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Exceptional algebraic curves for infinite groups of $\textrm{PGL}(n,\mathbb{C})$

Angel Cano, Luis Loeza, Rodrigo Dávila Figueroa

TL;DR

The paper classifies one-dimensional algebraic curves in complex projective space that are invariant under infinite subgroups of projective linear transformations. Using a projection-based strategy, normalization to $\mathbb{CP}^1$, and ramification/Plücker invariants, it proves that each invariant irreducible curve is projectively equivalent to a monomial curve unless a finite-index subgroup acts trivially on its span; in the discrete, infinite, non-virtually cyclic case, invariant curves are forced to be lines or non-singular rational curves. The authors then analyze the automorphism structure of monomial curves, showing that (depending on a symmetry condition on the exponents) the automorphism group is generated by a diagonal subgroup, possibly extended by an anti-diagonal element. These results bridge algebraic geometry with dynamics and foliation rigidity, providing a sharp dichotomy for invariant curves and a complete description of their symmetry groups.

Abstract

We classify algebraic curves in $\mathbb{CP}^{n}$ ($n \geq 2$) that are invariant under an infinite subgroup of $\operatorname{PGL}(n+1,\mathbb{C})$. In particular, we prove that any irreducible, non-degenerate, one-dimensional algebraic set in $\mathbb{CP}^{n}$ invariant under an infinite subgroup of $\operatorname{Aut}(\mathbb{CP}^{n})$ must be projectively equivalent to a monomial curve.

Exceptional algebraic curves for infinite groups of $\textrm{PGL}(n,\mathbb{C})$

TL;DR

The paper classifies one-dimensional algebraic curves in complex projective space that are invariant under infinite subgroups of projective linear transformations. Using a projection-based strategy, normalization to , and ramification/Plücker invariants, it proves that each invariant irreducible curve is projectively equivalent to a monomial curve unless a finite-index subgroup acts trivially on its span; in the discrete, infinite, non-virtually cyclic case, invariant curves are forced to be lines or non-singular rational curves. The authors then analyze the automorphism structure of monomial curves, showing that (depending on a symmetry condition on the exponents) the automorphism group is generated by a diagonal subgroup, possibly extended by an anti-diagonal element. These results bridge algebraic geometry with dynamics and foliation rigidity, providing a sharp dichotomy for invariant curves and a complete description of their symmetry groups.

Abstract

We classify algebraic curves in () that are invariant under an infinite subgroup of . In particular, we prove that any irreducible, non-degenerate, one-dimensional algebraic set in invariant under an infinite subgroup of must be projectively equivalent to a monomial curve.
Paper Structure (7 sections, 24 theorems, 68 equations, 2 figures)

This paper contains 7 sections, 24 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projective Geometry
  4. Algebraic Curves
  5. Geometry of exceptional curves
  6. Parametrization of the exceptional set
  7. Projective Automorphism Groups of Monomial Curves

Key Result

Theorem 1.1

Let $g \in \mathrm{PGL}(n+1,\mathbb{C})$, $n \geq 2$, be an element of infinite order, and let $S \subset \mathbb{CP}^n$ be a one-dimensional algebraic set invariant under $g$. For every irreducible component $S_0 \subset S$, one of the following holds:

Figures (2)

  • Figure 1: Examples of monomial sets
  • Figure 2: Example of projected monomial curve

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: Projection and Induced Homomorphism, cf. CNS
  • Proposition 2.2: See Lemmas 2.2 and 2.8 in CLU
  • Definition 2.3: Associated Curves
  • Definition 2.4: Ramification Indices
  • Theorem 2.5: Linear Plücker formulae, GH
  • Corollary 2.6: See page 3 in W
  • Corollary 2.7: Plücker identity, See Lemma 1.2 in W
  • Definition 2.8: Monomial Curves
  • ...and 38 more