Distinguishing Brill-Noether loci

Abstract: Brill-Noether theory has since the end of the 19th century studied linear systems on smooth projective curves (or, equivalently, compact Riemann surfaces). A degree $d$ linear system of dimension $r$ on a curve $C$, called a $g^r_d$, roughly corresponds to a non-degenerate morphism $C \to \mathbb{P}^r$ of degree $d$. In the moduli space $\mathcal{M}_g$ of curves of genus $g$ one can define the Brill-Noether loci \[ \mathcal{M}^r_{g,d}:= \{ C \in \mathcal{M}_g \; : \; C \; {\rm has~a} \; g^r_d \} , \] which are closed subvarieties.

The classical Brill-Noether-Petri theorem, proved first by D. Gieseker in 1982, states that a general smooth curve $C$ of genus $g$ admits a linear system of dimension $r$ and degree $d$ if and only if the Brill–Noether number \[ \rho(g, r, d): = g - (r+ 1)(g-d+r) \geq 0.\] This can be restated as $\mathcal{M}^r_{g,d}=\mathcal{M}_g$ if and only if $\rho(g, r, d) \geq 0$. When $\rho(g, r, d) < 0$ it is known that the codimension of any component of $\mathcal{M}^r_{g,d}$ is at most $-\rho(g, r, d)$. In general surprisingly little is known about the geometry of Brill–Noether loci when $\rho(g, r, d) < 0$, in particular about the containments between them.

I will describe results from a joint work with Asher Auel and Richard Haburcak (arXiv:2406.19993), where we determine all the maximal Brill-Noether loci (wrt containment) in terms of numerical conditions on $g,r,d$ and show that they are all distinct.

algebraic geometry

Audience: researchers in the discipline

( paper )

ODTU-Bilkent Algebraic Geometry Seminars