The geometry of Ulrich bundles on del Pezzo surfaces

• DOI: 10.1016/j.jalgebra.2012.08.032
• Publication Date: 2012-11-30T00:34:22Z
• Authors: Emre Coskun; Rajesh S. Kulkarni; Yusuf Mustopa
• Keywords: Complete intersection; Kernel (algebra); Degree (music); Rank (graph theory); Splitting principle

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-Musta����-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 $C.$<br/>23 pages, section added on the case of Arithmetically Gorenstein surfaces and minor revisions<br/>