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# parametrize(MultiprojectiveVariety) -- try to get a parametrization of a multi-projective variety

## Synopsis

• Function: parametrize
• Usage:
parametrize X
• Inputs:
• X, , a rational $k$-dimensional subvariety of $\mathbb{P}^{r_1}\times\cdots\times\mathbb{P}^{r_n}$
• Outputs:
• , a birational map from $\mathbb{P}^k$ to $X$ (or an error if it fails)

## Description

Currently, this function works in particular for linear varieties, quadrics, varieties of minimal degree, Grassmannians, Severi varieties, del Pezzo fivefolds, and some types of Fano fourfolds.

 i1 : K = ZZ/65521; i2 : X = PP_K^{2,4,1,3}; o2 : ProjectiveVariety, PP^2 x PP^4 x PP^1 x PP^3 i3 : f = parametrize X; o3 : MultirationalMap (rational map from PP^10 to X) i4 : Y = random({{1,0,0,0},{0,1,0,0},{0,1,0,0},{0,0,0,1}},0_X); o4 : ProjectiveVariety, 6-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 i5 : g = parametrize Y; o5 : MultirationalMap (rational map from PP^6 to Y) i6 : Z = random({{1,1,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1},{0,0,0,1}},0_X); o6 : ProjectiveVariety, 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 i7 : h = parametrize Z; o7 : MultirationalMap (rational map from PP^5 to Z) i8 : describe h o8 = multi-rational map consisting of 4 rational maps source variety: PP^5 target variety: 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 cut out by 5 hypersurfaces of multi-degrees (0,0,0,1)^2 (0,0,1,0)^1 (0,1,0,0)^1 (1,1,0,0)^1 base locus: threefold in PP^5 cut out by 6 hypersurfaces of degrees 2^1 4^5 dominance: true multidegree: {1, 6, 15, 31, 50, 50} degree: 1 degree sequence (map 1/4): [2] degree sequence (map 2/4): [2] degree sequence (map 3/4): [0] degree sequence (map 4/4): [2] coefficient ring: K i9 : describe inverse h o9 = multi-rational map consisting of one single rational map source variety: 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 cut out by 5 hypersurfaces of multi-degrees (0,0,0,1)^2 (0,0,1,0)^1 (0,1,0,0)^1 (1,1,0,0)^1 target variety: PP^5 base locus: threefold in PP^2 x PP^4 x PP^1 x PP^3 cut out by 23 hypersurfaces of multi-degrees (0,0,0,1)^2 (0,0,1,0)^1 (0,1,0,0)^1 (0,1,0,2)^1 (0,2,0,1)^3 (1,0,0,2)^1 (1,1,0,0)^1 (1,1,0,1)^6 (1,2,0,0)^3 (2,0,0,1)^2 (2,1,0,0)^2 dominance: true multidegree: {50, 50, 31, 15, 6, 1} degree: 1 degree sequence (map 1/1): [(1,1,0,1)] coefficient ring: K i10 : A = matrix pack(5,for i to 24 list random(1,ring PP_K^8)), A = A - transpose A; i11 : W = projectiveVariety pfaffians(4,A); o11 : ProjectiveVariety, 5-dimensional subvariety of PP^8 i12 : parametrize W o12 = multi-rational map consisting of one single rational map source variety: PP^5 target variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2 dominance: true degree: 1 o12 : MultirationalMap (birational map from PP^5 to W) i13 : parametrize (W ** (point W)) o13 = multi-rational map consisting of 2 rational maps source variety: PP^5 target variety: 5-dimensional subvariety of PP^8 x PP^8 cut out by 13 hypersurfaces of multi-degrees (0,1)^8 (2,0)^5 o13 : MultirationalMap (rational map from PP^5 to 5-dimensional subvariety of PP^8 x PP^8)