Borcherds’ method for Enriques surfaces
We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by into an even unimodular hyperbolic lattice of rank . These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds’ method.
Key words and phrases:Enriques surface, automorphism group, lattice
First we fix notation about lattices. A lattice is a free -module of finite rank with a non-degenerate symmetric bilinear form . A lattice is even if for all . An embedding of a lattice into a lattice is primitive if the cokernel is torsion-free. A lattice of rank is hyperbolic if is of signature . For a lattice and a non-zero integer , let denote the lattice with the same underlying -module as and with the symmetric bilinear form being the times that of . The automorphism group of a lattice is denoted by , and an element of is called an isometry. We let act on from the right. A lattice is embedded by into the dual lattice as a submodule of finite index. The cokernel is called the discriminant group of . We say that is unimodular if is trivial. For an integer , a vector of a lattice is called a -vector if . Let denote the set of -vectors of a lattice . A negative-definite root lattice is a negative-definite lattice generated by . It is well-known that negative-definite root lattices are classified by their -types (see, for example, Chapter 1 of ). Let be an even unimodular hyperbolic lattice of rank . It is also well-known (for example, see Section V of ) that exists if and only if , and that, if , then is unique up to isomorphism.
The lattice theory is a very strong tool in the study of and Enriques surfaces. Let be a or an Enriques surface defined over an algebraically closed field. We denote by the lattice of numerical equivalence classes of divisors on . Note that is even, and if , then is hyperbolic. Borcherds’ method [2, 3] is a procedure to calculate the automorphism group of a surface by embedding primitively into , and applying Conway’s result  on . After the work of Kondo , this method has been applied to many surfaces, and automatized for computer calculation (see  and the references therein).
Let be an Enriques surface in characteristic with the universal covering . Then we have a primitive embedding . Note that is isomorphic to . Hence, to extend Borcherds’ method to Enriques surfaces, it is important to study the primitive embeddings of into .
We identify with . We say that two embeddings and of into are equivalent up to the action of and if there exist isometries and such that, for all , one has . Our first main result is as follows.
Up to the action of and , there exist exactly equivalence classes of primitive embeddings of into , and they are given in Table 1.1.
Explanation of Table 1.1. Let denote the orthogonal complement of the image of a primitive embedding in . Note that is a negative-definite even lattice of rank with . The item rt is the -type of the negative-definite root lattice generated by . For the embedding infty, the lattice contains no -vectors. The item m4 is the number of -vectors in . The item og is the order of the group .
These embeddings have a remarkable property, which is very useful for the calculation of the automorphism group of an Enriques surface. In order to state this property, we need to explain the notion of tessellation by chambers. Let be an even hyperbolic lattice. A positive cone is one of the two connected components of the subspace of consisting of vectors such that . We fix a positive cone of , and denote by the stabilizer subgroup of in , which is of index in . A rational hyperplane is a subspace of defined by , where is a vector satisfying . Let be a locally finite family of rational hyperplanes of . A closed subset of is said to be an -chamber if is the closure in of a connected component of the complement
We say that a subset of has a tessellation by -chambers if is a union of -chambers. For example, if is a subfamily of , then every -chamber has a tessellation by -chambers.
Note that has a tessellation by -chambers. We say that this tessellation of is simple if there exists a subgroup of that preserves the family of hyperplanes (and hence the set of -chambers) and acts on the set of -chambers transitively.
We say that -chambers and are isomorphic if there exists an isometry such that . The automorphism group of an -chamber is defined to be
Let be an -chamber, and the closure of in . We say that is quasi-finite if is contained in a union of at most countably many half-lines , where are non-zero vectors of satisfying , is the closure of in , and .
Each -vector defines the reflection into the mirror , which is defined by . Let denote the subgroup of generated by reflections , where runs through .
We put , which is a locally finite family of rational hyperplanes. Then an -chamber is a standard fundamental domain of the action on of . Hence the tessellation of by -chambers is simple. Note that we have .
It is known that Vinberg chambers and Conway chambers are quasi-finite.
Let be an -chamber. A wall of is a closed subset of disjoint from the interior of satisfying the following; there exists a hyperplane such that is equal to and that contains a non-empty open subset of . We say that defines a wall of if is equal to and holds for all .
Let be as in Example 1.5. Then the group is generated by reflections with respect to the -vectors defining walls of .
Let be an -chamber, and a wall of . Then there exists a unique -chamber such that . We call the -chamber adjacent to across the wall .
Let be an embedding of an even hyperbolic lattice , the positive cone of that is mapped to by , and the induced inclusion. We put
Then is a locally finite family of rational hyperplanes of , and has a tessellation by -chambers. If all -chambers are quasi-finite, then so are all -chambers.
In the following, we identify with . If is a primitive embedding, then has a tessellation by -chambers. We call an -chamber an induced chamber associated with the embedding . Note that every induced chamber is quasi-finite.
In the application of Borcherds’ method for the calculation of of a surface , we embed into primitively and investigate the tessellation of by induced chambers. This tessellation is usually not simple, and in these cases, the computation of becomes very hard. See, for example, the case of the singular surface with transcendental lattice of discriminant treated in , or the case of the supersingular surface of Artin invariant in characteristic studied in .
