Constructing Reducible Brill--Noether Curves

TL;DR

This paper develops a method to construct reducible curves passing through general points in projective space and proves these are limits of nondegenerate curves, aiding in the proof of the Maximal Rank Conjecture.

Contribution

It introduces a construction technique for reducible BN-curves via gluing nonspecial curves through general points, expanding the toolkit for studying curve moduli.

Findings

Constructed reducible curves passing through general points

Proved these curves lie in the closure of nondegenerate curves

Applied results to the Maximal Rank Conjecture

Abstract

It was recently determined exactly through how many general points a nondegenerate curve with nonspecial hyperplane section can pass. This gives rise to a method of constructing reducible curves $C_1 \cup_\Gamma C_2 \to \mathbb{P}^r$ with general nodes: We take a finite set $\Gamma \subset \mathbb{P}^r$ of general points, and find nondegenerate nonspecial curves $C_1$ and $C_2$ in $\mathbb{P}^r$ of specified degrees and genera which pass through $\Gamma$, and glue together along $\Gamma$. The goal of this paper is to show that, subject to certain mild assumptions, stable maps constructed in this manner lie in the closure of the locus of nondegenerate stable maps from curves of general moduli, i.e. are BN-curves. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.