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Automorphisms of odd dimensional $(2,2)$-complete intersections in characteristic $2$

Yang Zhang

TL;DR

The paper classifies the automorphism scheme of generic odd-dimensional $(2,2)$-complete intersections in characteristic $2$, showing the nontrivial identity component is unique to this non-hypersurface case (besides quadrics and genus $1$ curves). It reduces the problem to a Pfaffian analysis of pencils of quadrics and obtains a concrete decomposition $ ext{Aut}_X\cong \text{Aut}^0_X\rtimes \pi_0(\text{Aut}_X)$, with $ ext{Aut}^0_X\cong \mu_2^{M}$ for $M\ge 2$ and $\,\pi_0(\text{Aut}_X)\cong (\mathbb{Z}/2\mathbb{Z})^{M+1}$ generically. A central result is the exact sequence $1\to (\mathbb{Z}/2\mathbb{Z})^{M+1}\to \pi_0(\text{Aut}_X)\to \text{Aut}(P^1;\varphi(Z))\to 1$, with the latter group typically trivial, yielding a complete arithmetic description of the automorphism structure. The findings advance moduli considerations for this class and provide explicit criteria for lifting automorphisms from the base to the full automorphism group, along with constructive examples.

Abstract

We compute the automorphism scheme of a generic odd dimensional $(2,2)$-complete intersection in characteristic $2$. This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from quadric hypersurfaces and genus $1$ curves.

Automorphisms of odd dimensional $(2,2)$-complete intersections in characteristic $2$

TL;DR

The paper classifies the automorphism scheme of generic odd-dimensional -complete intersections in characteristic , showing the nontrivial identity component is unique to this non-hypersurface case (besides quadrics and genus curves). It reduces the problem to a Pfaffian analysis of pencils of quadrics and obtains a concrete decomposition , with for and generically. A central result is the exact sequence , with the latter group typically trivial, yielding a complete arithmetic description of the automorphism structure. The findings advance moduli considerations for this class and provide explicit criteria for lifting automorphisms from the base to the full automorphism group, along with constructive examples.

Abstract

We compute the automorphism scheme of a generic odd dimensional -complete intersection in characteristic . This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from quadric hypersurfaces and genus curves.
Paper Structure (9 sections, 24 theorems, 50 equations)

This paper contains 9 sections, 24 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finiteness of $\mathop{\mathrm{Aut}}\nolimits_X$
  4. Pfaffian of pencils of quadratic forms in characteristic $2$
  5. The singular locus of a pencil
  6. The lifting group with respect to a line bundle
  7. Pushforward of automorphisms
  8. Study of $\mathop{\mathrm{Aut}}\nolimits^0$
  9. Study of $\pi_0(\mathop{\mathrm{Aut}}\nolimits_X)$

Key Result

Theorem 1.1

Fix integers $N\geq 2, c\geq 1, d_1\geq \dots\geq d_c\geq 2$. The Stack $\mathcal{M}_{N,c,(d_i)_{1\leq i\leq c}}$ classifying smooth $(d_1,\dots, d_c)$-complete intersections in $\mathbb{P}^N$ polarized by $\mathcal{O}(1)$ is Deligne-Mumford over $\mathop{\mathrm{Spec}}\nolimits \mathbb{Z}$ except f

Theorems & Definitions (40)

  • Theorem 1.1: Benoist2011SparationEP
  • Theorem 1.2: cf. Theorem \ref{['thm:structure of Aut^0']}
  • Theorem 1.3: cf. Theorem \ref{['thm: exact sequence for pi_0']} and Corollary \ref{['cor: pi_0 for a generic X']}
  • Proposition 2.1: Benoist2011SparationEP
  • Proposition 2.2: brion2018notes
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 30 more