Our second main result is as follows.
Let be a primitive embedding that is not of type . Then the number of walls of an induced chamber is finite, and each wall of is defined by a -vector of . If defines a wall of , then the reflection with respect to preserves the family of hyperplanes and hence the set of induced chambers. In particular, the induced chamber adjacent to across the wall is equal to .
If is not of type , then the tessellation of by induced chambers is simple.
The data of the induced chambers are given in Table 1.2. Before explaining the contents of Table 1.2, we recall two classical results about automorphism groups of Enriques surfaces. Let be an Enriques surface. We denote by the positive cone of containing an ample class. We then put
Then has a tessellation by Vinberg chambers, because is bounded by the hyperplanes defined by the classes of smooth rational curves on and every smooth rational curve on has the self-intersection number .
Let be a complex generic Enriques surface. Then we have . Barth and Peters  showed that is canonically identified with the kernel of the -reduction homomorphism . Since a Vinberg chamber has no automorphism group, the group is equal to the subgroup . Since the -reduction homomorphism above is surjective (see  and Section 2.3 of this paper), there exists a union of
Vinberg chambers such that (i) is the union of , where runs through , and (ii) if is not the identity, then the interiors of and of are disjoint.
Kondo  and Nikulin  classified all complex Enriques surfaces with finite automorphism group. This classification was extended to odd characteristics by Martin . It turns out that Enriques surfaces in characteristic with finite automorphism group are divided into classes I, …, VII. An Enriques surface with finite automorphism group has only a finite number of smooth rational curves , and is bounded by the hyperplanes defined by these curves. The configurations of these smooth rational curves are explicitly depicted in .
Explanation of Table 1.2. The item walls is the number of walls of an induced chamber . Since every wall of is defined by a -vector of , it follows that is a union of Vinberg chambers. The item volindex shows that the number of Vinberg chambers contained in is equal to
The item gD is the order of the automorphism group of . The item orb describes the orbit decomposition of the set of walls under the action of . The item isom shows that, for example, the induced chambers of the primitive embeddings 20C and 20D are isomorphic. The item NK shows that, for example, the induced chamber of the primitive embedding 12A is, under a suitable isomorphism , equal to of an Enriques surface of type I.
Since all types I, …, VII appear in the column NK, our results on the induced chamber can be applied to for an arbitrary Enriques surface with finite automorphism group in characteristic .
Borcherds’ method has been applied to Enriques surfaces in  and  without using the facts proved in this paper. These facts actually give us a big advantage in the calculation of the automorphism group of an Enriques surface by Borcherds’ method, as is exemplified in . We can also enumerate all polarizations of with a fixed degree modulo by means of the method in . These applications will be treated in other papers.
For the computation, the first author used a mixture of SageMath, PARI, GAP [30, 29, 28], and the second author used GAP . The explicit computational data is available at the second author’s webpage .
Thanks are due to Professor Igor Dolgachev and Professor Shigeyuki Kondo for their interests in this work and many comments.
Notation. To avoid possible confusions between and , we put
We identify the underlying -modules of and , and choose positive cones so that . We also have a natural identification . We denote by , and the symmetric bilinear forms of , , and , respectively.
2.1. Discriminant form
Let be an even lattice. Recall that is the discriminant group of . The quadratic form
defined by for is called the discriminant form of . Let denote the automorphism group of the finite quadratic form . Then we have a natural homomorphism
See Nikulin  for the basic properties of discriminant forms. Among these properties, the following is especially important for us:
Let and be even lattices. We consider the following sets:
the set of even unimodular lattices contained in , containing , and containing each of and primitively, and
the set of isomorphisms between the finite quadratic forms and .
Let be an isomorphism from to , let denote the graph of , and let be the pull-back of by the natural projection . Then the mapping gives rise to a bijection from to . This bijection is compatible with the natural actions of on and on .
Suppose that , so that is the orthogonal complement of the primitive sublattice . Let be the isomorphism corresponding to , and the induced isomorphism. We put
and let and denote the restriction homomorphisms from to and , respectively. We say that is a lift of if .
Let be an isometry of . Then the homomorphism induces a bijection from the set of lifts of to the set of all isometries of such that is mapped to by .
2.2. Kneser’s neighbor method
Recall that two lattices and are in the same genus if we have an isomorphisms
of - or -valued quadratic modules for every prime , where denotes the ring of -adic integers. Suppose that and are in the same genus. Then, by the Hasse–Minkowski theorem, we have . Thus we may and will assume that . Let be an odd prime which does not divide the determinant of . We say that two lattices and are -neighbors if
Suppose that and are -neighbors. Then for all primes . Moreover, since does not divide , both and are unimodular -lattices isomorphic over the field of -adic rationals . Thus and are in fact isomorphic. We have proved that the -neighbors and are in the same genus.
For a given genus , we denote by the set of isomorphism classes of lattices in this genus. Let be an odd prime. Set
Then is called the -neighbor graph of . Assume further that represents for a lattice in this genus. This is certainly the case if the rank of is at least . In general each connected component of this graph is the union of several so called proper spinor genera. In the case relevant to us, the genus consists of a single proper spinor genus, so this does not concern us.
