On Cox ring generators of K3 surfaces of Picard number three

Claudia Correa
Universidad de Concepci\'on
The Cox ring of a normal projective algebraic variety
, defined on
, 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 
is finitely generated, the variety 
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.