# Minimal log discrepancies of determinantal varieties via jet schemes

## TL;DR

This paper calculates the minimal log discrepancies of determinantal varieties and related pairs, confirming a conjecture and providing explicit formulas using jet schemes, which advances understanding in algebraic geometry.

## Contribution

It introduces explicit computations of minimal log discrepancies for determinantal varieties and pairs, confirming the semicontinuity conjecture using jet scheme techniques.

## Key findings

- Confirmed semicontinuity conjecture for these pairs
- Provided explicit generators for canonical forms and Nash ideals
- Enhanced understanding of jet scheme computations in determinantal varieties

## Abstract

We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum \alpha_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal subvarieties. Our result implies the semicontinuity conjecture for minimal log discrepancies of such pairs. For these computations, we use the description of minimal log discrepancies via codimensions of cylinders in the space of jets; this necessitates the computations of an explicit generator for the canonical differential forms and the Nash ideal of determinantal varieties, which may be of independent interest.