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 0.0800078 seconds
o2 : MultirationalMap (birational map from PP^4 to X)
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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
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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
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