Some special cubic fourfolds are known to be rational. In this case, the function 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 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1 |
i2 : time phi = parametrize X; -- used 0.196973 seconds o2 : MultirationalMap (birational map from PP^4 to X) |
i3 : describe phi o3 = multi-rational map consisting of one single rational map source variety: PP^4 target variety: hypersurface in PP^5 defined by a form of degree 3 base locus: surface in PP^4 cut out by 6 hypersurfaces of degree 4 dominance: true multidegree: {1, 4, 7, 6, 3} degree: 1 degree sequence (map 1/1): [4] coefficient ring: ZZ/65521 |
i4 : describe phi^-1 o4 = multi-rational map consisting of one single rational map source variety: hypersurface in PP^5 defined by a form of degree 3 target variety: PP^4 base locus: surface in PP^5 cut out by 5 hypersurfaces of degree 2 dominance: true multidegree: {3, 6, 7, 4, 1} degree: 1 degree sequence (map 1/1): [2] coefficient ring: ZZ/65521 |