Brill-Noether Theory for Stable Vector Bundles

Abstract

Abstract. This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves.

Brill–Noether theory on Hirzebruch surfaces

“…Indeed, as in [10] we can carry out the constructions locally, show the independence of the choice of the locally universal sheaf and conclude that the construction glue as a global algebraic object.…”

“…The kernel of the idea of the classical Brill-Noether theory is found at least as early as the work of Brill and Noether in the 19th century and deals with line bundles on algebraic curves (see [1]). Its first natural generalization concerns higher rank vector bundles on algebraic curves and in this context, the theory has been extensively developed during the last decades by several authors (see the overview [10] and references quoted there). Very recently, the foundation of a generalized Brill-Noether theory for moduli space of rank r ≥ 2 stable vector bundles on higher-dimensional varieties has been formulated [3,17,18,20] and very interesting problems have been settled.…”

Singularities of Brill‐Noether loci for vector bundles on a curve

“…It is still not fully understood when these subschemes have the expected dimension, or even when they are nonempty. We refer the reader to 12 for more details on this question.…”

“…In another direction, Riemann's result was generalized in 21 by Laszlo, where it was shown that if d = r ( g − 1), and [ E ] ∈ B 1 r , r ( g − 1) , then \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\rm {mult}_{[E]} B^1_{r,r(g-1)} =h^0(C,E)$\end{document}. We point out that the Zariski tangent spaces to the B k r , d are well understood: at a point E ∈ B k r , d , if E ∉ B k + 1 r , d , the tangent space to B k r , d at E is where is the cup product mapping, and if E ∈ B k + 1 r , d , then \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$T_EB^k_{r,d}=T_E\mathscr{U}(r,d)$\end{document} (see 12 for more details). Using work of Kempf 16 as motivation, the following extends the results above by giving a description of C E B k r , d , the tangent cone to B k r , d at E .…”

Coherent Systems on Algebraic Curves

“…In general, I have not included papers concerned solely with Brill-Noether theory except where they are needed for reference, but I have included some which have clear relevance for coherent systems but have not been fully developed in this context. For another survey of coherent systems, concentrating on structural results and including an appendix on the cases g = 0 and g = 1, see [13]; for a survey of higher rank Brill-Noether theory, see [27].…”