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SpecialFanoFourfolds :: parametrize(SpecialCubicFourfold)

parametrize(SpecialCubicFourfold) -- rational parametrization

Synopsis

Description

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 0.149381 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

See also

Ways to use this method: