On the cone of effective 2-cycles on $\overline{M}_{0,7}$

TL;DR

This paper investigates the structure of the cone of effective 2-cycles on the moduli space , demonstrating that known boundary strata and lifted divisors do not generate all effective 2-cycles, by providing explicit counterexamples.

Contribution

It proves that the cone of effective 2-cycles on  is strictly larger than the cone generated by boundary strata and lifted Keel-Vermeire divisors, introducing new effective cycles.

Findings

Boundary strata and lifted divisors do not generate the entire cone.

Explicit examples of effective 2-cycles not generated by known classes.

Counterexamples inspired by Castravet and Tevelev's blow-up construction.

Abstract

Fulton's question about effective $k$-cycles on $\overline{M}_{0,n}$ for $1<k<n-4$ can be answered negatively by appropriately lifting to $\overline{M}_{0,n}$ the Keel-Vermeire divisors on $\overline{M}_{0,k+1}$. In this paper we focus on the case of $2$-cycles on $\overline{M}_{0,7}$, and we prove that the $2$-dimensional boundary strata together with the lifts of the Keel-Vermeire divisors are not enough to generate the cone of effective $2$-cycles. We do this by providing examples of effective $2$-cycles on $\overline{M}_{0,7}$ that cannot be written as an effective combination of the aforementioned $2$-cycles. These examples are inspired by a blow up construction of Castravet and Tevelev.