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Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C to mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on $mathbb{P}^1$.
We compute the integral 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
We slightly extend a previous result 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 consider the Brill-Noether locus $W_{underline d}(C)$ of line bundles of multidegree $underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $ggeq2$. We give an explicit description of the
We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a hypergeometric differential eq
Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold tot