The Cox ring of a Del Pezzo surface

Victor V. Batyrev; Oleg N. Popov

paper

The Cox ring of a Del Pezzo surface

arXiv:math/0309111

Abstract

Let $X_{r}$ be a smooth Del Pezzo surface obtained from $¶^{2}$ by blowing up $r \leq 8$ points in general position. It is well known that for $r \in {3, 4, 5, 6, 7, 8}$ the Picard group $\Pic(X_r)$ contains a canonical root system $R_{r} \in {A_{2} \times A_{1}, A_{4}, D_{5}, E_{6}, E_{7}, E_{8}}$ . We prove some general properties of the Cox ring of $X_{r}$ ( $r \geq 4$ ) and show its similarity to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group $G$ associated with the root system $R_{r}$ .

AMS-LaTeX, 15 pages