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On the generalised Kummer fourfold of the Jacobian of a genus two curve

Samuel Boissiere, Marc Nieper-Wisskirchen, Gregory Sankaran

Abstract

We construct a birational model of the generalised Kummer fourfold of the Jacobian of a genus two curve, based on a geometric interpretation of the addition law on this Jacobian, obtained by the properties of the linear system of cubics on that curve. We show that our model has mild singularities and that it admits a finite ramified covering to the four-dimensional projective space.

On the generalised Kummer fourfold of the Jacobian of a genus two curve

Abstract

We construct a birational model of the generalised Kummer fourfold of the Jacobian of a genus two curve, based on a geometric interpretation of the addition law on this Jacobian, obtained by the properties of the linear system of cubics on that curve. We show that our model has mild singularities and that it admits a finite ramified covering to the four-dimensional projective space.

Paper Structure

This paper contains 30 sections, 21 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A rational contraction to the generalised Kummer fourfold
  3. The second generalised Kummer variety of an abelian surface
  4. The birational model
  5. The divisorial contraction to the Chow quotient
  6. The rational contraction
  7. The genus two curves
  8. The Jacobian of a genus two curve
  9. The linear system of cubics
  10. Interpretation of the linear system of cubics as a degeneracy locus
  11. The linear system of conics
  12. A degree 15 covering of the linear system of cubics
  13. Covering maps and Galois closure
  14. Organising six points on the curve into three pairs
  15. The degree 15 covering
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $C$ be a smooth genus two curve. The linear system of cubics embeds $C$ in $(\mathbb P^4)^\vee$ and the dual variety $C^\ast\subset \mathbb P^4$ of $C$ is a degree $14$ irreducible hypersurface. The second generalised Kummer variety ${\mathop{\mathrm{Kum}}\nolimits}^{2}({\rm Jac}(C))$ of the Jac

Figures (2)

  • Figure 1: A genus $2$ curve (in blue) and a cubic interpolation (in red) intersecting in $6$ points on the affine chart $y=1$ of coordinates $x$ (abscissa) and $z$ (ordinate).
  • Figure 2: A genus $2$ curve (in blue) and a cubic interpolation (in red) intersecting consisting in three vertical lines intersecting in $6$ points on the affine chart $y=1$ of coordinates $x$ (abscissa) and $z$ (ordinate).

Theorems & Definitions (53)

  • Theorem 1.1: Corollary \ref{['cor:main1']}
  • Proposition 1.2: Proposition \ref{['prop:multiple_4plane']}
  • Proposition 1.3: Propositions \ref{['prop:birational_F1']} and \ref{['prop:birational_F2']}
  • Proposition 1.4: Proposition \ref{['prop:marc']}
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • ...and 43 more