New invariants of singularities

In commutative algebra, there are many numerical invariants that measure the singularity of a local ring.

These invariants are typically based on the properties of the maximal ideal, and the possibility of using all primary ideals was previously overlooked. I propose to study two natural measures defined as the supremum over all primary ideals.

First, Lech-Mumford slope is based on the ratio of two fundamental invariants of primary ideals: colength and multiplicity. Lech´s inequality connects the two invariants, but it is never sharp in dimension at least two.

During the first three years, I will investigate several approaches for improving the inequality, organized as Projects I & II, one of which is conceptualized by Lech-Mumford slope. Lech-Mumford slope first appeared in Mumford´s treatment of GIT, but has not received an extensive study before.

I will study its properties and how it is affected by the singularity: tighter bounds should correspond to milder singularities.

Second, F-volume, is a positive characteristic analogue of the normalized volume, a recent notion and an active topic in birational geometry.

In the last two years, I will initiate an algebraic study of this invariant and study its algebraic and geometric properties (Project III).

Beyond the immediate goals, this proposal will build new and strengthen the existing connections of commutative algebra with algebraic geometry and combinatorics. In particular, both main measures have geometric origin.