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On linear sections of the spinor tenfold II

Yingqi Liu, Laurent Manivel

TL;DR

The paper develops a Lie-theoretic framework for linear sections of the spinor tenfold $X\subset {\mathbb P}^{15}$, linking codimension-four sections to Kummer surfaces through the spinor quadratic complex and Vinberg's graded ${\mathfrak e}_8$-structure. It provides explicit models for generic and very special smooth sections, classifies their automorphism groups, and identifies the GIT moduli with the Cartan-subspace quotient ${\mathbb P}(\mathfrak c)/W_{\mathfrak c}$, which is isomorphic to the weighted projective space ${\mathbb w}{\mathbb P}(2,3,5,6)$ and to the moduli of Kummer surfaces. The study ties together classical Kummer theory (including the ${16_6}$ spinorial configuration, the Segre cubic ${\mathrm S}_3$, and the Igusa quartic ${\mathrm CR}_4$) with modern invariant theory via quadratic line complexes and Coble quadrics, producing a coherent picture of the moduli and birational geometry of these sections. It also demonstrates the rich interplay between representation theory, algebraic geometry of Fano varieties, and classical Kummer geometry, yielding concrete descriptions of orbit stratifications, Hilbert schemes, and quantum cohomology in this context.

Abstract

Following previous work by A. Kuznetsov, we study the Fano manifolds obtained as linear sections of the spinor tenfold in $\mathbb{P}^{15}$. Up to codimension three there are finitely many such sections, up to projective equivalence. In codimension four there are three moduli, and this family is particularly interesting because of its relationship with Kummer surfaces on the one hand, and a grading of the exceptional Lie algebra $\mathfrak{e}_8$ on the other hand. We show how the two approaches are intertwined, and we prove that codimension four sections of the spinor tenfolds and Kummer surfaces have the very same GIT moduli space. The Lie theoretic viewpoint provides a wealth of additional information. In particular we locate and study the unique section admitting an action of $SL_2\times SL_2$; similarly to the Mukai-Umemura variety in the family of prime Fano threefold of genus 12, it is a compactification of a quotient by a finite group.

On linear sections of the spinor tenfold II

TL;DR

The paper develops a Lie-theoretic framework for linear sections of the spinor tenfold , linking codimension-four sections to Kummer surfaces through the spinor quadratic complex and Vinberg's graded -structure. It provides explicit models for generic and very special smooth sections, classifies their automorphism groups, and identifies the GIT moduli with the Cartan-subspace quotient , which is isomorphic to the weighted projective space and to the moduli of Kummer surfaces. The study ties together classical Kummer theory (including the spinorial configuration, the Segre cubic , and the Igusa quartic ) with modern invariant theory via quadratic line complexes and Coble quadrics, producing a coherent picture of the moduli and birational geometry of these sections. It also demonstrates the rich interplay between representation theory, algebraic geometry of Fano varieties, and classical Kummer geometry, yielding concrete descriptions of orbit stratifications, Hilbert schemes, and quantum cohomology in this context.

Abstract

Following previous work by A. Kuznetsov, we study the Fano manifolds obtained as linear sections of the spinor tenfold in . Up to codimension three there are finitely many such sections, up to projective equivalence. In codimension four there are three moduli, and this family is particularly interesting because of its relationship with Kummer surfaces on the one hand, and a grading of the exceptional Lie algebra on the other hand. We show how the two approaches are intertwined, and we prove that codimension four sections of the spinor tenfolds and Kummer surfaces have the very same GIT moduli space. The Lie theoretic viewpoint provides a wealth of additional information. In particular we locate and study the unique section admitting an action of ; similarly to the Mukai-Umemura variety in the family of prime Fano threefold of genus 12, it is a compactification of a quotient by a finite group.
Paper Structure (31 sections, 63 theorems, 156 equations)

This paper contains 31 sections, 63 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. The spinor tenfold
  3. Codimension two sections
  4. Codimension three sections
  5. Classification of smooth sections
  6. A model for the generic section
  7. A model for the special smooth section
  8. A brief reminder on Kummer surfaces
  9. The spinor Kummer complex
  10. Codimension four sections: Kummer surfaces
  11. A grading of ${\mathfrak e}_8$
  12. From $\mathrm{Spin}_{10}$ to the Kummer configuration
  13. The Cartan subspace and the little Weyl group
  14. Quadratic complexes of lines from ${\mathbb{C}}^4\otimes\Delta_+$
  15. Coble quadrics from ${\mathbb{C}}^4\otimes\Delta_+$
  16. ...and 16 more sections

Key Result

Proposition 2.1

(i) Lines in $X$ are parametrized by $OG(3,10)$, planes by $OF(2,5^\vee,10)$. (ii) There are two types of ${\mathbb{P}}^3$'s in $X$: maximal ones parametrized by $OG(2,10)$, extendable ones parametrized by $OF(1,5^\vee,10)$. (iii) ${\mathbb{P}}^4$'s in $X$ are parametrized by $X^\vee$ and are all ma

Theorems & Definitions (71)

  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.3
  • ...and 61 more