Matrix Factorizations and Curves in ${\mathbb{P}}^4$

Abstract

Let $C$ be a curve in ${\mathbb{P}}^4$ and $X$ be a hypersurface containing it. We show how it is possible to construct a matrix factorization on $X$ from the pair $(C,X)$ and, conversely, how a matrix factorization on $X$ leads to curves lying on $X$. We use this correspondence to prove the unirationality of the Hurwitz space ${\mathcal{H}}_{12,8}$ and the uniruledness of the Brill-Noether space ${\mathcal{W}}_{13,9}^1$. Several unirational families of curves of genus $16\le g\le 20$ in ${\mathbb{P}}^4$ are also exhibited.