The strong maximal rank conjecture and moduli spaces of curves
Abstract
Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus and . This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus and are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.
Keywords
Cite
@article{arxiv.1808.01290,
title = {The strong maximal rank conjecture and moduli spaces of curves},
author = {Fu Liu and Brian Osserman and Montserrat Teixidor i Bigas and Naizhen Zhang},
journal= {arXiv preprint arXiv:1808.01290},
year = {2024}
}
Comments
59 pages. v3: Farkas has informed us that his divisor class computation does not suffice to reduce the moduli spaces being of general type to the corresponding cases of the strong maximal rank conjecture: a second type of excess degeneracy also has to be ruled out. We have accordingly removed the implications on the moduli spaces pending further study of the second form of degeneracy