Resonance, syzygies, and rank-$3$ Ulrich bundles on the del Pezzo threefold V₅

We investigate a geometric criterion for a smooth curve C of genus $14$ and degree $18$ to be described as the zero locus of a section in an Ulrich bundle of rank $3$ on a del Pezzo threefold V₅ ⊂ P⁶. The main challenge is to read off the Pfaffian quadrics defining V₅ 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. This answers a question posed by Ciliberto-Flamini-Knutsen, in 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.