For given and with , the lattice
is called the -neighbor of with respect to . One can show that and are indeed -neighbors, that depends only on (as long as stays divisible by ), and that every -neighbor of arises in this fashion.
Thus one can classify lattices in the genus by iteratively computing the neighbors of the lattices in and testing for isomorphism (see ). One can speed this up by computing the neighbors of a given lattice only up to the action of the orthogonal group. When we are interested only in the vertices and not in the edges, we can break the computation when we have “explored” all vertices. The mass of the genus is defined as
It can be calculated from the invariants of alone as described in . We can break the computation as soon as the sum of the reciprocals of reaches .
This procedure is implemented for example in Magma . In the example relevant to us, the computation with Magma simply exhausted all memory available. Thus we had to resort to a modified strategy: A random walk through the neighbor graph.
2.3. Proof of Theorems 1.1
Let be a basis of consisting of -vectors that form the configuration in Figure 2.1. Then
is a Vinberg chamber, and each is a wall of (see Vinberg ).
Since the graph in Figure 2.1 has no non-trivial automorphisms, the group is generated by the reflections with respect to . Recall from the paragraph Notation at the end of Introduction that we put . In , we have , and the mapping gives an isomorphism of -modules, which gives rise to an isomorphism
of finite quadratic forms. Hence we see that is of order by Proposition 1.7 of . Since we have explicit generators of , we can confirm that restricted to is surjective.
The surjectivity of can also be proved by Theorem 7.5 and Lemma 7.7 of Chapter VIII of .
Let be a primitive embedding, and let be the orthogonal complement of the image of in . Then is of signature . By Proposition 2.1, the discriminant form is isomorphic to . Since is surjective, Proposition 2.1 implies that, if a primitive embedding satisfies , then is equivalent to up to the action of and . Hence the proof of Theorem 1.1 is reduced to the classification of isomorphism classes of even lattices with signature such that . Note that these conditions on signature and discriminant form determine the genus of . By the mass formula , we see that the mass of this genus is
Let be defined by . Then one calculates that
and that and , where and are the negative-definite root lattices of -type and , respectively. Thus we have found a first lattice in . To find representatives up to isomorphism, we use a variant of Kneser’s neighbor method for . Start by inserting into a list . Then enter the following loop. Pick a random in and a random with divisible by , replace by for such that is divisible by . Calculate the -neighbor and check if it is isomorphic to any lattice in the list . If not add it to . Break the loop when the mass of the lattices in matches . By this computation, it turns out that is constituted by isomorphism classes in Table 1.1, and hence Theorems 1.1 follows.
2.4. Conway theory
Let be a non-zero primitive vector of contained in . Note that . We put
Then has a natural structure of an even unimodular negative-definite lattice, and hence is isomorphic to , where is one of the Niemeier lattices (see, for example, Chapter 16 of ).
We say that is a Weyl vector if is isomorphic to the negative-definite Leech lattice.
Since the Leech lattice is characterized as the unique Niemeier lattice with no roots, we can determine whether is a Weyl vector or not by calculating the set of -vectors in .
For a Weyl vector , we put
The following theorem is very important.
Theorem 2.5 (Conway ).
The mapping gives a bijection from the set of Weyl vectors to the set of Conway chambers.
Let be a Weyl vector. Since is primitive and is unimodular, there exists a vector such that and . Then every -vector of with is written as
Since implies , we see that is an interior point of .
2.5. Proof of Theorem 1.10
In Section 2.3, we have calculated the primitive embeddings explicitly. As was said in Notation, we identify and , and denote by the induced inclusion.
Our first task is to find a Weyl vector such that is an induced chamber, that is, contains a non-empty open subset of . Recall that we have fixed a basis of . We put , where are the basis of dual to . Then is an interior point of the Vinberg chamber defined by (2.1), and we have . By direct calculation, we confirm the equality
which means that, if a hyperplane of defined by passes through , then contains the image of . (Note that the second set in (2.2) is identified with by the embedding .) Therefore is an interior point of an induced chamber .
Let be an even hyperbolic lattice. Suppose that are vectors of . We say that a -vector separates and if . We can calculate the set of -vectors of separating and by the algorithm given in Section 3.3 of .
We perturb to in a general direction so that is also an interior point of the same induced chamber as , that is, the equality (2.2) remains true with replaced by and there exist no -vectors of separating and . We choose an arbitrary Weyl vector of , and calculate a vector in the interior of by Remark 2.6. We then calculate the set of -vectors of separating and . We sort these -vectors in such a way that the line segment from to intersects the hyperplanes in this order. Since is a result of general perturbation, these intersection points are distinct. Let be the reflection with respect to . We move by in this order and obtain a new Weyl vector . Then contains in its interior, and there exist no -vectors of separating and . Therefore is the induced chamber containing in its interior.
Next we calculate the set of walls of the induced chamber . We denote by and the orthogonal projections and , and let denote the symmetric bilinear form of . It turns out that holds except for the case where is of type . Henceforth we assume that is not of type . We put