parametrize(HodgeSpecialFourfold) -- rational parametrization

Synopsis

• Function: parametrize
• Usage:
parametrize X
• Inputs:
• Outputs:
• , a birational map from a rational fourfold to X

Description

Some Hodge-special 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 = specialFourfold surface {3,4}; o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1 i2 : phi = parametrize X; 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 : Y = specialFourfold "tau-quadric"; o4 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0 i5 : psi = parametrize Y; o5 : MultirationalMap (birational map from PP^4 to Y) i6 : describe psi o6 = multi-rational map consisting of one single rational map source variety: PP^4 target variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2 base locus: surface in PP^4 cut out by 5 hypersurfaces of degrees 3^1 4^4 dominance: true multidegree: {1, 4, 8, 10, 10} degree: 1 degree sequence (map 1/1): [4] coefficient ring: ZZ/65521
 i7 : Z = specialFourfold "plane in PP^7"; o7 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 1 and sectional genus 0 i8 : eta = parametrize Z; o8 : MultirationalMap (birational map from PP^4 to Z) i9 : describe eta o9 = multi-rational map consisting of one single rational map source variety: PP^4 target variety: 4-dimensional subvariety of PP^7 cut out by 3 hypersurfaces of degree 2 base locus: surface in PP^4 cut out by 4 hypersurfaces of degrees 3^1 4^3 dominance: true multidegree: {1, 4, 7, 8, 8} degree: 1 degree sequence (map 1/1): [4] coefficient ring: ZZ/65521