Brill-Noether-type Theorems with a Movable Ramification Point

A simple characteristic-free proof of the Brill-Noether theorem

We describe how the use of a different degeneration from that considered by Eisenbud and Harris leads to a simple and characteristic-independent proof of the Brill-Noether theorem using limit linear series. As suggested by the degeneration, we prove an extended version of the theorem allowing for imposed ramification at up to two points. Although experts in the field have long been aware of the main ideas, we address some technical issues which arise in proving the full version of theorem.

An Effective Divisor in $\overline{M_{g}}$ Defined by Ramification Conditions

We define an effective divisor of the moduli space of stable curves $\overline{M_g}$, which is denoted $\overline{S^{2}W}$. Writing the class of $\overline{S^{2}W}$ in the Picard group of the moduli functor Pic$_{\text{fun}}(\overline{M_{g}})\otimes \mathbb{Q}$ in terms of the so-called Harer basis $λ,δ_0,\ldots,δ_{[g/2]}$, we prove that the relations among the coefficients of $δ_1,\ldots,δ_{[g/2]}$ are the same relations on coefficients as the Brill-Noether divisors. We present a result on effective divisors of $\overline{M_g}$ which could be useful to get the same relations on coefficients f…

投射线的完全分枝覆盖的Brill-Noether理论

Brill-Noether theory for totally ramified covers of the projective line

研究完全分枝覆盖的线束空间及其分枝特性，通过扩展的k-仿射对称群参数化传输领域，证明Pflueger关于传输领域的Brill-Noether猜想。

Brill-Noether旗的通用性与双标记曲线的退化轨迹

Versality of Brill-Noether flags and degeneracy loci of twice-marked curves

本文研究一般双标记曲线上Brill-Noether退化轨迹的几何性质，通过证明一对旗的通用性定理，确定了其维数、奇异轨迹和相交类，并给出相交类的组合解释。

二元曲线的Brill-Noether一般性

Brill-Noether Generality of Binary Curves

本文证明一般二元曲线上多度数线性级数空间具有预期维数，推广至任意秩，并验证施加分歧条件后维数不变。

Euler characteristics of Brill-Noether varieties

We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramification profiles specified at up to two general points. Up to sign, this Euler characteristic is the number of standard set-valued tableaux of a certain skew shape with g labels. We use a flat degeneration via the Eisenbud-Harris theory of limit linear series, relying on moduli-theoretic advances of Osserman and Murray-Osserman; the count of set-valued tableaux is an explicit enumeration of strata of this degene…

Ulrich sheaves and higher-rank Brill-Noether theory

An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory, generalized Clifford algebras, and for an approach to Lech's conjecture in commutative algebra. In this note, we give a reduction of the construction of Ulrich sheaves on a projective variety X to the construction of an Ulrich sheaf for a finite map of curves, which is in turn equivalent to a higher-rank Brill-Noether problem for any of a certain class of c…

Genera of Brill-Noether curves and staircase paths in Young tableaux

In this paper, we compute the genus of the variety of linear series of rank $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $α$ and $β$ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.

Special divisors on marked chains of cycles

We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill-Noether theorem for general marked curve…