Macaulay2 » Documentation
Packages » SpecialFanoFourfolds :: parametrize(HodgeSpecialFourfold)
next | previous | forward | backward | up | index | toc

parametrize(HodgeSpecialFourfold) -- rational parametrization

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

See also

Ways to use this method:

The source of this document is in SpecialFanoFourfolds.m2:3499:0.