Real normalized differentials and Arbarello's conjecture

TL;DR

This paper proves Arbarello's conjecture by employing meromorphic differentials with real periods to show that certain cycles in the moduli space of curves intersect loci with specific Weierstrass points.

Contribution

It introduces a novel approach using real normalized differentials to prove a longstanding conjecture in algebraic geometry.

Findings

Proof of Arbarello's conjecture for all genera

Establishment of a link between real differentials and intersection properties

Advancement in understanding the geometry of moduli spaces

Abstract

Using meromorphic differentials with real periods, we prove Arbarello's conjecture: any compact complex cycle of dimension $g-n$ in the moduli space $\M_g$ of smooth genus $g$ algebraic curves must intersect the locus of curves having a Weierstrass point of order at most $n$.