Constructing Reducible Brill--Noether Curves II

TL;DR

This paper establishes criteria for the existence of reducible Brill--Noether curves with specific properties, advancing the understanding of stable maps and contributing to the proof of the Maximal Rank Conjecture.

Contribution

It provides new criteria for the existence of reducible Brill--Noether curves with given degrees and genera, refining previous results and aiding in the Maximal Rank Conjecture proof.

Findings

Criteria for existence of Brill--Noether curves of a certain type

Sharpened conditions for Brill--Noether space curves

Application to the Maximal Rank Conjecture

Abstract

In this paper, we study maps from reducible curves $f : C \cup_\Gamma D \to \mathbb{P}^r$. We restrict our attention to two cases: first, when $f|_D$ factors through a hyperplane $H$ and $f|_C$ is transverse to $H$; and second, when $r = 3$. Degeneration to stable maps of this type have played a crucial role in works of Hartshorne, Ballico, and others, on special cases of the maximal rank conjecture. However, the general problem of studying when such stable maps with specified combinatorial types exist remains open. Here, we give criteria for such Brill--Noether curves of this first type to exist, of specified degree $d$ and genus $g$, such that $f|_C$ is of specified degree $d'$ and genus $g'$. We also give criteria, sharpening earlier results of the author, for the existence of Brill--Noether space curves of specified combinatorial types. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.