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