Matrix factorizations and curves in P^4

Let C be a curve in 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 H12,8 and the uniruledness of the Brill-Noether space W13,91. Several unirational families of curves of genus 16 ≤ g ≤ 20 in 4 are also exhibited.