Measuring ideals in a singularity
Objective:
This proposal concerns singularities arising in the solution spaces of systems of polynomial equations. Such systems are ubiquitous within mathematics and its applications, and equally ubiquitous are singularities of its solutions, where a small change of parameter may result in an unpredictable change in behavior. Singularity measures are numbers that assign a numerical value to each singularity which aims to describe a part of its complexity. They give us a convenient way to compare and to study singularities. I will develop two novel singularity measures. These invariants are defined algebraically in the local ring of a singularity and the unifying theme is to consider not just the maximal ideal but rather all ideals primary to the maximal ideal. This allows to capture more information, but also makes the invariants harder to compute and work with, since, in principle, the definition will involve an infinite number of objects. Thus, a special attention will be given to finding good examples in order to guide the development and connect the proposal to established areas of research.
My first direction is the theory of the Lech-Mumford constant that we develop with Linquan Ma. This invariant represents an optimal version of the fundamental Lechs inequality and originates from the work of Mumford which used it as a restriction on singularities appearing on Mumfords compactification of the moduli of smooth varieties, i.e., on limits of smooth varieties. First, we aim to connect Mumfords restriction with the later developed classes of singularities from birational geometry. Second, we will develop an algebraic theory of this invariant, focusing on various deformation properties. Third, we will search for ways to compute and bound the invariant with a hope of connecting with other methods of studying curve and surface singularities. The three directions presented in this proposal will lead to at least a partial advance on the long- standing conjecture classifying the surface singularities that satisfy Mumfords restriction. This classification was initiated by Mumford and continued by his student Shah who gave a conjectural list.
The second part is in the theory of singularities in positive characteristic, where several classes and numerical invariants of singularities are defined by exploiting the properties of Frobenius. Strongly F-regular singularities are a central class - one of the reasons is that they possess a very useful singularity measure called F-signature. Recently, there were several attempts to relax F-signature so that it works in a wider class of F-rational singularities: Hochster-Yao and Sannai gave two very distinct definitions and, in a recent preprint with Kevin Tucker, we proposed a better-behaving modification of the Hochster-Yao definition. In this proposal, I aim to continue to develop this theory: my main goal is to show that our invariant is equal Sannais. The importance of this equality is two-fold: first, it represents equivalent definitions of F-signature and, second, it immediately implies many desirable properties due to complementary natures of the theories. I will study further properties of both invariants, regardless of the equality, it will benefit to have different methods. I will also aim to push further computation of this invariants since any theory needs good examples.
