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Matrix factorizations and families of curves of genus 15

In this note, we explain how certain matrix factorizations on cubic threefolds lead to families of curves of genus 15 and degree 16 in P^4. We prove that the moduli space M={(C,L) | C a curve of genus 15, L a line bundle on C of degree 16 with 5 sections} is birational to a certain space of matrix factorizations of cubics, and that M is uniruled. Our attempt to prove the unirationality of this space failed with our methods. Instead one can interpret our findings as evidence for the conjecture that the basis of the maximal rational connected fibration of M has a three dimensional base.