point, and, for each in , the set of -rational points on are decomposed into the union of disjoint four sets , , and , and with the following properties.
, , .
For any point in and each , there exists exactly one curve in passing through . For any point in , there exists exactly one curve in passing through . For any point in resp. , there exists exactly one curve in (resp. ) passing through .
There exists an isomorphism defined over such that , , and .
We give three different proofs of the existence of the smooth rational curves mentioned in Theorem 1.2. We do not know whether such sets of curves coincide under the action of the group of automorphisms of .
By using the Borcherds method [3, 4], the groups of automorphisms of some surfaces were calculated (Kondo , Keum and Kondo , Dolgachev and Kondo , Kondo and Shimada , Ujigawa ). In all cases, the Néron-Severi lattice of each surface is isomorphic to the orthogonal complement of a root lattice in , where is an even unimodular lattice of signature . See Lemma 5.1 of , in which Borcherds gave a sufficient condition for the restrictions of standard fundamental domains of the reflection group of to the positive cone of the surface to be conjugate to each other under the action of the orthogonal group of the Néron-Severi lattice. Contrary to these cases, a new phenomenon occurs in the present case of the supersingular surface in characteristic with Artin invariant : there exist at least three non-conjugate chambers obtained by the restriction of fundamental domains (see also Section 4.6). The projective models in Theorem 1.1 correspond to these three non-conjugate chambers. This phenomenon also happens in the case of the complex Fermat quartic surface.
The plan of this paper is as follows. In Section 2, we recall some lattice theory which will be used in this paper. Section 3 is devoted to the explanation of the Borcherds method for finding a finite polyhedron in the positive cone of a hyperbolic lattice primitively embedded into the even unimodular lattice of signature . In Section 4, we apply this method to the case of the supersingular surface in characteristic with Artin invariant . In particular, by using computer, we give a proof of Theorems 1.1 and 1.2. In Section 5, by using a geometry of Leech lattice, we give another proof of Theorems 1.1 and 1.2 without using computer. In Section 6, we recall some facts on the supersingular elliptic curve in characteristic , and in Section 7 we investigate -rational points on the Kummer surface associated with the product of two supersingular elliptic curves. Section 8 is devoted to give another proof of Theorem 1.2 by using the Kummer surface structure of . Moreover we study the intersection between the smooth rational curves and prove Theorem 1.3.
A -lattice is a pair of a free -module of finite rank and a non-degenerate symmetric bilinear form . We omit the bilinear form or the subscript in if no confusions will occur. If takes values in , is called a lattice. For , we call the norm of . A lattice is even if holds for any .
Let be a lattice of rank . The signature of is the signature of the real quadratic space . We say that is negative definite if is negative definite, and is hyperbolic if the signature is . A Gram matrix of is an matrix with entries , where is a basis of . The determinant of a Gram matrix of is called the discriminant of .
Let be an even lattice, and let be naturally identified with a submodule of with the extended symmetric bilinear form. We call this -lattice the dual lattice of . The discriminant group of is defined to be the quotient , and is denoted by . The order of is equal to the discriminant of up to sign. A lattice is called unimodular if is trivial, while is called -elementary if is -elementary.
For an even lattice , the discriminant quadratic form of
is defined by .
A submodule of is called primitive if is torsion free. A non-zero vector is called primitive if the submodule of generated by is primitive.
Let be the orthogonal group of a lattice ; that is, the group of isomorphisms of preserving . We assume that acts on from the right, and the action of on is denoted by . Similarly denotes the group of isomorphisms of preserving . There is a natural homomorphism .
Let be a hyperbolic lattice. A positive cone of is one of the two connected components of the set
Let be a positive cone of . We denote by the group of isometries of preserving . Then . For a vector with , we define
which is a real hyperplane of . An isometry is called a reflection with respect to or a reflection into if is of order and fixes each point of . For a lattice , the set of -vectors is denoted by . Any element of defines a reflection
with respect to . We denote by the group generated by the set of reflections . Since preserves , is a subgroup of . It is obvious that is normal in .
A negative definite even lattice is said to be a root lattice if is generated by .
