On Cox ring generators of K3 surfaces of Picard number three |
Claudia Correa |
Universidad de Concepci\'on |
Abstract: |
The Cox ring of a normal projective algebraic variety \[X\] , defined on \[\mathbb{C}\] , with divisor class group \[\rm Cl{(X)}\] finitely generated and free, is the following multigraduate algebra \[R(X):=\bigoplus_{[D]\in \rm Cl(X)}H^0(X,\mathcal{O}_X(D)).\] |
When the Cox ring of \[X\] is finitely generated, the variety \[X\] is called Mori dream, for example toric varieties are Mori dream spaces whose Cox ring is a ring of polynomials. |
The research on Cox rings of K3 surfaces is initiated in 2009 with the works of Artebani, Hausen, and Laface, and McKernan, who show that a K3 surface has finitely generated Cox ring if and only if its effective cone is polyhedral. The K3 surfaces with this property have been classified according to their Picard number. |
In this talk we will see some techniques that have allowed to determine a generator set of Cox ring for K3 surfaces of Picard number three, and we will illustrate it with some examples. |