Abstract
We study (−1)-classes on the blow-up X of ℙ n at points in very general position; in the case n = 2, we give a new proof of the equivalence of two conjectures about the dimension of the class of a divisor on X, and we prove that h 1(D) = 0 for any nef D which is a nonnegative sum of (−1)-curves. For n = 3, we fill a gap in the formulation of a conjecture about the dimension of a class. We provide algorithms and Maple code based on these conjectures.
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