Brill Noether Loci Research Articles

New examples of twisted Brill–Noether loci I

Our purpose in this paper is to construct new examples of twisted Brill–Noether loci on curves of genus [Formula: see text]. Many of these examples have negative expected dimension. We deduce also the existence of a new region in the Brill–Noether map, whose points support non-empty standard Brill–Noether loci.

On the local structure of the Brill–Noether locus of locally free sheaves on a smooth variety

We study the functor \operatorname{Def}_{E}^{k} of infinitesimal deformations of a locally free sheaf E of \mathcal{O}_{X} -modules on a smooth variety X , such that at least k independent sections lift to the deformed sheaf. We deduce some information on the k th Brill–Noether locus of E , such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor \operatorname{Def}_{E}^{k} and the smoothness of some well-known deformation functors and their associated moduli spaces. As a tool for the investigation of \operatorname{Def}_{E}^{k} , we study infinitesimal deformations of the pairs (E,U) , where U is a linear subspace of sections of E . We generalise many classical results concerning the moduli space of coherent systems to the case where E has any rank and X any dimension. This includes a description of its tangent space and the link between smoothness and the injectivity of the Petri map.

Brill–Noether Theory of Hilbert Schemes of Points on Surfaces

Abstract We show that Brill–Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$, where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino’s description [9] of the Hilbert–Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill–Noether loci given by nested Hilbert schemes.

Euler characteristics of Brill–Noether loci on Prym varieties

In this article we compute formulas for the connected K-theory class of the pointed Brill–Noether loci in Prym varieties, which extend the result by De Concini and Pragacz. Applying the formulas, we compute the holomorphic Euler Characteristics of the loci.

Higher-rank Brill-Noether loci on nodal reducible curves

In this paper, we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve (C,{{,mathrm{underline{w}},}}) following the ideas of Brambila-Paz et al. (J Algebraic Geom 6(4): 645–669, 1997). We study the Brill-Noether loci of {{,mathrm{underline{w}},}}-stable depth one sheaves on C having rank r on all irreducible components and having small slope. In analogy with what happens in the smooth case, we prove that these loci are closely related to BGN extensions. Moreover, we produce irreducible components of the expected dimension for these Brill-Noether loci.

Higher Rank Brill–Noether Theory on ℙ2

Abstract Let $M_{{\mathbb {P}}^{2}}(\textbf {v})$ be a moduli space of semistable sheaves on ${\mathbb {P}}^{2}$, and let $B^{k}(\textbf {v}) \subseteq M_{{\mathbb {P}}^{2}}(\textbf {v})$ be the Brill–Noether locus of sheaves $E$ with $h^{0}({\mathbb {P}}^{2}, E) \geq k$. In this paper, we develop the foundational properties of Brill–Noether loci on ${\mathbb {P}}^{2}$. Set $r = r(E)$ to be the rank and $c_{1}, c_{2}$ the Chern classes. The Brill–Noether loci have natural determinantal scheme structures and expected dimensions $\dim B^{k}(\textbf {v}) = \dim M_{{\mathbb {P}}^{2}}(\textbf {v}) - k(k - \chi (E))$. When $c_{1}> 0$, we show that the Brill–Noether locus $B^{r}(\textbf {v})$ is nonempty. When $c_{1} = 1$, we show all of the Brill–Noether loci are irreducible and of the expected dimension. We show that when $\mu = c_{1}/r> 1/2$ is not an integer and $c_{2} \gg 0$, the Brill–Noether loci are reducible and describe distinct irreducible components of both expected and unexpected dimension.

Brill-Noether loci with ramification at two points

Brill-Noether loci with ramification at two points

Classifying the irreducible components of moduli stacks of torsion free sheaves on K3 surfaces and an application to Brill-Noether theory

In this article, we classify the irreducible components of moduli stacks of torsion free sheaves of rank 2 on K3 surfaces of Picard number 1. For ruled surfaces, the components of moduli stacks of torsion free sheaves were classified by Walter ([18]). Moreover, by virtue of our result, we classify the irreducible components of Brill-Noether loci of Hilbert schemes of points on K3 surfaces.

