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The Hasse principle for 9-nodal cubic threefolds

N. i. Shepherd-Barron

Abstract

We prove that the Hasse principle holds for cubic threefolds with 9 singular points over a number field.

The Hasse principle for 9-nodal cubic threefolds

Abstract

We prove that the Hasse principle holds for cubic threefolds with 9 singular points over a number field.

Paper Structure

This paper contains 7 sections, 23 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Universal torsors over the smooth locus of a Perazzo cubic
  3. Geometry
  4. The Hasse principle
  5. Weak approximation
  6. Computing the Brauer group and obstructions
  7. An aside: weak approximation for 10 nodes

Key Result

lemma 1

The automorphism group scheme $G=\operatorname{Aut} (P,{\mathcal{O}}(1))$ over ${\mathbb Q}$ is a split extension of $\Gamma$ by $T$.

Theorems & Definitions (49)

  • lemma 1
  • proof
  • proposition 1
  • proof
  • definition 1
  • proposition 2
  • proof
  • lemma 2
  • proof
  • remark 1
  • ...and 39 more