# ℂ*-actions on rational homogeneous varieties and the associated birational maps

**Franceschini Alberto** - PhD in Mathematics, University of Trento

**Abstract:**

Given a birational map among projective varieties, it is known that there exists a variety Z with a one-dimensional torus action such that the birational map is induced from two geometric quotients of Z. We proceed in the opposite direction: given a smooth projective variety X with a one-dimensional torus action, one can define a birational map associated to the action and study the properties of the map via the geometry of X. Rational homogeneous varieties admit natural torus actions, so they are a good class of examples to test the general theory. In the thesis, we obtain and discuss some results about the birational maps associated to some one-dimensional torus actions on rational homogeneous varieties.

Supervisor: **Luis Eduardo Solà Conde** (University of Trento)