On the classification of non-big Ulrich vector bundles on surfaces and threefolds

Abstract

In this paper, we classify Ulrich vector bundles that are not big on smooth complex surfaces and threefolds.

Non-big Ulrich bundles: the classification on quadrics and the case of small numerical dimension

“…Then E is an Ulrich vector bundle for (X, H), E it is not big, E |f ∼ = Ω f (2) and, in many cases, c 1 (E) 4 > 0. This is shown for b = 1 in [LM,Lemma 4.1]. With the same method it can be shown for b = 2.…”

“…We can assume furthermore that Φ has a fiber F x 0 that is a linear P k with 2 ≤ k ≤ 3. In fact if all fibers of Φ were one-dimensional it would follow, as in [LM,proof of Prop. 5.3], that we are in case (ii).…”

On the classification of non-big Ulrich vector bundles on fourfolds

“…If (X, O X (1)) ∼ = (P n , O P n (1)) we are in case (i1) by Remark 2.9. If (X, O X (1)) ∼ = (P n , O P n (1)) then c 1 (E) n > 0 by [Lo,Lemma 3.2] and we are in case (i2) by [LM,Thms. 1 and 2] and Theorem 1.…”