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Projective Hypersurfaces of High Degree Admitting an Induced Additive Action

Ivan Beldiev

TL;DR

The paper classifies non-degenerate projective hypersurfaces $X\subseteq \mathbb{P}^n$ admitting an induced additive action for degrees $d=n-1, n-2, n-3$ by translating the problem into $H$-pairs $(A,U)$ via a generalized Hassett–Tschinkel correspondence. It systematically analyzes local Gorenstein algebras and their Hilbert–Samuel sequences, using orbit arguments to prove independence from the choice of $U$ and providing explicit algebraic models and defining equations for each case. The main results yield a unique $X$ for $d=n-1$ with algebra $A=\mathbb{K}[x,y]/(xy, y^2-x^{n-1})$, three distinct algebras for $d=n-2$ (with corresponding ideals) and a detailed classification for $d=n-3$ across several Hilbert–Samuel types, including infinite families in low dimensions. This work completes the high-degree classification of hypersurfaces with induced additive actions and extends the Hassett–Tschinkel framework to a broader class of projective varieties.

Abstract

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group $\mathbb G_a^m$ with an open orbit that can be extended to a regular action on the ambient projective space. It is known that the degree of a hypersurface $X\subseteq\mathbb{P}^n$ admitting an induced additive action cannot be greater than $n$ and there is a unique such hypersurface of degree $n$. We give a complete classification of hypersurfaces $X\subseteq \mathbb{P}^n$ admitting an induced additive action of degrees from $n-1$ to $n-3$.

Projective Hypersurfaces of High Degree Admitting an Induced Additive Action

TL;DR

The paper classifies non-degenerate projective hypersurfaces admitting an induced additive action for degrees by translating the problem into -pairs via a generalized Hassett–Tschinkel correspondence. It systematically analyzes local Gorenstein algebras and their Hilbert–Samuel sequences, using orbit arguments to prove independence from the choice of and providing explicit algebraic models and defining equations for each case. The main results yield a unique for with algebra , three distinct algebras for (with corresponding ideals) and a detailed classification for across several Hilbert–Samuel types, including infinite families in low dimensions. This work completes the high-degree classification of hypersurfaces with induced additive actions and extends the Hassett–Tschinkel framework to a broader class of projective varieties.

Abstract

We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group with an open orbit that can be extended to a regular action on the ambient projective space. It is known that the degree of a hypersurface admitting an induced additive action cannot be greater than and there is a unique such hypersurface of degree . We give a complete classification of hypersurfaces admitting an induced additive action of degrees from to .

Paper Structure

This paper contains 9 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results on Gorenstein algebras
  4. Hypersurfaces in $\mathbb{P}^n$ of degree $n-1$
  5. Hypersurfaces in $\mathbb{P}^n$ of degree $n-2$
  6. Hypersurfaces in $\mathbb{P}^n$ of degree $n - 3$
  7. Hilbert-Samuel sequence $(1,4,1,\ldots, 1)$
  8. Hilbert-Samuel sequence $(1,3,2,1,\ldots, 1)$
  9. Hilbert-Samuel sequence $(1,2,2,2,1,1,\ldots, 1)$

Key Result

Lemma 1

AZa A finite-dimensional algebra $A$ is local if and only if $A$ is the direct sum of its subspaces $\mathbb{K} \oplus \mathfrak{m}$, where $\mathfrak{m}$ is the ideal consisting of all nilpotent elements of $A$.

Theorems & Definitions (25)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Definition 6
  • Theorem 2
  • Proposition 1
  • ...and 15 more