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The Cox ring of a del Pezzo surface. (English) Zbl 1075.14035
Poonen, Bjorn (ed.) et al., Arithmetic of higher-dimensional algebraic varieties. Proceedings of the workshop on rational and integral points of higher-dimensional varieties, Palo Alto, CA, USA, December 11–20, 2002. Boston, MA: Birkhäuser (ISBN 0-8176-3259-X/hbk). Progress in Mathematics 226, 85-103 (2004).
For a projective variety $$X$$ over a field $$k$$ with $$\text{Pic}(X)$$ generated by a $$\mathbb{Z}$$-basis $$L_1, \dots, L_r$$, Hu and Keel defined the Cox ring (named after D. A. Cox) as $\text{Cox}(X) = R(X, L_1, \dots, L_r) = \bigoplus_{(m_1, \dots, m_r) \in \mathbb{Z}^r} H^0(X, {\mathcal O}(m_1L_1 + \dots + m_r L_r)).$ In the current paper, the authors define a Cox ring using a set of generators, instead. They prove certain general properties for their Cox rings for $$X = X_r$$ (a smooth del Pezzo surface obtained from $$\mathbb{P}^2$$ by blowing up $$r \leq 8$$ points in general position). They also observe and show that the above properties have 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 $$R_r$$ of $$X_r$$. Some interesting conjectures, remarks and corollaries are included.
For the entire collection see [Zbl 1054.11006].

##### MSC:
 14J26 Rational and ruled surfaces
##### Keywords:
torsor; homogeneous space; algebraic group
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