Some reducible and irreducible Brill-Noether loci

TL;DR

The paper develops a constructive framework for Brill-Noether loci via limit linear series on chains of elliptic curves, enabling explicit creation of theta-characteristics and analysis of locus reducibility. It provides a streamlined proof of the Farkas conjecture on $r$-dimensional theta-characteristics for $g\ge{r+2\choose 2}$ and derives new reducible Brill-Noether loci, including components of $\mathcal{M}^r_{g,g-1}$ arising from half-canonical and non-half-canonical cases. It also establishes irreducibility results for $r=2$ in broad parameter ranges, with sharp counterexamples, and shows the existence of at least three components in certain Brill-Noether loci for specific $r$ and $g$. Altogether, the work exposes rich, multi-component structures in Brill-Noether theory and highlights the utility of limit linear series on elliptic chains for moduli problems and Hilbert-scheme components.

Abstract

We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera. Using the additional structure afforded by considering limit linear series on chains of elliptic curves, we find examples of reducible Brill-Noether loci, admitting at least two components, with and without a theta-characteristic respectively. This allows us to display reducible Hilbert schemes for $r\ge 3$ and the largest possible value of $d$, namely $d=g-1$. We also give examples of Brill-Noether loci with three components. On the positive side, we provide optimal bounds on the degree under which Brill-Noether loci are irreducible when $r=2$.

Figures (5)

• Figure 1: Example of the filling of a rectangle corresponding to a limit linear series (a limit $g^{3}_{9}$) on a chain of 10 elliptic curves.
• Figure 2: Naive fillings of the rectangle when $r=2, 3, 4,5,6, 7$.
• Figure 3: Replacing a pair of elliptic curves with a curve of genus two.
• Figure 4: Filling of the rectangle when $r=3, 4, 5, 6, 7, 8$ giving $N^r_{g,g-1}$.
• Figure 5: Admissible filling giving a $g^{2}_{7}$.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1.1
• Theorem 2.1
• proof
• Definition 2.2
• Definition 3.1
• Theorem 3.2
• proof