Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective cone.The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.
Laface, A., Ugaglia, L. (2024). Effective cone of the blow up of the symmetric product of a curve. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 11, 229-242 [10.1090/bproc/196].
Effective cone of the blow up of the symmetric product of a curve
Laface, Antonio;Ugaglia, Luca
2024-01-01
Abstract
Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective cone.The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.
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