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Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities

Yairon Cid-Ruiz

TL;DR

This work extends the SUV multiplicities to a multigraded setting by introducing relative mixed multiplicities for inclusions of standard $\mathbb{N}^p$-graded algebras. The authors develop a multigraded version of Kleiman--Thorup's joint blow-up construction and define a framework of mixed Buchsbaum--Rim multiplicities that capture the asymptotic behavior of lengths in multigraded graded components. A key result is that the relative mixed multiplicities stabilize to the classical mixed Buchsbaum--Rim multiplicities, and their vanishing detects integral dependence and birationality without requiring extra hypotheses on $B$; they also provide new joint blow-up constructions to measure deviations from the stable values. These invariants yield concrete criteria for integrality and birationality in the multigraded context and connect to geometric objects via multiprojective schemes and intersection theory, with potential applications to the study of multigraded blow-ups and rational maps.

Abstract

We define and study the natural multigraded extension of the relative multiplicities introduced by Simis, Ulrich and Vasconcelos. We call these new invariants relative mixed multiplicities. We show that they have a stable value equal to the mixed Buchsbaum-Rim multiplicity of Kleiman and Thorup. Furthermore, we prove that integral dependence and birationality can be detected via the vanishing of relative mixed multiplicities.

Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities

TL;DR

This work extends the SUV multiplicities to a multigraded setting by introducing relative mixed multiplicities for inclusions of standard -graded algebras. The authors develop a multigraded version of Kleiman--Thorup's joint blow-up construction and define a framework of mixed Buchsbaum--Rim multiplicities that capture the asymptotic behavior of lengths in multigraded graded components. A key result is that the relative mixed multiplicities stabilize to the classical mixed Buchsbaum--Rim multiplicities, and their vanishing detects integral dependence and birationality without requiring extra hypotheses on ; they also provide new joint blow-up constructions to measure deviations from the stable values. These invariants yield concrete criteria for integrality and birationality in the multigraded context and connect to geometric objects via multiprojective schemes and intersection theory, with potential applications to the study of multigraded blow-ups and rational maps.

Abstract

We define and study the natural multigraded extension of the relative multiplicities introduced by Simis, Ulrich and Vasconcelos. We call these new invariants relative mixed multiplicities. We show that they have a stable value equal to the mixed Buchsbaum-Rim multiplicity of Kleiman and Thorup. Furthermore, we prove that integral dependence and birationality can be detected via the vanishing of relative mixed multiplicities.
Paper Structure (6 sections, 18 theorems, 81 equations)

This paper contains 6 sections, 18 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. An extension of a joint blow-up construction of Kleiman and Thorup
  3. Mixed $j$-multiplicities
  4. New joint blow-up constructions
  5. Relative mixed multiplicities
  6. Applications

Key Result

Theorem A

Fix a tuple $\mathbf{t} = (t_1,\ldots,t_p) \in \mathbb{Z}_+^p$ of positive integers. Then the following function coincides with a polynomial $P_{\mathbf{t}}^{A,B}(n_1,\ldots,n_p)$ for all $n_i \gg 0$. Moreover, $P_{\mathbf{t}}^{A,B}$ has total degree at most $\dim\left(\normalfont\text{MultiProj}(B)\right)$ and its normalized leading coefficients are nonnegative integers.

Theorems & Definitions (44)

  • Theorem A: \ref{['thm_relative_mixed_mult']}
  • Theorem B: \ref{['thm_criteria']}
  • Definition 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 2.1
  • proof
  • Definition 2
  • Definition 3
  • ...and 34 more