Matías Vásquez (Universidad de Talca) : Twisted forms of classical hypersurfaces

Given an algebraic variety X defined over $\mathbb{C}$, it is natural to ask whether it descends to a subfield, such as $\mathbb{R}$, and, if so, in how many essentially different ways. This geometric question has an algebraic answer: real forms of X correspond to certain descent data attached to the automorphism group of X, via Weil's theorem. Making this correspondence explicit is, in general, difficult. 

In this talk we count real forms for three classical families of hypersurfaces (Fermat, Delsarte, and Klein) exploiting the structure of their automorphism groups. In particular, we recover a recent result of Sasaki for Fermat hypersurfaces. Time permitting, we will comment on the analogous problem over finite fields