Resonance, syzygies, and rank-$3$ Ulrich bundles on the del Pezzo threefold $V_5$

Abstract

We investigate a geometric criterion for a smooth curve $C$ of genus $14$ and degree $18$ to be described as the zero locus of sections in an Ulrich bundle of rank $3$ on a del Pezzo threefold $V_5 \subset \mathbb{P}^6$. The main challenge is to read off the Pfaffian quadrics defining $V_5$ from geometric structures of $C$. We find that this problem is related to the existence of a special rank-two vector bundle on $C$ with trivial resonance. It gives a description of the image of the rational map that appeared in a work of Ciliberto-Flamini-Knutsen, for the case of degree $5$ del Pezzo threefolds. From an explicit calculation of the Betti table of such a curve, we also deduce the uniqueness of the del Pezzo threefold containing a given curve.