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

We investigate a geometric criterion for a smooth curve [Formula: see text] of genus [Formula: see text] and degree [Formula: see text] to be described as the zero locus of sections in an Ulrich bundle of rank [Formula: see text] on a del Pezzo threefold [Formula: see text]. The main challenge is to read off the Pfaffian quadrics defining [Formula: see text] from geometric structures of [Formula: see text]. We find that this problem is related to the existence of a special rank-2 vector bundle on [Formula: see text] 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 [Formula: see text] 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.