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The canonical ideal and the deformation theory of curves with automorphisms

Aristides Kontogeorgis, Alexios Terezakis

TL;DR

This work reframes the deformation theory of curves with automorphisms through the canonical (relative) ideal, replacing the subtle automorphism group of formal power series by linear representation theory to analyze liftings. It proves a relative Petri theorem ensuring the relative canonical ideal remains generated by quadrics under deformation and shows that embedded deformations capture all deformations under mild hypotheses. A central result is a compatibility condition between the base representation $\rho$ and its lift $\rho^{(1)}$ that must hold for a lifted automorphism to act on a deformation $X_A$, with obstructions governed by $H^2(G,\cdot)$ and a tangent-space condition linking $H^1(X,\mathcal{T}_X)$ and $H^0(X,\mathcal{N}_{X/\mathbb{P}^{g-1}})$. Collectively, the paper provides a practical, linear-algebraic route to assess liftability of automorphisms in families of curves and clarifies when obstructions appear in characteristic $p$.

Abstract

The deformation theory of curves is studied by using the canonical ideal. The problem of lifting curves with automorphisms is reduced to a lifting problem of linear representations.

The canonical ideal and the deformation theory of curves with automorphisms

TL;DR

This work reframes the deformation theory of curves with automorphisms through the canonical (relative) ideal, replacing the subtle automorphism group of formal power series by linear representation theory to analyze liftings. It proves a relative Petri theorem ensuring the relative canonical ideal remains generated by quadrics under deformation and shows that embedded deformations capture all deformations under mild hypotheses. A central result is a compatibility condition between the base representation and its lift that must hold for a lifted automorphism to act on a deformation , with obstructions governed by and a tangent-space condition linking and . Collectively, the paper provides a practical, linear-algebraic route to assess liftability of automorphisms in families of curves and clarifies when obstructions appear in characteristic .

Abstract

The deformation theory of curves is studied by using the canonical ideal. The problem of lifting curves with automorphisms is reduced to a lifting problem of linear representations.

Paper Structure

This paper contains 14 sections, 22 theorems, 134 equations, 1 table.

Table of Contents

  1. Introduction
  2. Deformation theory of curves with automorphisms
  3. Global deformation functor
  4. Lifting of representations
  5. Deforming representations
  6. Relative Petri's theorem.
  7. Embedded deformations
  8. Embedded deformations and small extensions
  9. Embedded deformations and tangent spaces
  10. Some matrix computations
  11. Study of $H^0(X,N_f)$
  12. Invariant spaces
  13. On the deformation theory of curves with automorphisms
  14. A tangent space condition

Key Result

Theorem 1

Let $X$ be a non-singular, non-hyperelliptic curve of genus $g\geq 3$, defined over an algebraically closed field. Let $\Omega_X$ be the sheaf of differentials of $X$. There is the following short exact sequence: where $I_X$ is generated by elements of degree $2$ and $3$. Also if $X$ is not a non-singular quintic of genus $6$ or $X$ is not a trigonal curve, then $I_X$ is generated by elements of

Theorems & Definitions (54)

  • Theorem 1: Petri's theorem
  • Proposition 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Proposition 9
  • Remark 10
  • ...and 44 more