Borcherds' method for Enriques surfaces

Abstract

We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds' method.

Automorphism groups of certain Enriques surfaces

“…We fix positive half-cones P 10 of L 10 and P 26 of L 26 . In [6], we have proved the following. Recall the notion of being reflexively simple from Definition 2.9.…”

“…In [6], we apply this method to S = L 10 (2). We fix positive half-cones P 10 of L 10 and P 26 of L 26 .…”

“…Our new idea is the lattice theoretic result obtained in [6] (see also Dolgachev-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 −, − .…”

“…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). This property enables us to calculate the automorphism group Aut(Y ) efficiently and explicitly for (τ, τ )-generic Enriques surfaces Y .…”

“…In Section 4, we present a computational procedure on a graph (Procedure 4.1), which is an abstraction of the generalized Borcherds' method formulated in [28]. Then we recall the classification of primitive embeddings L 10 (2) ֒→ L 26 obtained in [6], and construct primitive embeddings S Y (2) ֒→ S X ֒→ L 26 for (τ, τ )-generic Enriques surfaces Y . In Section 5, we prepare some geometric algorithms used in the application of the generalized Borcherds' method to (τ, τ )-generic Enriques surfaces.…”