beauvilleMap -- construction of Beauville for complete intersections of three quadrics in P^7

Synopsis

• Usage:

  f = beauvilleMap X

• Inputs:
  • X, a complete intersection of three quadrics in PP^7
• Outputs:
  • f, a multi-rational map, a birational map from $X$ to a fourfold $Y\subset\mathbb{P}^2\times\mathbb{P}^5$ whose first projection $Y\to\mathbb{P}^2$ is a quadric surface bundle

Description

Smooth intersections of three quadrics in $\mathbb{P}^7$ are birational to quadric surface bundles over $\mathbb{P}^2$ with discriminant curve of degree 8. This is a construction of Beauville; see e.g. Proposition 6 in the paper Intersections of three quadrics in P7, by B. Hassett, A. Pirutka, and Y. Tschinkel.

i1 : X = specialFourfold random({5:{1}},0_(PP_(ZZ/33331)^7));

o1 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 1 and sectional genus 0

i2 : f = beauvilleMap X;

o2 : MultirationalMap (birational map from X to 4-dimensional subvariety of PP^2 x PP^5)

i3 : Y = target f;

o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^5

i4 : inverse f;

o4 : MultirationalMap (birational map from Y to X)

i5 : first projectionMaps target f;

o5 : QuadricFibration (dominant rational map from Y to PP^2)