We develop a theory of Brill-Noether divisors on the moduli space of stable spin curves of genus g, and compute the classes of these loci. A spin Brill-Noether cycle is defined in terms of the relative position of the spin structure with respect to certain Raynaud theta-divisors in the Jacobian of the curve.
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.
We extend a previous result of Feyzbakhsh concerning the injectivity of a map of moduli spaces and we use this result to construct curves whose Brill-Noether loci have unexpected dimension.
In this paper we deal with Brill-Noether theory for higher-rank sheaves on a polarized nodal reducible curve $(C,\underline{w})$ following the ideas of [arXiv:alg-geom/9511003v1]. We study the Brill-Noether loci of $\underline{w}$-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.
We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety $G^{r}_{d}(C)$ that parametrizing linear series of degree $d$ and dimension $r$ on a general smooth projective curve $C$ is indecomposable when $d\leq g(C)-1$.
Let $M_{\mathbb{P}^2}(v)$ be a moduli space of semistable sheaves on $\mathbb{P}^2$, and let $B^k(v) \subseteq M_{\mathbb{P}^2}(v)$ be the \textit{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 $dim B^k(v) = dim M_{\mathbb{P}^2}(v) - k(k - χ(E))$. When $c_1 > 0$, we show that the Brill-Noether locus $B^r(v)$ is nonempty. Wh…
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 $r$.
We show that a general curve in an explicit class of what we call Du Val pointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, we prove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noether divisors on the universal curve. This provides explicit examples of smooth pointed curves of arbitrary genus defined over Q which are Brill-Noether general. A similar result is proved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces.
We prove that the pullbacks of the virtual fundamental classes of the Brill-Noether loci under any Abel-Jacobi section lie in the tautological ring of the moduli space of stable curves. This resolves a conjecture of Pagani, Ricolfi and van Zelm, and is part of a broader program to understand the logarithmic intersection theory of $\Mgn$.
We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill--Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai bundles of minimal $c_2$.
We use results of M. Aprodu and E. Sernesi to extend a result by Fulton--Harris--Lazarsfeld in Brill--Noether theory of line bundles %and, as well, a result by Aprod-Sernesi in theory of Secant Loci, to Brill--Noether loci of stable bundles inside the moduli space of rank two stable vector bundles on a smooth projective algebraic curve. As a consequence; if $B^k_{2, d}$ is of expected dimension, then we prove that its smoothness is equivalent with the smoothness of $B^{k}_{2, d+1}$.
We prove the injectivity of the Petri map for linear series on a general curve with given ramification at two generic points. We also describe the components of such a set of linear series on a chain of elliptic curves.
We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear series of degree d and rank r on a general curve of genus g is an embedding if r is at least 3. If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of \(f_* L\) provides the appropriate generalization of the pair (r, d) in classical Brill--Noether theory. In…
