A very short proof of the Borisov-Nuer conjecture

TL;DR

This paper proves Borisov and Nuer's conjecture that elements in a specific lattice can be expressed as differences of vectors with squared length -2, leading to the existence of Ulrich line bundles on certain surfaces.

Contribution

It provides a short proof of a conjecture relating lattice elements to geometric properties of Enriques surfaces.

Findings

Every element in the lattice can be written as a difference of two vectors with squared length -2.

Every unnodal Enriques surface admits an Ulrich line bundle.

The proof confirms a key lattice-theoretic conjecture with geometric implications.

Abstract

We prove a conjecture of Borisov and Nuer, which states that every element in the even unimodular lattice of signature (1,9) can be expressed as the difference of two elements of squared length -2. As a consequence, every unnodal Enriques surface admits an Ulrich line bundle.