Rational powers, invariant ideals, and the summation formula

TL;DR

The paper develops explicit polyhedral descriptions of rational powers and Rees valuations for several classes of invariant ideals, and introduces the Rees package as a combinatorial framework to encode these invariants. It proves a Mustaţă–Takagi type summation formula for rational powers in the setting of tensor products $R\otimes_k S$ when the factors’ ideals admit a Rees package, with broad applicability to monomial, determinantal, Pfaffian, symmetric, and Hankel-based invariant ideals. The results provide concrete, computable descriptions of Rees valuations and rational powers via lattice points and supporting hyperplanes, and establish an asymptotic version of the formula over algebraically closed fields. These contributions offer new computational tools for integral closures, multiplier ideals, and related filtrations in algebraic geometry and commutative algebra. The framework clarifies how valuations interact under tensor products and birational transformations, with implications for symbolic powers and asymptotic invariants in invariant-theoretic contexts.

Abstract

We provide explicit descriptions for the rational powers and Rees valuations of several classes of ideals invariant under natural actions of tori and products of general linear groups, in terms of polyhedra and lattice points. This allows us to show that a version of Mustaţă-Takagi's summation formula for multiplier ideals also holds for the rational powers of these ideals. Moreover, for arbitrary ideals in normal domains that are finitely generated over algebraically closed fields, we prove a weaker version of this formula that holds for sufficiently large rational numbers.

Figures (3)

• Figure 1: The semigroup $\mathscr S= {\mathbb Z}_{\geqslant 0}(2,1)+ {\mathbb Z}_{\geqslant 0}(1,3)$ and ideal $I=\left(x_1^4x_2^2,\, x_1^3x_2^4\right)\subset \mathbb{k}[\mathscr S]$.
• Figure 2: The polyhedron $\Gamma={\operatorname{conv}}(\underline{\gamma}(\Lambda))+\mathbb R_{\geqslant 0}^2={\operatorname{conv}}\left((2,1), (3,0)\right)+\mathbb R^2_{\geqslant 0}$.
• Figure 3: The convex hull ${\operatorname{conv}}(\Gamma_1, \Gamma_2)$.

Theorems & Definitions (91)

• Theorem A
• Conjecture 1
• Theorem B: \ref{['main_Rees_P']}, \ref{['cor_invs_all']}
• Theorem C: \ref{['thm_asymp_main']}
• Definition 1: Rational powers
• Remark 1
• Proposition 1
• proof
• Definition 2: Rees valuations
• Remark 2