The geometry of Ulrich bundles on del Pezzo surfaces

Abstract

Given a smooth del Pezzo surface $X_d\subseteq {\mathbb{P}}^d$ of degree $d,$ we show that a smooth irreducible curve $C$ on $X_d$ represents the first Chern class of an Ulrich bundle on $X_d$ if and only if its kernel bundle $M_C$ admits a generalized theta-divisor. This result is applied to produce new examples of complete intersection curves with semistable kernel bundle, and also combined with work of Farkas-Mustaţǎ-Popa to relate the existence of Ulrich bundles on $X_d$ to the Minimal Resolution Conjecture for curves lying on $X_d.$ In particular, we show that a smooth irreducible curve $C$ of degree $3r$ lying on a smooth cubic surface $X_3$ represents the first Chern class of an Ulrich bundle on $X_3$ if and only if the Minimal Resolution Conjecture holds for $C.$

23 pages, section added on the case of Arithmetically Gorenstein surfaces and minor revisions

References in corpus (2)

• n-dimensional Fano varieties of wild representation type
• Stable Ulrich bundles

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• Characterization of Ulrich bundles on Hirzebruch surfaces
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• Equivariant Ulrich bundles on flag varieties
• Ulrich bundles on the degree six Segre fourfold
• Stable Ulrich Bundles on Fano Threefolds with Picard Number 2
• A remark on Ulrich and ACM bundles
• Ulrich bundles on rational surfaces with an anticanonical pencil
• Ulrich modules over cyclic quotient surface singularities