Seminario SIPo: A glimpse about blow-up algebras

Abstract:

Blow-up algebras is a classical and central topic in both commutative algebra and algebraic geometry. From the geometric point of view, they appear related to the Theory of Resolution of Singularities. On the algebraic side, they are mostly used to capture the algebraic relations between generators of some ideal. This is a very intuitive problem that can be roughly translated in terms of computing the implicit equations of a parametric curve or a parametric surface.

In this talk, we present through examples two of these Blow-up Algebras: The Rees Algebra and the special fiber of an ideal. We discuss the challenging problem of finding the defining equations of these algebras, highlighting some new results and open problems.