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On the Cox ring of Del Pezzo surfaces

Ulrich Derenthal
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Abstract

Let S_r be the blow-up of P^2 in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. For r <= 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S_8) is calculated up to radical. As conjectured by Batyrev and Popov, all the generating relations are quadratic.

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Ulrich Derenthal. 2006-03-04. On the Cox ring of Del Pezzo surfaces. https://arxiv.org/abs/math/0603111

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