Special Divisors on Real Trigonal Curves

TL;DR

This work addresses the topology of real Brill-Noether loci $W_d^r$ on real trigonal curves, focusing on the number of connected components of their real points. The authors leverage the Maroni invariant $m$ and the real Hirzebruch scroll $Y=\Sigma_{g-2-2m}$ to decompose $W_d^r$ into two natural loci $U_d^r$ and $V_d^r$, enabling combinatorial counts via Harris–Gross type arguments. They establish sharp bounds $s_{n(X)}(d-3r)\le n(W_d^r)\le s_{n(X)}(d-3r)+\binom{n(X)-1}{2d-g-3r+1}$ under $0<g-d+r-1\le m\le d-2r-1$, with an exact value when $2g-3d+3r=0$, and discuss how admissible $(d,r)$ pairs align along a line $2g-3d+3r+1=0$. A concrete genus $5$ example with $n(W_4^1)$ is presented to illustrate the intersection structure of components. These results quantify real Brill-Noether topology on trigonal curves and connect component counts to explicit geometric data such as base points of $K_X-(g-d+2r-1)T$.

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.

Figures (1)

• Figure 1: Admissible $(d,r)$ pairs

Theorems & Definitions (29)

• Definition 1: Huisnonspe, Page 23
• Example 1
• Lemma 1
• proof
• Proposition 1
• proof
• Remark 1
• Lemma 2
• proof
• Theorem 1