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The Weil Decoration of the Horrocks-Mumford Bundle

Klaus Altmann, Andreas Hochenegger, Frederik Witt

Abstract

For a normal algebraic variety we generalise the relation between reflexive rank one sheaves and Weil divisors to reflexive sheaves of arbitrary rank and so-called Weil decorations. As an application, we define and study a natural generalisation of the celebrated Horrocks-Mumford bundle.

The Weil Decoration of the Horrocks-Mumford Bundle

Abstract

For a normal algebraic variety we generalise the relation between reflexive rank one sheaves and Weil divisors to reflexive sheaves of arbitrary rank and so-called Weil decorations. As an application, we define and study a natural generalisation of the celebrated Horrocks-Mumford bundle.
Paper Structure (29 sections, 20 theorems, 177 equations, 1 figure)

This paper contains 29 sections, 20 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weil decorations
  3. Pre-Weil decorations
  4. Weil decorations and their reflexive sheaf
  5. The Weil decoration of a reflexive sheaf
  6. Slices
  7. P-orthogonal bases
  8. The dual of a reflexive sheaf
  9. Toric slice s
  10. Morphisms of Weil decorations
  11. The category of Weil decorations
  12. Kernels
  13. Quotients
  14. Example: the Euler diagram
  15. The Horrocks-Mumford bundle
  16. ...and 14 more sections

Key Result

Proposition 2.3

Let $\mathcal{W}\colon\mathcal{V}\to\operatorname{\operatorname{Div}}(X)$ be a pre-Weil decoration. Then defines the quasi-coherent sheaf $\mathcal{O}_X(\mathcal{W})$ associated with $\mathcal{W}$. Its generic stalk is $\mathcal{V}$.

Figures (1)

  • Figure 1: Visualisation of the mixed terms in $\mathcal{W}_{\mathcal{H}\space{\mathcal{M}}}(f,g)_{P}(\underline{h})$ as coloured edges of a pentagon.

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Example 2.4
  • proof : Proof of Proposition \ref{['prop:QCSPWD']} .
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • proof : Proof of Proposition \ref{['prop:SemiNorm']}
  • Definition 2.8
  • ...and 48 more