  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 $V_5 \subset \mathbb{P}^6$. 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. 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.
