Automorphism Groups of Certain Enriques Surfaces

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-021-09530-y Automorphism Groups of Certain Enriques Surfaces Simon Brandhorst1 · Ichiro Shimada2 Received: 4 January 2021 / Accepted: 4 June 2021 © The Author(s) 2021 Abstract We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general n-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations. Keywords Enriques surface · K3 surface · Hyperbolic lattice Mathematics Subject Classification 14J28 · 14J50 1 Introduction A central theme in algebraic geometry is to study varieties using convex geometry. The cone of curves of a variety is the convex hull of the numerical equivalence classes of curves. Its dual is the cone of nef line bundles. Much of the birational geometry of a variety is encoded in these cones and their interplay with the canonical divisor. While for Fano varieties the nef cone is rational polyhedral [15, Theorem 3.7], in general the nef cone is not well understood. For instance it can have infinitely many faces or be round. Communicated by Christophe Ritzenhaler. Simon Brandhorst gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 286237555—TRR 195. Ichiro Shimada gratefully acknowledges financial support by JSPS KAKENHI Grant Nos. 16H03926 and 20H01798. B Simon Brandhorst Ichiro Shimada 1 Fachbereich Mathematik, Saarland University, Campus E2.4, 66123 Saarbrücken, Germany 2 Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan 123 Foundations of Computational Mathematics The Morrison–Kawamata cone conjecture [12,20] gives a clear picture of the effective nef cone of a Calabi–Yau variety. It predicts that the action of the automorphism group on the effective nef cone admits a fundamental domain which is a rational polyhedral cone. The conjecture is wide open in dimension three and beyond [18]. But it has been verified for K 3 surfaces by Sterk [33], and for Enriques surfaces by Namikawa [21]. It follows that an Enriques surface admits up to the action of the automorphism group only finitely many smooth rational curves, finitely many elliptic fibrations, finitely many projective models of a given degree and its automorphism group is finitely generated and in fact finitely presented [19, Corollaries 4.15, 4.16]. Naturally, enumerative questions arise: • Can one explicitly describe a fundamental domain? • How many smooth rational curves, elliptic fibrations or projective models are there up to the action of the automorphism group? • Can one give generators for the automorphism group? Barth and Peters [2] noted that very general Enriques surfaces do not contain smooth rational curves. Hence their nef cone is round—it is the entire positive cone, and they proceed to answer the three questions for very general Enriques surfaces. Enriques surfaces containing a smooth rational curve are called nodal. They form a subset of codimension one in the moduli space of Enriques surfaces. Very general nodal Enriques surfaces are treated by Cossec–Dolgachev [8] (see also the works of Allcock [1] and Peters and Sterk [25]). When an Enriques surface is deformed to one containing more rational curves several phenomena working against each other occur. On the one hand the nef cone gets smaller and on the other hand the automorphism group may change drastically. Barth and Peters [2, p. 395] write that they do not know whether one can control these effects. Albeit the behaviour of the nef cone and the automorphism group may be erratic, the cone conjecture promises that the fundamental domain on the nef cone stays of finite volume at least. Our first main result (Theorem 3.4) states that we can control the (change of) volume in a precise way under mild assumptions. To generalize the aforementioned results of Barth, Peters, Cossec and Dolgachev to Enriques surfaces with more nodes, we introduce the notion of (τ, τ̄ )-generic Enriques surfaces, which is closely related to the root invariant introduced by Nikulin [24]. See the next subsection for the precise definition. For instance the very general Enriques surface is (0, 0)-generic, a very general nodal Enriques surface is (A1 , A1 )-generic and if Y is an Enriques surface that is very general in the moduli of Enriques surfaces containing n disjoint smooth rational curves, then Y is (n A1 , n A1 )-generic. If Y is very general in the moduli of Enriques surfaces containing two smooth rational curves whose dual graph is c c (that is, Y is a very general cuspidal Enriques surface), then Y is (A2 , A2 )-generic. Next we give algorithms to compute generators for the automorphism group Aut(Y ), a fundamental domain for Aut(Y ) on the nef and big cone Nef(Y ) and orbit representatives for its action on R(Y ) := the set of smooth rational curves on Y , 123 Foundations of Computational Mathematics E(Y ) := the set of elliptic fibrations Y → P1 . We apply Theorem 3.4 and the aforementioned algorithms to (τ, τ̄ )-generic Enriques surfaces. This results in our second, series of main results: Theorem 1.18 expresses the volume of the fundamental domain of Aut(Y ) on the nef cone Nef(Y ) in terms of the Weyl group of τ , Theorem 1.19 relates the orbits of Aut(Y ) on the set of smooth rational curves R(Y ) to the connected components of the Dynkin diagram τ and Theorem 1.21 counts the Aut(Y )-orbits of the set of elliptic fibrations E(Y ) and their fiber types. Our new idea is the lattice theoretic result obtained in [6] (see also Dolgachev and Kondo [9, Chapter 10]). For a lattice L with the intersection form −, −, let L(m) denote the lattice with the same underlying Z-module as L and with the intersection form m −, −. A lattice L of rank n > 1 is said to be hyperbolic if the signature is (1, n −1). For a positive integer n with n mod 8 = 2, let L n denote an even unimodular hyperbolic lattice of rank n, which is unique up to isomorphism. Borcherds [4,5] developed a method to calculate the orthogonal group of an even hyperbolic lattice S by embedding S primitively into L 26 and using the result of Conway [7]. This method has been applied to the study of automorphism groups of K 3 surfaces by many authors. However, the method often requires impractically heavy computation (see, for example, [11,28]). On the other hand, in [6], we have classified all primitive embeddings of L 10 (2) into L 26 and showed that they have a remarkable property (see Theorems 4.2 and 4.3) which enables us to calculate automorphism groups of Enriques surfaces efficiently and explicitly for the first time. The resulting speed up (roughly by a factor of 1020 in the best situation see Remark 6.1) over a more direct approach, allows us to (...truncated)