Related papers: On Treibich-Verdier curves

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…

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…

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…

This article classifies Knutsen K3 surfaces all of whose hyperplane sections are irreducible and reduced. As an application, this gives infinite families of K3 surfaces of Picard number two whose general hyperplane sections are…

We proved that the general members of Severi varieties on an Atiyah ruled surface over a general elliptic curve have nodes and ordinary triple points as singularities.

Mukai showed that projective models of Brill-Noether general polarized K3 surfaces of genus $6-10$ and $12$ are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill-Noether general…

In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\em singular} Brill-Noether number $\rho_{sing}$ and show that if the Picard group of the K3 surface…

We completely describe the Brill-Noether theory of pencils on general primitive covers of elliptic curves of any degree.

In this paper we study examples of P^r-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from classical Lazarsfeld's…

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…

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.

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…

Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two…

In \cite{kim11} we have generalized a tangency condition in the Treibich-Verdier theory \cite{trei89,tv90,trei97} about elliptic solitons to a Hitchin system. As an application of this generalization, we will define, so-called, Hitchin…

We use the Brill-Noether theory to prove the Green conjecture for exceptional curves on K3 surfaces. Such curves count among the few ones having Clifford dimension at least three. We obtain our result by adopting an infinitesimal approach…

Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether…

Under the assumption that the adjusted Brill-Noether number $\widetilde{\rho}$ is at least $-g$, we prove that the Brill-Noether loci in $\mathcal{M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked…

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 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…

We study the geometry of a class of $n$-dimensional smooth projective varieties constructed by Schreieder for their noteworthy Hodge-theoretic properties. In particular, we realize Schreieder's surfaces as elliptic modular surfaces and…