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# RAT MultiprojectiveVariety -- rational map defined by a linear system of hypersurfaces through a variety

## Synopsis

• Operator: SPACE
• Usage:
H(Z,d,m)
H(Z,d)
H(Z)
• Inputs:
• , the hom-set of rational maps between two varieties X and Y
• Z, , a subvariety of $X$; d is a (multi-)degree and e is a multiplicity (by default, e=1 and d is the maximum degree of the generators)
• Outputs:
• , the map $X\dashrightarrow Y$ defined by the linear system of hypersurfaces of degree $d$ having points of multiplicity $e$ along $Z$

## Description

This is another way of calling the method (rationalMap,MultiprojectiveVariety,List,ZZ).

 i1 : X = random(3,0_(PP_(ZZ/17)^2)); o1 : ProjectiveVariety, curve in PP^2 i2 : Y = PP_(ZZ/17)^4; o2 : ProjectiveVariety, PP^4 i3 : H = Hom(X,Y); o3 : Hom(X,Y) i4 : Z = point X + point X; o4 : ProjectiveVariety, 0-dimensional subvariety of PP^2 i5 : H(Z,3,2) -- map defined by the cubics with double points along Z o5 = multi-rational map consisting of one single rational map source variety: curve in PP^2 defined by a form of degree 3 target variety: PP^4 o5 : MultirationalMap (rational map from X to Y) i6 : show oo o6 = -- multi-rational map -- ZZ source: subvariety of Proj(--[x , x , x ]) defined by 17 0 1 2 { 3 2 2 3 2 2 2 2 3 x - 3x x - 7x x - 7x - 4x x - 7x x x - 5x x + 6x x - 4x x - 7x 0 0 1 0 1 1 0 2 0 1 2 1 2 0 2 1 2 2 } ZZ target: Proj(--[x , x , x , x , x ]) 17 0 1 2 3 4 -- rational map 1/1 -- map 1/1, one of its representatives: { 2 2 2 2 x x + 7x x - 5x x + 3x x x - 3x x , 0 1 0 1 0 2 0 1 2 0 2 2 3 2 2 x x + 7x - 5x x x + 3x x - 3x x , 0 1 1 0 1 2 1 2 1 2 2 2 2 3 x x x + 7x x - 5x x + 3x x - 3x , 0 1 2 1 2 0 2 1 2 2 2 3 2 2 x x + 3x + 3x x x - 4x x - 6x x , 0 1 1 0 1 2 1 2 1 2 2 2 2 2 3 x x + 3x x + 3x x - 4x x - 6x 0 2 1 2 0 2 1 2 2 }

As always, you can invoke the functions check and isWellDefined to check if the map is valid.

 i7 : X = PP_(ZZ/17)^{1,1}; o7 : ProjectiveVariety, PP^1 x PP^1 i8 : H = Hom(X,image segreEmbedding X); o8 : Hom(X,surface in PP^3) i9 : check H(0_X,{1,1}) -- Segre embedding of X o9 = multi-rational map consisting of one single rational map source variety: PP^1 x PP^1 target variety: surface in PP^3 defined by a form of degree 2 o9 : MultirationalMap (rational map from X to surface in PP^3) i10 : show oo o10 = -- multi-rational map -- ZZ ZZ source: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 ]) 17 0 1 17 0 1 ZZ target: subvariety of Proj(--[t , t , t , t ]) defined by 17 0 1 2 3 { t t - t t 1 2 0 3 } -- rational map 1/1 -- map 1/1, one of its representatives: { x0 x1 , 0 0 x0 x1 , 0 1 x0 x1 , 1 0 x0 x1 1 1 }