Related papers: Du Val curves and the pointed Brill-Noether Theore…
We completely describe the Brill-Noether theory of pencils on general primitive covers of elliptic curves of any degree.
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…
In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of…
The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct…
We completely describe the Brill-Noether theory for curves in the primitive linear system on generic abelian surfaces, in the following sense: given integers $d$ and $r$, consider the variety $V^r_d(|H|)$ parametrizing curves $C$ in the…
We prove a generalization of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve $C$ of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$. We build on…
A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we…
Severi varieties and Brill-Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface $S$ with polarization $L$ of type $(1,n)$, we prove…
Let $p_1,\dots, p_9$ be the points in $\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)$ with coordinates $$(-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr)$$ respectively.…
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…
We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general \'etale double covers of curves of genus g>=2. We then specialize to curves on Nikulin surfaces and…
In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with…
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.
Lazarsfeld proved Brill--Noether generality of any smooth curve in the linear system $|H|$ where $(X,H)$ is a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb{Z}\cdot H$. Mukai introduced the notion of Brill--Noether generality for…
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,…
This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves.
We initiate the study of Prym-Brill-Noether theory for ramified double covers, extending several key results from classical Prym-Brill-Noether theory to this new framework. In particular, we improve Kanev's results on the dimension of…
We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3…
The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and…
This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…