On the Cox ring of Del Pezzo surfaces

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.

17 pages

Cited by in corpus (6)

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• On the 10-web by conics on the quartic del Pezzo surface
• The blow-up of ${\mathbb{P}}^4$ at 8 points and its Fano model, via vector bundles on a del Pezzo surface
• Hilbert functions of Cox rings of del Pezzo surfaces