Irreducibility of a Universal Prym–Brill–Noether Locus

Abstract For genus $g = \frac {r(r+1)}{2}+1$, we prove that via the forgetful map, the universal Prym–Brill–Noether locus $\mathcal {V}^r_g$ has a unique irreducible component dominating the moduli space $\mathcal {R}_g$ of Prym curves.

Extension groups of tautological bundles on symmetric products of curves

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if Ene mathcal O_X is simple, then the natural map {{,mathrm{mathsf {Ext}},}}^1(E,E)rightarrow {{,mathrm{mathsf {Ext}},}}^1(E^{[n]},E^{[n]}) is injective for every n. Along with previous results, this implies that Emapsto E^{[n]} defines an embedding of the moduli space of stable bundles of slope mu notin [-1,n-1] on the curve X into the moduli space of stable bundles on the symmetric product X^{(n)}. The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X^{(n)} where the dimension of the tangent space jumps. We also prove that E^{[n]} is simple if E is simple.

Geometry of certain Brill–Noether locus on a very general sextic surface and Ulrich bundles

Let \(X \subset {\mathbb {P}}^3\) be a very general sextic surface over complex numbers. In this paper, we study certain Brill–Noether problems for moduli of rank 2 stable bundles on X and its relation with rank 2 weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill–Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank 2 weakly Ulrich bundles \({\mathcal {E}}\) on X with \(c_1({\mathcal {E}}) =5H\) and \(c_2 \ge 91\) and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill–Noether loci. Finally, using the existence of rank 2 Ulrich bundles on X we obtain some more non-empty Brill–Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on X.

Betti numbers of Brill–Noether varieties on a general curve

We compute the rational cohomology groups of the smooth Brill–Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill–Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.

Linear series on a curve of compact type bridged by a chain of elliptic curves

In the present paper we investigate conditions for the non-existence of a smoothable limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. Combining this work with results on the existence of a smoothable limit linear series on such a curve, we show relations among Brill–Noether loci of codimension at most two in the moduli space of complex curves. Specifically, Brill–Noether loci of codimension two have mutually distinct supports.

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

Birational geometry of the Mukai system of rank two and genus two

Using the techniques of Bayer–Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface S of genus two. We study the (essentially unique) birational map to S^{[5]} and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill–Noether loci.

A NOTE ON RANK TWO STABLE BUNDLES OVER SURFACES

Abstract.Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

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

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.

A refined Brill–Noether theory over Hurwitz spaces

Let $$f:C \rightarrow \mathbb {P}^1$$ be a degree k genus g cover. The stratification of line bundles $$L \in {{\,\mathrm{Pic}\,}}^d(C)$$ by the splitting type of $$f_*L$$ is a refinement of the stratification by Brill–Noether loci $$W^r_d(C)$$ . We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $$W^r_d(C)$$ for a general k-gonal curve (there are often components of different dimensions), extending results of Pflueger (Adv Math 312:46–63, 2017) and Jensen and Ranganathan (Brill–Noether theory for curves of a fixed gonality, arXiv:1701.06579 , 2017). The results here apply over any algebraically closed field.

Brill–Noether loci on moduli spaces of symplectic bundles over curves

The symplectic Brill–Noether locus $${{{\mathcal {S}}}}_{2n, K}^k$$ associated to a curve C parametrises stable rank 2n bundles over C with at least k sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of $${{{\mathcal {S}}}}_{2n, K}^k$$ . We show the nonemptiness of several $${{{\mathcal {S}}}}_{2n, K}^k$$ , and in most of these cases also the existence of a component which is generically smooth and of the expected dimension. As an application, for certain values of n and k we exhibit components of excess dimension of the standard Brill–Noether locus $$B^k_{2n, 2n(g-1)}$$ over any curve of genus $$g \ge 122$$ . We obtain similar results for moduli spaces of coherent systems.

Components of Brill–Noether Loci for Curves with Fixed Gonality

We describe a conjectural stratification of the Brill–Noether variety for general curves of fixed genus and gonality. As evidence for this conjecture, we show that this Brill–Noether variety has at least as many irreducible components as predicted by the conjecture and that each of these components has the expected dimension. Our proof uses combinatorial and tropical techniques. Specifically, we analyze containment relations between the various strata of tropical Brill–Noether loci identified by Pflueger in his classification of special divisors on chains of loops.