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1. Brill–Noether Theory on ℙ2 for Bundles with Many Sections

   https://doi.org/10.1093/imrn/rnaf064  

   Coskun, Izzet; Huizenga, Jack; Raha, Neelarnab (March 2025, International Mathematics Research Notices)

   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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2. 𝐾-classes of Brill–Noether Loci and a Determinantal Formula

   https://doi.org/10.1093/imrn/rnab025  

   Anderson, Dave; Chen, Linda; Tarasca, Nicola (April 2021, International Mathematics Research Notices)

   Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $$\rho $$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $$\rho =1$$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $$K$$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux. 

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3. Birational geometry of Beauville–Mukai systems III: Asymptotic behavior

   https://doi.org/10.1112/blms.13158  

   Qin, Xuqiang; Sawon, Justin (September 2024, Bulletin of the London Mathematical Society)

   Abstract Suppose that a Hilbert scheme of points on a K3 surface of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of . We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme. 

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4. The moduli spaces of sheaves on surfaces, pathologies and brill-noether problems

   https://doi.org/10.1007/978-3-319-94881-2_4  

   Cosun, I.; Huizenga, J. (November 2018, Abel symposia)

   Brill-Noether Theorems play a central role in the birational geometry of moduli spaces of sheaves on surfaces. This paper surveys recent work on the Brill-Noether problem for rational surfaces. In order to highlight some of the difficulties for more general surfaces, we show that moduli spaces of rank 2 sheaves on very general hypersurfaces of degree d in P3 can have arbitrarily many irreducible components as d tends to infinity. 

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5. Interpolation for Brill–Noether curves

   https://doi.org/10.1017/fmp.2023.22  

   Larson, Eric; Vogt, Isabel (October 2023, Forum of Mathematics, Pi)

   Abstract In this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed. 

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