Purdue Commutative Algebra Seminar (Fall 2025)

Residual intersections provide a natural extension of the classical notion of algebraic links. We establish a general upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian ring, and we show that this bound is sharp under specific characteristic assumptions. This work is joint with Kesavan Mohana Sundaram, Taylor Murray, and Vaibhav Pandey.

A reduced and irreducible one-dimensional local complete domain R over an algebraically closed field k can be parametrized in many different ways as a subring of its integral closure R = k[[t]]. In this talk we will show that there is a unique sequence of positive integers a₁ < ... < a_n such that any set x₁, ..., x_n in R with valuations v_R(x_i) = a_i defines a minimal parametrization of R. We will then use this result to study the torsion of differential forms and to prove a theorem about modifications of parametrizations of R.

We present an affirmative answer to this problem for complete intersections in every positive characteristic, improving a theorem by Enescu and Shimomoto, thus settling the conjecture for this family of singularities. The proof relies on advanced characteristic-dependent applications of a technique developed by Han and Monsky, and critically includes a explicit calculation needed to fill the gaps for the often-overlooked characteristic 2 case.