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 $22$ and $23$. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus $22$ and $23$ 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.

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

Cited by in corpus (4)

• Limit linear series and ranks of multiplication maps
• The Strong Maximal Rank Conjecture and higher rank Brill--Noether theory
• On the strong maximal rank conjecture in genus 22 and 23
• Some reducible and irreducible Brill-Noether loci