On intersections of symmetric determinantal varieties and theta characteristics of canonical curves

TL;DR

This paper explores the geometric relationships between symmetric determinantal varieties and theta characteristics of canonical curves, providing new insights into their singularities and constructions for specific genera.

Contribution

It introduces a novel characterization of singularities of symmetric determinantal varieties and generalizes classical constructions of theta characteristics for certain canonical curves.

Findings

Characterization of accidental singularities of symmetric determinantal hypersurfaces

Construction of theta characteristics for genus 3, 4, and 5 curves

Generalization of Cayley's classical construction

Abstract

From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of Cayley.