Frobenius powers of some monomial ideals

TL;DR

This paper characterizes Frobenius powers and critical exponents of specific monomial ideals in positive characteristic, making these invariants computable and revealing their uniform variation with the characteristic's congruence class.

Contribution

It provides explicit characterizations of Frobenius powers, test ideals, and F-jumping exponents for certain monomial ideals, linking positive characteristic invariants with characteristic zero counterparts.

Findings

Frobenius powers of monomial ideals are explicitly characterized.

Test ideals and F-jumping exponents are computable for these ideals.

Test ideals in positive characteristic match reductions of multiplier ideals for many primes.

Abstract

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes.