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

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.

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