Variety of Singular Quadrics Containing a Projective Curve

TL;DR

This paper investigates the geometry of quadrics containing a general projective curve, constructs new divisor classes in moduli spaces, and proves that ar{al M}_{15,9} is of general type.

Contribution

It introduces new divisor classes in moduli spaces via the study of quadrics containing curves, and establishes the general type of ar{al M}_{15,9}.

Findings

Variety of quadrics has expected dimension in certain ranges.

Constructed new divisor classes in ar{al M}_{g,n}.

Proved ar{al M}_{15,9} is of general type.

Abstract

We study the variety of rank $\leq k$ quadrics containing a general projective curve and show that it has the expected dimension in the range $g-d+r\leq 1$. By considering the loci where this expectation is not true, we construct new divisor classes in $\overline{\mathcal{M}}_{g,n}$. We use one of these classes to show that $\overline{\mathcal{M}}_{15,9}$ is of general type.