Ulrich bundles on Del Pezzo threefolds

Abstract

We prove that for any r 2 the moduli space of stable Ulrich bundles of rank r and determinant OX (r) on any smooth Fano threefold X of index two is smooth of dimension r 2 + 1 and that the same holds true for even r when the index is four, in which case no odd-rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties. A preliminary lemmaThis short section is devoted to the proof of the following: Lemma 2.1. Let X ⊂ P m be a smooth, irreducible, projective variety with deg(X) > 1 and let E be an Ulrich bundle on X. Then h 0 (E * ) = 0.Proof. Assume by contradiction that h 0 (E * ) > 0 and pick a non-zero section s * ∈ H 0 (E * ). Since h 0 (E) = deg(X) rk(E) deg(X) > 1 by [Be1, (3.1)], we have

Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

“…In this section we make some remarks about rank 2 Ulrich bundles on prime Fano threefolds X of genus g and degree 2g − 2 in P g+1 . By [CFK,Ex. 3.8] any such X carries a rank 2 Ulrich bundle E if and only if det(E) = O X (3) and there is a smooth, irreducible, non-degenerate, linearly and quadratically normal curve C in X such that More precisely, given a rank 2 Ulrich bundle E on X and a general section s ∈ H 0 (E), the curve C is the zero locus of s, and conversely, if there is such a curve C ⊂ X, there is a rank 2 Ulrich bundle E on X and a non-zero section s ∈ H 0 (E), such that C is the zero locus of s.…”

“…There is however only a short list of varieties for which ACM or Ulrich bundles are completely classified. For works regarding ACM and Ulrich bundles on Fano threefolds, mostly of low ranks, we refer to [AC,Be1,Be2,Be3,Be4,BF1,BF2,BF3,CH,CKL,CM,CFaM1,CFaM2,CFaM3,CFiM,CFK,FP,LMS,LP,MT].…”

“…This result is optimal in the sense that a prime Fano threefold X does not carry any Ulrich bundles of odd rank by [CFK,Cor. 3.7] and any Ulrich bundle of rank 2h on X is forced to have determinant O X (3h) by [CFK,Thm. 3.5].…”

“…Finally, in §7, taking profit from a correspondence we found in [CFK,Thm. 3.5] between rank two Ulrich bundles and certain curves on prime Fano threefolds, we propose an application to the moduli space of curves of genus 5g with theta-characteristics with g + 2 independent global sections, for 3 g 5.…”