Related papers: Special Divisors on Real Trigonal Curves

We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for…

For $r\geq 3$ and $g= \frac{r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta)$ associated to Prym curves $[C,\eta]$. The locus $\mathcal{R}_g^r$ in $\mathcal{R}_g$ parametrizing Prym curves $(C, \eta)$ with nonempty…

Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute…

The Brill-Noether Theorem gives necessary and sufficient conditions for the existence of a linear series. Here we consider a general n-fold, etale cyclic cover p of a curve C of genus g and investigate for which numbers r,d a linear series…

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on…

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 $g\geq2$. We give an explicit…

Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill--Noether theory addresses this question via linear series,…

The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification…

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…

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the…

We investigate the Brill-Noether theory of rank-two, degree-$d$ stable vector bundles of speciality $3$ on a general $\nu$-gonal curve of genus $g$, $3 \leq \nu < \lfloor \frac{g+3}{2} \rfloor$. Our approach leverages universal extension…

We study special linear systems called "very special" whose dimension does not satisfy a Clifford type inequality given by Huisman. We classify all these very special linear systems when they are compounded of an involution. Examples of…

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence…

We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give…

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general $\nu$-gonal curve. We classify its reduced components whose dimensions are at least the…

A refined Brill--Noether theory seeks to determine which linear series are admitted by a ``general'' curve in a particular Brill--Noether locus. However, as Brill--Noether loci are not irreducible in general, a coarse answer is given by the…

We construct curves carrying certain special linear series and not others, showing many non-containments between Brill-Noether loci in the moduli space of curves. In particular, we prove the Maximal Brill-Noether Loci conjecture in full…

The gonality sequence $(d_r)_{r\geq1}$ of a smooth algebraic curve comprises the minimal degrees $d_r$ of linear systems of rank $r$. We explain two approaches to compute the gonality sequence of smooth curves in $\mathbb{P}^1 \times…

We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then…

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^2$ of degree $n$. This map assigns to every curve…