  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.
