The Cox Ring of a Del Pezzo Surface

By V. Batyrev, O. N. Popov. 2003

116 citations recorded by the source corpus.

Let Xr be a smooth Del Pezzo surface obtained from ℙ(2 by blowing up r ⩽8 points in general position. It is well known that for r e {3,4,5,6,7,8} the Picard group Pic(X r ) contains a canonical root system R r € A 2 × A1, A4, D5, E6, E7, E8. We prove some general properties of the Cox ring of X r (r ⩾ 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 .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/15cac4bc4bb12cdb594e8336433f2fbd6d9d9c56

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Cox Rings and Combinatorics II (2008, background): “n aﬃne neighbourhood. Introduction The remarkable success of Cox’s homogeneous coordinate ring in toric geometry [6] has stimulated many authors to study related rings for more general varieties, see [1], [2], [7], [12] and [13]. We consider here the version discussed in [7]: Let X be a normal varie”
On Cox rings of K3 surfaces (2009, background): “Moreover, for del Pezzo surfaces X there are recent results on generators and relations of the Cox ring, see [5], [9], [14], [25], [26], and [27].”
Singular del Pezzo surfaces whose universal torsors are hypersurfaces (2006, background): “Generators and relations in the Cox rings of smooth Del Pezzo surfaces can be found in [BP04] and [Der06].”