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

Abstract

We classify non-big Ulrich vector bundles on quadrics. On any smooth n-dimensional variety we give a pretty precise picture of rank r Ulrich vector bundles with numerical dimension at most n 2 + r − 1. Turning to fourfolds, we first classify, with some exceptions, non-big Ulrich vector bundles on them. This allows a complete classification in the case of rank one fourfolds, in the case of Mukai fourfolds and in the case of Del Pezzo n-folds for n ≤ 4. We also classify Ulrich bundles E with det E non-big on Del Pezzo and Mukai n-folds, n ≥ 2.

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

“…By (3.2) it remains to study the case ν(E) = r + 2. Then, with the same proof given in [LMS,Lemma 4.4], the incidence correspondence…”

“…Now E ∼ = p * (G(2)) where G is a rank r vector bundle on Q such that H j (G ⊗ S k F * ) = 0 for j ≥ 0, 0 ≤ k ≤ n − 2. Hence H := G(1) is an Ulrich vector bundle on Q and we get that E ∼ = q * (H(1)), where q : P 1 × Q → Q is the second projection and H is a direct sum of spinor bundles on Q by [LMS,Lemma 3.2(iv)].…”

“…Suppose that (X, O X (1), E) is as in (i)-(xi). Then, using, in the corresponding cases, Remark 2.9, [LMS,Prop. 3.3(iii)], [Lo,Lemma 4.1], [B, (3.5)] and Remark 2.8(iv), we see that E is Ulrich not big.…”

“…If (X, O X (1)) is as in (a), we are in case (i) by Remark 2.9. If (X, O X (1)) is as in (b.1), we are in case (xi) by [LMS,Prop. 3.3(iii)].…”

“…] implies that we are in case (c.3), a contradiction. Therefore we can apply [LMS,Prop. 4.6] and deduce that there is a morphism ψ : P r−1 → F 1 (X, x) that is finite onto its image and…”