Vector bundles on bielliptic surfaces: Ulrich bundles and degree of irrationality
Edoardo Mason
TL;DR
The paper addresses Ulrich bundles on bielliptic surfaces and the degree of irrationality. It develops a Mukai-lattice framework to classify Ulrich vector characters and proves weak Brill-Noether results for line bundles and certain isotropic Mukai vectors, yielding a detailed type-dependent existence theory for Ulrich bundles with vectors ${\bf v}^{\mathrm{Ulrich}}(r,k)=(r,kaA_0+(3r-k)bB_0,2rab)$. It then applies a Moretti-style construction via stable rank-2 bundles to bound the degree of irrationality, proving $\mathrm{irr}(S)\le 3$ for all bielliptic surfaces and recovering the known $\mathrm{irr}(S)=2$ for types 1 and 2. Together, these results provide a uniform, birationally-informed account of Ulrich cohomology vanishing and the birational geometry of bielliptic surfaces across all topological types.
Abstract
This paper deals with two problems about vector bundles on bielliptic surfaces. The first is to give a classification of Ulrich bundles on such surfaces $S$, which depends on the topological type of $S$. In doing so, we study the weak Brill-Noether property for moduli spaces of sheaves with isotropic Mukai vector. Adapting an idea of Moretti, we also interpret the problem of determining the degree of irrationality of bielliptic surfaces in terms of the existence of certain stable vector bundles of rank 2, completing the work of Yoshihara.