Maximally Nodal Sextic Surfaces and Linear Determinantal Representations

Source 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.