UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 09, 2008

939 Evans Hall

3:45PM: The Cox rings of big anticanonical rational surfaces

Mauricio Velasco

The Cox ring of a smooth projective variety X is the Pic(X)-graded algebra consisting of all global sections of all invertible sheaves on X. When X is projective space P^n , this construction yields the usual polynomial ring k[x_0,..,x_n]. It is known that for some choices of X (e.g. for Del Pezzo surfaces) this ring is a finitely generated algebra and in these cases the Cox ring should be thought of as a "geometric" generalization of the polynomial ring. The purpose of this talk is to shed some light on the question of which rational surfaces have finitely generated Cox rings: We will show that rational surfaces with big and effective anticanonical divisor have this property and discuss explicit presentations of these algebras as quotients of polynomial rings. This class of rational surfaces contains all blow-ups of P^2 at any configuration of at most 8 points and several families containing surfaces of abritrarily large rank. The results in this talk are joint work with D. Testa and T. Varilly-Alvarado

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