Special Divisors on Real Trigonal Curves

TL;DR

This paper investigates the topology of Brill-Noether varieties on real trigonal curves, focusing on counting the connected components of their real loci under specific numerical conditions.

Contribution

It provides a new count of connected components of real Brill-Noether varieties for trigonal curves satisfying certain relations between invariants.

Findings

Count of connected components under given conditions

Topological characterization of real loci

Extension of classical Brill-Noether theory to real trigonal curves

Abstract

In this paper we examine the topology of Brill-Noether varieties associated to real trigonal curves. More precisely, we aim to count the connected components of the real locus of the varieties parametrizing linear systems of degree $d$ and dimension at least $r$. We do this count when the relations $m=g-d+r-1\leq d-2r-1$ are satisfied, where $m$ is the Maroni invariant and $g$ is the genus of the curve.