Symmetries in Algebraic Geometry

Abstract: In this talk I will discuss different notions of symmetry of algebraic varieties. First we examine automorphisms. The structure of the group of automorphisms of a projective variety encodes some relevant geometric properties. However, the notion of automorphism is too rigid in the scope of birational geometry. We are then led to consider another class of symmetries of projective varieties, its birational self-maps. Birational self-maps of the projective space $\mathbb{P}^n$ are called Cremona transformations. Describing the structure of the group of Cremona transformations of the plane is a classical problem that goes back to the 19th century. In higher dimensions, not so much is known, and a natural problem is to construct interesting subgroups of the Cremona group. I will end the talk by discussing a recent work with Alessio Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of symmetries preserving special geometric structures.

Mathematics

Audience: general audience

VCU Mathematics and Applied Mathematics Colloquium Series

Series comments: A series of public lectures in Mathematics and Applied Mathematics from Virginia Commonwealth University.

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