Maximally nodal sextic surfaces and linear determinantal representations

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.