Research Projects of Giuliano Gagliardi

Toric varieties are a widely-studied class of algebraic varieties, where difficult problems in the language of algebraic geometry are translated into elementary problems involving combinatorial objects. Spherical varieties are a natural generalization of toric varieties to the setting of a noncommutative acting group.

A natural question is what geometrical properties of spherical varieties can be deduced from their combinatorial description. For example, a toric variety is smooth if and only if every cone in its fan is generated by a subset of a basis. For spherical varieties over an algebraically closed field of characteristic 0, there is a rather involved generalization of this criterion. We investigate how this criterion can be simplified.

Spherical varieties can also be studied over nonclosed fields. We study the distribution of rational points (Manin's conjecture) on particular examples of spherical varieties over the rational numbers.

On the other hand, it would be useful to have a transparent description of models of spherical varieties over arbitrary fields. Combining combinatorial methods, Galois chomology, and invariant Hilbert schemes, we have obtained a general result on their existence in characteristic 0.