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Birational equivalence of exponential matrices

Ryuji Tanimoto

TL;DR

This work classifies exponential matrices up to birational equivalence, framing them as $ obreak abla G_a$-actions on projective spaces. In characteristic zero, every exponential matrix is birationally equivalent to either the identity or a single canonical form, giving a clean two-type classification. In positive characteristic, the $3 imes3$ case splits into three families, with the birational landscape depending on the characteristic: $p=2$ yields two primary families, while $p e 2$ includes an additional family, and indecomposable variants are described via GL-orbits of $p$-polynomial data. The results connect matrix exponential structures to birational $ abla G_a$-actions and provide explicit canonical representatives and orbit descriptions for the classifications.

Abstract

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size $n$-by-$n$ $(n \geq 2)$, which consists of two types. And in positive characteristic, we give birational classifications of exponential matrices of sizes two-by-two and three-by-three, respectively.

Birational equivalence of exponential matrices

TL;DR

This work classifies exponential matrices up to birational equivalence, framing them as -actions on projective spaces. In characteristic zero, every exponential matrix is birationally equivalent to either the identity or a single canonical form, giving a clean two-type classification. In positive characteristic, the case splits into three families, with the birational landscape depending on the characteristic: yields two primary families, while includes an additional family, and indecomposable variants are described via GL-orbits of -polynomial data. The results connect matrix exponential structures to birational -actions and provide explicit canonical representatives and orbit descriptions for the classifications.

Abstract

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size -by- , which consists of two types. And in positive characteristic, we give birational classifications of exponential matrices of sizes two-by-two and three-by-three, respectively.

Paper Structure

This paper contains 13 sections, 13 theorems, 106 equations.

Table of Contents

  1. Preliminaries
  2. Proof of Theorem 0.1
  3. Proof of Theorem 0.2
  4. Proof of Theorem 0.3
  5. On $\mathfrak{A}_{1, \, 2}^E$, $\mathfrak{A}_{2, \, 1}^E$, $\mathfrak{J}_{[3]}^E$
  6. Proof of (1)
  7. Proof of (2)
  8. Proof of (3)
  9. Birational equivalence of exponential matrices of $(\, \mathfrak{A}_{1, \, 1}^E \,)^{\rm indep}$ and $(\, \mathfrak{A}_{1, \, 2}^E \,)^{\rm indep}$
  10. Birational equivalence of exponential matrices of $(\, \mathfrak{A}_{1, \, 1}^E \,)^{\rm indep}$
  11. Birational equivalence of exponential matrices of $(\, \mathfrak{A}_{1, \, 2}^E \,)^{\rm indep}$
  12. $(\, \mathfrak{A}_{1, \, 1}^E \,)^{\rm indep}$ and $( \, \mathfrak{A}_{1, \, 2}^E \,)^{\rm indep}$ are not birationally equivalent
  13. Proof of Theorem 0.3

Key Result

Theorem 1

Assume $p = 0$ and let $n \geq 2$. Then there exists a one-to-one correspondence between the set and the set

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • ...and 12 more