3. Borcherds method
We define some notions and fix some notation. Let be an even hyperbolic lattice with a fixed positive cone . Let be a set of vectors with . Suppose that the family of hyperplanes
is locally finite in . By a -chamber, we mean a closure in of a connected component of
Let be a -chamber. A hyperplane is said to be a wall of if is disjoint from the interior of and contains a non-empty open subset of .
Recall that is the set of vectors with . Then each -chamber is a fundamental domain of the action of on .
3.1. Conway-Borcherds theory
Let be an even unimodular hyperbolic lattice of rank , which is unique up to isomorphisms, and let be a positive cone of . An -chamber will be called a Conway chamber. A non-zero primitive vector with is called a Weyl vector if is contained in the closure of in and the even negative-definite unimodular lattice is isomorphic to the (negative-definite) Leech lattice (that is, contains no -vectors). For a Weyl vector , we put
If is a Weyl vector, then
is a Conway chamber, and is the set of walls of . For any Conway chamber , there exists a unique Weyl vector such that .
Let be an even hyperbolic lattice of rank . Suppose that is primitively embedded into . Let be the positive cone of that is contained in . Let denote the orthogonal complement of in . For , we denote by
the projections to and , respectively. Note that, if , then and . We assume the following:
The negative-definite lattice cannot be embedded into the Leech lattice. (For example, this condition is satisfied if .)
The natural homomorphism is surjective.
It is easy to see that the family of hyperplanes is locally finite in . A Conway chamber is said to be -nondegenerate if contains a non-empty open subset of . If is an -nondegenerate Conway chamber, then is an -chamber of , which is called an induced chamber. Since is tessellated by Conway chambers, is tessellated by induced chambers. Since is a subset of , any -chamber is a union of induced chambers. We have the following. See .
(1) Any induced chamber has only a finite number of walls.
(2) The automorphism group of an induced chamber is a finite group.
a Weyl vector such that is -nondegenerate, and
a wall of the induced chamber ,
then we can calculate a Weyl vector such that is the induced chamber adjacent to along the wall .
3.2. Periods and automorphisms of supersingular surfaces
Let be a supersingular surface defined over an algebraically closed field of odd characteristic with Artin invariant , and let denote the Néron-Severi lattice of . Since is -elementary, we have . Consider the -dimensional -vector space
on which we have an -valued quadratic form defined by
Let be the Chern class map. Then is a -dimensional isotropic subspace of . Let denote the map , where is the Frobenius of .
The period of is defined to be .
Note that acts on naturally. We put
We denote by the positive cone of containing an ample class of . Let denote the intersection of with the nef cone of ;
Now suppose that is embedded into in such a way that the conditions (i) and (ii) in Section 3.1 are satisfied and that the image of is contained in . It is well-known that is an -chamber in . (See, for example, Rudakov and Shafarevich .) Hence is tessellated by induced chambers. For an induced chamber contained in , we put
Then is a finite subgroup of . More precisely, if is a vector in the interior of , then
is an ample class, and is the automorphism group of the polarized surface . We have an algorithm to make the complete list of elements of . Hence, in order to calculate , all we have to do is to calculate the action of on the period .
We say that two induced chambers and are -congruent if there exists such that . The number of -congruence classes is finite. If we obtain the list of all -congruence classes, we can determine the automorphism group of . (As is explained in Introduction, in the previous works of computing automorphism groups of surfaces using this technique, there exists only one -congruence class.) See  and Section 4.6.
4. Proof of Theorems by computer
4.1. The Néron-Severi lattice and the period of
Using the projective model , we calculate the Néron-Severi lattice and the period of explicitly.
As is explained in Introduction, the surface contains smooth rational curves such that . We call these smooth rational curves -lines.
The -lines are labelled as follows. Let denote the double covering. Part of the -rational points on the Fermat curve of degree are given explicitly in Table 4.1. Let be the line on tangent to at . We put
which is an irreducible component of , and let denote the other irreducible component. For , let be the irreducible component of such that , and let be the other irreducible component. Consider the following twenty-two -lines.
Their intersection matrix is of determinant . Hence the classes of these -lines form a basis of . The Gram matrix of with respect to this basis is given in Table 4.2.