It is an open problem to describe the shape of the effective cone of an algebraic surface. Nagata conjecture predicts part of this shape when the surface is the blow-up of the projective plane at general points. More recently Ciliberto and Kouvidakis proved that Nagata conjecture implies that the two-dimensional effective cone of the symmetric product C_2 of a general, genus g > 9, curve C is open on one side whenever g is not a square. In this talk I will show that the effective cone of the blow-up of C_2 at a general point is non-polyhedral for a general positive genus curve C. This result generalizes previous statements of J.F. Garcia and G. McGrat about the genus 1 case. To prove the statement we first show that having polyhedral effective cone is a closed property for families of surfaces having the same Picard group and then we prove it in the hyperelliptic case. This is joint work with Luca Ugaglia.

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