Examples of surfaces which are Ulrich–wild

Abstract

We give examples of surfaces which are Ulrich-wild, i.e. that support families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p.2010 Mathematics Subject Classification. 14J60. Key words and phrases. Vector bundle, Ulrich bundle, Ulrich-wild, Surfaces of low degree. The author is a member of GNSAGA group of INdAM and is supported by the framework of PRIN 2015 'Geometry of Algebraic Varieties', cofinanced by MIUR.Unfortunately, the above result is not sharp. E.g. if S is a del Pezzo surface, h S K S = −d and χ(O S ) = 1, thus the first member of Inequality (1.1) is strictly negative. Nevertheless del Pezzo surfaces are Ulrich-wild as shown in [41] and [36]: see also Section 5.3 of [24] where the authors prove the Ulrich-wildness of each 2-dimensional maximally del Pezzo variety (see [11] and the references therein for details about such varieties).Anyhow, Theorem 1.2 yields some interesting examples. As a consequence of results by A. Beauville and D. Faenzi, we show in Example 3.1 that abelian surfaces (i.e. surfaces S with K S = 0 and q(S) = 2) are Ulrich-wild. A similar result has been recently proved in [23] for K3 surfaces (i.e. surfaces S with K S = 0 and q(S) = 0) extending the previous works [19] and [4]. Thus each minimal surface S with Kodaira dimension κ(S) = 0 is Ulrich-wild (see Proposition 6 of [8]).We also deal with surfaces of general type (i.e. minimal surfaces S with κ(S) = 2: seeIn order to find other examples, we prove the following helpful theorem which holds without restrictions on the characteristic of k: it makes slightly more precise the results from [27].

25 open questions about vector bundles and their moduli

“…There are many papers describing Ulrich sheaves on specific choices of (𝑋, 𝐿) such as [24,[50][51][52]97] and references therein. The reader may be interested in this list, because usually such sheaves give good insights about the derived categories of coherent sheaves 𝐷 𝑏 (𝑋).…”

Special Ulrich bundles on regular surfaces with non–negative Kodaira dimension

“…In [15, Theorems 1.4 and 1.5] the author deals with the existence of special Ulrich bundles on surfaces of low degree on surfaces S ⊆ P N . We start this section by improving [15,Theorem 1.4]. We work over the complex field C, hence S 0 is dense for each surface S with κ(S) ≥ 0 by Remark 2.3.…”

“…If κ(S) = 1, taking into account of the classification in [15,Table A] and the results in [41,42], we know that S ⊆ P 4 is a properly elliptic surface of degree either 7 or 8. The existence of special Ulrich bundles on such surfaces has been proved in Example 6.4.…”

“…We now extend [15,Theorem 1.5]. Here π(S) denotes the genus of a general plane section of S. [13,14], because S is neither a rational normal scroll nor a plane by degree reasons.…”