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Generic Triviality of Automorphism Groups of Complete Intersections

Renjie Lyu, Dingxin Zhang

TL;DR

The paper investigates when a general smooth complete intersection in projective space has no nontrivial linear automorphisms. It introduces a general criterion that ties the triviality of the linear automorphism group to finiteness and unramifiedness of the relative automorphism scheme, faithfulness of the automorphism action on cohomology, and large monodromy (via symplectic or orthogonal groups) in a Lefschetz-type setting. Applying this framework to complete intersections, it proves that in characteristic not equal to 2, general smooth complete intersections of the specified multidegrees have trivial automorphism groups, with the cubic curve as the sole exception; in the Fermat-type case the automorphism action on cohomology is shown to be faithful, enabling the deduction of triviality for general members. The work builds on and extends monodromy-based approaches (à la Katz–Sarnak), connects automorphisms to cohomology representations, and has implications for moduli separateness and the structure of automorphism groups in families of complete intersections.

Abstract

We prove in most cases that a general smooth complete intersection in the projective space has no non-trivial automorphisms.

Generic Triviality of Automorphism Groups of Complete Intersections

TL;DR

The paper investigates when a general smooth complete intersection in projective space has no nontrivial linear automorphisms. It introduces a general criterion that ties the triviality of the linear automorphism group to finiteness and unramifiedness of the relative automorphism scheme, faithfulness of the automorphism action on cohomology, and large monodromy (via symplectic or orthogonal groups) in a Lefschetz-type setting. Applying this framework to complete intersections, it proves that in characteristic not equal to 2, general smooth complete intersections of the specified multidegrees have trivial automorphism groups, with the cubic curve as the sole exception; in the Fermat-type case the automorphism action on cohomology is shown to be faithful, enabling the deduction of triviality for general members. The work builds on and extends monodromy-based approaches (à la Katz–Sarnak), connects automorphisms to cohomology representations, and has implications for moduli separateness and the structure of automorphism groups in families of complete intersections.

Abstract

We prove in most cases that a general smooth complete intersection in the projective space has no non-trivial automorphisms.
Paper Structure (2 sections, 13 theorems, 66 equations)

This paper contains 2 sections, 13 theorems, 66 equations.

Table of Contents

  1. A criterion of generic triviality of automorphisms
  2. Automorphism of complete intersection in projective spaces.

Key Result

Theorem 1

Let $k$ be an algebraically closed field of characteristic $p \neq 2$. Suppose $(d_{1},\ldots,d_{c})$ is a sequence of positive integers satisfying one of the following Then, for a general smooth complete intersection $X\subset \mathbf{P}^{n}_{k}$ of multidegree $(d_{1},\ldots,d_{c})$ with $c < n$, we have $\mathop{\mathrm{\mathrm{Aut}}}\nolimits_L(X)=\{id\}$, unless $X$ is a cubic curve.

Theorems & Definitions (23)

  • Theorem 1: = Theorem \ref{['theorem:gen-trivial-auto']}
  • Theorem 2: = Theorem \ref{['theorem:general-type-hypersurface']}
  • Theorem 3: = Theorem \ref{['theorem:mere']}
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['theorem:mere']}
  • Lemma 1.3
  • Theorem 1.4
  • proof
  • Theorem 2.1
  • Remark 2.3
  • ...and 13 more