Special Ulrich bundles on non-special surfaces with pg = q = 0

Abstract

Let [Formula: see text] be a surface with [Formula: see text] and endowed with a very ample line bundle [Formula: see text] such that [Formula: see text]. We show that [Formula: see text] supports special (often stable) Ulrich bundles of rank [Formula: see text], extending a recent result by A. Beauville. Moreover, we show that such an [Formula: see text] supports families of dimension [Formula: see text] of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large [Formula: see text] except for very few cases. We also show that the same is true for each linearly normal non-special surface with [Formula: see text] in [Formula: see text] of degree at least [Formula: see text], Enriques surface and anticanonical rational surface.

On the existence of Ulrich vector bundles on some surfaces of maximal Albanese dimension

“…3.3 and Thm. 3.4] and in all cases by [C4,Cor. 3.3] (observing that, as above, one always has that H 1 (H) = 0).…”

“…Since H − K S is ample we see by Remark 4.1(i) that (a) of Theorem 1 is satisfied. As a matter of fact the Ulrich vector bundle obtained is the same as the one in [C4,Cor. 3.3] and essentially the same as the one in [B1,Prop.…”

“…In several cases the main idea is to construct a vector bundle E on the surface via a zero-dimensional Cayley-Bacharach subscheme. One strong condition is given by Riemann-Roch since one has that χ(E(−p)) = 0 for p = 1, 2: the Chern classes have to satisfy two conditions [C1,Prop. 2.1].…”

Theta divisors and Ulrich bundles on geometrically ruled surfaces

“…In this section, we focus our attention on the existence of special stable rank 2 Ulrich bundles. The existence is known for g = 0 and a ≥ 2 see [5], Theorem 1.2. When a = 1 and g = 0 there are no such bundles, actually it is known (see [9], Corollary to Theorem B), that each Ulrich bundle of rank at least 2 is in this case strictly µ-semistable.…”

“…Proof. For the proof we refer the interested reader to [5], Corollary 2.2 or [1], Lemma 3.2. The proof therein is given under the apparently more restrictive hypothesis that F is initialized, i.e.…”