Characterization of affine surfaces by their automorphism groups

Alvaro Liendo
Universidad de Talca 
Abstract:
In this talk we show that affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of affine normal irreducible varieties by their automorphism group as ind-group. More precisely, let
\[X\]
be an affine toric variety different from the algebraic torus and let
\[Y\]
be an normal irreducible affine variety. If the groups
\[{\rm Aut}(X)\]
and
\[{\rm Aut}(Y)\]
are isomorphic as ind-groups, then
\[X\]
and
\[Y\]
are isomorphic as abstract varieties. We also provide counter-examples in the case where
\[X\]
is the algebraic torus.