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Hyperplane sections of cubic threefolds

Arnaud Beauville

TL;DR

Let $S$ be a general cubic surface and $X$ a smooth cubic threefold. The result proves that $S$ is isomorphic to a hyperplane section of $X$; the approach reduces to a weak Lefschetz property for the Jacobian ring $\mathfrak J=R/J$ of the defining cubic $F$, via a tangent-space criterion that the map $s_X$ is étale at $H$ iff $\times L: \mathfrak J_2\to\mathfrak J_3$ is injective. Under the weak Lefschetz property (A-R), this injectivity holds for general $L$, implying $s_X$ is étale at a general point and thus dominant. The paper also discusses extensions to degree $d$ hypersurfaces in $\mathbb{P}^n$ and notes limitations in positive characteristic, highlighting how the moduli of cubic surfaces relate to algebraic properties of the Jacobian ring.

Abstract

Let X be a smooth cubic hypersurface. We prove that a general cubic surface is isomorphic to a hyperplane section of X .

Hyperplane sections of cubic threefolds

TL;DR

Let be a general cubic surface and a smooth cubic threefold. The result proves that is isomorphic to a hyperplane section of ; the approach reduces to a weak Lefschetz property for the Jacobian ring of the defining cubic , via a tangent-space criterion that the map is étale at iff is injective. Under the weak Lefschetz property (A-R), this injectivity holds for general , implying is étale at a general point and thus dominant. The paper also discusses extensions to degree hypersurfaces in and notes limitations in positive characteristic, highlighting how the moduli of cubic surfaces relate to algebraic properties of the Jacobian ring.

Abstract

Let X be a smooth cubic hypersurface. We prove that a general cubic surface is isomorphic to a hyperplane section of X .
Paper Structure (2 sections, 2 theorems, 7 equations)

This paper contains 2 sections, 2 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Proposition 1

Let $H\in \mathbb{P}^*\smallsetminus X^*$, given by a linear form $L\in R_1$. The map $s^{}_X: \mathbb{P}^*\smallsetminus X^* \rightarrow \mathscr{M}_3$ is étale at $H$ if and only if the multiplication map $\times L: \mathfrak{J}_2\rightarrow \mathfrak{J}_3$ is injective.

Theorems & Definitions (2)

  • Proposition
  • Theorem