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Abstract The Brill–Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill–Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill–Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $$E$$ be a semistable sheaf on $${\mathbb{P}}^{2}$$. In this paper, we give an upper bound $$\beta _{r, \mu }$$ for $$h^{0}(E)$$ in terms of the rank $$r$$ and the slope $$\mu $$ of $$E$$. We show that the bound is achieved precisely when $$E$$ is a twist of a Steiner bundle. We classify the sheaves $$E$$ such that $$h^{0}(E)$$ is within $$\lfloor \mu (E) \rfloor + 1$$ of $$\beta _{r, \mu }$$. We determine the nonemptiness, irreducibility and dimension of the Brill–Noether loci in the moduli spaces of sheaves on $${\mathbb{P}}^{2}$$ with $$h^{0}(E)$$ in this range. When they are proper subvarieties, these Brill–Noether loci are irreducible though almost always of larger than the expected dimension.
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This content will become publicly available on October 1, 2026
Maximal Brill–Noether Loci via K3 Surfaces
Abstract The Brill–Noether loci $$\mathcal{M}^{r}_{g,d}$$ parameterize curves of genus $$g$$ admitting a linear system of rank $$r$$ and degree $$d$$. When the Brill–Noether number is negative, they are proper subvarieties of the moduli space of genus $$g$$ curves. We explain a strategy for distinguishing Brill–Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill–Noether loci. Via an analysis of the stability of Lazarsfeld–Mukai bundles, we obtain new lifting results for line bundles of type $$g^{3}_{d}$$ that suffice to prove the maximal Brill–Noether loci conjecture in genus $$3$$–$19$, $22$, $23$, and infinitely many cases.
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- Award ID(s):
- 2200845
- PAR ID:
- 10650604
- Publisher / Repository:
- https://arxiv.org/abs/2206.04610
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 20
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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