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 $\mathcal{W}^1_{13,9}$. Several unirational families of curves of genus $16 \leq g \leq 20$ in $\mathbb{P}^4$ are also exhibited.

14H10, 14M20, 14Q05, 13D02

matrix factorization, moduli of curves, unirationality, Hurwitz space

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Mathematik und Informatik, Universität des Saarlandes, Campus E2.4, D-66123 Saarbrücken, Germany

Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques, Université de Lille, 59655 Villeneuve d'Ascq CEDEX, France