Maximally nodal sextic surfaces and linear determinantal representations

TL;DR

This paper proves that maximally nodal sextic surfaces with 65 nodes have a symmetric determinantal representation, linking geometric properties to explicit matrix constructions and Ulrich sheaves.

Contribution

It establishes the existence of symmetric linear determinantal representations for maximally nodal sextic surfaces and provides explicit examples.

Findings

Maximally nodal sextic surfaces contain a symmetric half-even set of nodes.

Such surfaces admit a symmetric $6 imes 6$ linear matrix representation.

An explicit example of a matrix defining the Barth sextic surface is provided.

Abstract

We prove that every maximally nodal sextic surface\,(with 65 nodes) $X \subset \mathbb{P}_{\mathbb{C}}^3$ contains a symmetric half-even set of nodes of cardinality 35. It follows that the associated half-quadratic sheaf is the cokernel of a symmetric $6 \times 6$ matrix of linear forms, yielding a linear determinantal representation of $X$. In particular, after a suitable Serre twist, the half-quadratic sheaf is an Ulrich sheaf of rank 1. As an example, we exhibit an explicit $6 \times 6$ matrix of linear forms whose determinant defines the Barth sextic surface.