On the cone of effective 2-cycles on M₀,₇

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Fulton's question about effective k-cycles on M₀,ₙ for 1<k<n-4 can be answered negatively by appropriately lifting to M₀,ₙ the Keel-Vermeire divisors on M₀,ₖ₊₁. In this paper we focus on the case of 2-cycles on M₀,₇, 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 M₀,₇ 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.