An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this motivating paper, Eisenbud-Schreyer asked a question whether a given projective variety has an Ulrich bundle, and what is the minimal possible rank of an Ulrich bundle if exists. The answer is still widely open for algebraic surfaces and higher dimensional varieties.
Thanks to a number of studies, the answer for the above question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. In this talk, we first recall a construction of rank-3 Ulrich bundle on a cubic threefold by Geiss and Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We also discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 does not belong to V_5. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.
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