On the Brill--Noether Theory of Sheaves on Regular Surfaces

@misc{geiger2026brillnoether,
  title = {{On the Brill--Noether Theory of Sheaves on Regular Surfaces}},
  author = {Samir Geiger and Andreas Kretschmer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06824},
  eprint = {2609.06824},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06824},
}

TY  - UNPB
TI  - On the Brill--Noether Theory of Sheaves on Regular Surfaces
AU  - Samir Geiger
AU  - Andreas Kretschmer
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06824
UR  - https://arxiv.org/abs/2609.06824
AB  - We study the Brill--Noether theory of higher rank sheaves on surfaces X with H^1(X,𝒪_X) = 0 and its interactions with the classical Brill--Noether theory of line bundles on curves lying on X. Denoting by Spl(v) the moduli space of simple sheaves on X with Chern character v = (r,c_1,ch_2), we introduce the Brill--Noether loci BN^k(v)^∘ ⊆ Spl(v) of \emph{generically k-generated} sheaves and show, under assumptions on c_1 and K_X, certain well-behavedness results. In particular, if c_1 is \emph{indecomposable} and X is a K3 surface, we generalize a Brill--Noether theorem for sheaves on K3s due to Yoshioka. This is achieved by employing He's deformation theory of the moduli space of coherent systems and a globalization of the Lazarsfeld--Mukai construction, providing a correspondence between the Brill--Noether theories of generically r-generated sheaves on X and that of line bundles on curves C ∈ |c_1|. The same techniques yield upper bounds on the dimension of 𝒲^r,bpf_d(|c_1|) ∖ 𝒲^r+1_d(|c_1|) consisting of basepoint-free line bundles in terms of the number of endomorphisms of associated Lazarsfeld--Mukai bundles, for arbitrary c_1. This generalizes a result of Aprodu and Farkas in the context of Green's conjecture beyond K3 surfaces. In the case r = 1 of pencils, we obtain smoothness and expected-dimension results under an effectivity condition on −K_X|_C for C ∈ |c_1|.
ER  -