next | previous | forward | backward | up | top | index | toc | Macaulay2 website
SpecialFanoFourfolds :: parametrize(SpecialCubicFourfold)

parametrize(SpecialCubicFourfold) -- rational parametrization

Synopsis

Description

Some special cubic fourfolds and GM fourfolds are known to be rational. In this case, the method tries to obtain a birational map from $\mathbb{P}^4$ (or, e.g., from a quadric hypersurface in $\mathbb{P}^5$) to the fourfold.

i1 : X = specialCubicFourfold "quintic del Pezzo surface";

o1 : SpecialCubicFourfold (Cubic fourfold containing a surface of degree 5 and sectional genus 1)
i2 : time phi = parametrize X;
     -- used 0.178709 seconds

o2 : RationalMap (birational map from PP^4 to hypersurface in PP^5)
i3 : describe phi

o3 = rational map defined by forms of degree 4
     source variety: PP^4
     target variety: smooth cubic hypersurface in PP^5
     dominance: true
     birationality: true (the inverse map is already calculated)
     coefficient ring: ZZ/65521
i4 : X' = specialGushelMukaiFourfold "tau-quadric";

o4 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 2 and sectional genus 0)
i5 : time phi' = parametrize X';
     -- used 0.198318 seconds

o5 : RationalMap (birational map from PP^4 to 4-dimensional subvariety of PP^8)
i6 : describe phi'

o6 = rational map defined by forms of degree 4
     source variety: PP^4
     target variety: 4-dimensional variety of degree 10 in PP^8 cut out by 6 hypersurfaces of degree 2
     dominance: true
     birationality: true (the inverse map is already calculated)
     coefficient ring: ZZ/65521

See also