Matrix factorizations and curves in $\mathbb{P}^4$

TL;DR

This paper establishes a correspondence between curves in projective 4-space and matrix factorizations on hypersurfaces, leading to new results on the geometry of certain moduli spaces and families of curves.

Contribution

It introduces a novel method linking matrix factorizations to curves in projective space, enabling proofs of unirationality and uniruledness of specific moduli spaces.

Findings

Proves the unirationality of the Hurwitz space _{12,8}

Shows the uniruledness of the Brill-Noether space ^1_{13,9}

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