Title: Singularities of the (extended) Rees Algebra
Abstract: Blow-up is a fundamental operation in Algebraic Geometry. Multiplier (test) modules and ideals measure singularities present in a variety. In this talk, I'll explain what these modules look like for the Rees algebra. Extended Rees algebra is a useful and extensively studied notion both in commutative algebra and in algebraic geometry ("deformation to the normal cone"). I'll explain how extended Rees algebra can be used to understand various singularities (Rational, KLT, strongly F-regular, etc) and compute (in Macaulay2) test ideals of non-principal ideals as predicted by Schwede-Tucker. In the process, I'll answer a question of Hara-Watanabe-Yoshida (2002) in full generality. I'll also mention a technical result relating canonical modules of Rees and Extended Rees algebra, which might be of independent interest.
Some part of this work is joint with Hunter Simper.