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

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.

22 pages, 4 figures. Final version. Minor corrections. To appear in the European Journal of Mathematics

References in corpus (3)

• Extremal higher codimension cycles on moduli spaces of curves
• A simplicial approach to effective divisors in ${\overline{M}}_{0,n}$
• Effective curves on ${\overline{M}}_{0,n}$ from group actions