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 |