The Brill-Noether theory of the moduli spaces of sheaves on surfaces (Q6607618)

The main objects of Brill-Noether theory are families of coherent sheaves on a projective variety having prescribed cohomological behavior. When the variety is a smooth curve of genus at least \(2,\) one has that for all \((\chi,\mu) \in (\mathbb{Z}-\{0\}) \times \mathbb{Q}\) the moduli space of stable bundles with Euler characteristic \(\chi\) and slope \(\mu\) is nonempty, irreducible, and has the \textit{weak Brill-Noether property}, i.e. its general member has at most one nonzero cohomology group.\N\NThe paper under review gives an up-to-date survey of what is known when the variety is a smooth surface; here, the situation is more complicated. Not only does the notion of stability have to be specified, but numerical invariants of stable sheaves now have nontrivial restrictions, and the moduli spaces may be reducible or even disconnected. Weak Brill-Noether need not hold for the components of these moduli spaces, and the authors' choice of topics is centered around the problem of determining when it does.\N\NAfter briefly reviewing the curve case, the authors present examples demonstrating how weak Brill-Noether on surfaces can fail even for sheaves of rank 1. They then discuss the geometry (and to a lesser extent, the topology) of the higher-rank moduli spaces before turning to weak Brill-Noether theorems for surfaces. Prioritary sheaves play a key role in the results on rational surfaces, while Bridgeland stability plays a key role in the results on K-trivial surfaces. Though the weak Brill-Noether problem for surfaces remains open in general, one has an asymptotic statement for the case of large discriminant.\N\NThe final two sections treat the classification of stable Chern characters, the tensor product problem (which simultaneously generalizes the weak Brill-Noether and interpolation problems), the construction of Ulrich sheaves, and cohomology jumping loci in moduli spaces for which weak Brill-Noether holds.\N\NThe authors have succeeded in covering a diverse array of material while keeping the essential issues within view, and in providing a clear path from basic theory to recent results and open questions.\N\NFor the entire collection see [Zbl 1545.14003].