This method is mainly based on results contained in the classical paper Algebraic varieties with canonical curve sections, by L. Roth. In some examples, more strategies are available. For instance, if $X\subset\mathbb{P}^7$ is a 4-dimensional linear section of $\mathbb{G}(1,4)\subset\mathbb{P}^9$, then by passing Strategy=>1 (which is the default choice) we get the inverse of the projection from the plane spanned by a conic contained in $X$; while with Strategy=>2 we get the projection from the unique $\sigma_{2,2}$-plane contained in $X$ (Todd's result).
i1 : G'1'4 = ideal Grass(1,4,ZZ/65521); I = G'1'4 + ideal(random(1,ring G'1'4),random(1,ring G'1'4)) ZZ o1 : Ideal of -----[p ..p , p , p , p , p , p , p , p , p ] 65521 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 o2 = ideal (p p - p p + p p , p p - p p + p p , p p - p p + p p , p p - p p + 2,3 1,4 1,3 2,4 1,2 3,4 2,3 0,4 0,3 2,4 0,2 3,4 1,3 0,4 0,3 1,4 0,1 3,4 1,2 0,4 0,2 1,4 ---------------------------------------------------------------------------------------------------------------------------- p p , p p - p p + p p , - 32646p - 28377p + 26433p - 29566p + 3783p + 26696p + 8570p 0,1 2,4 1,2 0,3 0,2 1,3 0,1 2,3 0,1 0,2 1,2 0,3 1,3 2,3 0,4 ---------------------------------------------------------------------------------------------------------------------------- + 16659p + 8444p - 11230p , - 22394p - 8214p + 23752p - 30100p + 5071p + 25800p - 20388p - 1,4 2,4 3,4 0,1 0,2 1,2 0,3 1,3 2,3 0,4 ---------------------------------------------------------------------------------------------------------------------------- 14258p + 22523p - 12700p ) 1,4 2,4 3,4 ZZ o2 : Ideal of -----[p ..p , p , p , p , p , p , p , p , p ] 65521 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 |
i3 : (dim I -1, degree I,(genera I)_3) o3 = (4, 5, 1) o3 : Sequence |
i4 : time f = parametrizeFanoFourfold I; -- used 1.1993 seconds o4 : RationalMap (cubic rational map from PP^4 to 4-dimensional subvariety of PP^9) |
i5 : describe f! o5 = rational map defined by forms of degree 3 source variety: PP^4 target variety: 4-dimensional variety of degree 5 in PP^9 cut out by 7 hypersurfaces of degrees (1,1,2,2,2,2,2) dominance: true birationality: true (the inverse map is already calculated) projective degrees: {1, 3, 5, 5, 5} number of minimal representatives: 1 dimension base locus: 2 degree base locus: 4 coefficient ring: ZZ/65521 |
The object parametrizeFanoFourfold is a method function with options.