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# RAT List -- define a multi-rational map

## Synopsis

• Operator: SPACE
• Usage:
H {F1,F2,...}
check H {F1,F2,...}
• Inputs:
• H, , the hom-set of rational maps between two multi-projective varieties X and Y
• F, a list, a list of homogeneous row matrices over the ring of X of lengths compatible with the shape of Y (these matrices can be replaced by the lists of their entries)
• Outputs:
• , the map from X to Y defined on coordinates by sending a point p to the point (F1(p),F2(p),...)

## Description

This is a more controlled way of constructing rational maps than just using the multirationalMap function.

 i1 : R = ring PP_(ZZ/3)^{1,2}; i2 : F = {apply(2,i -> random({1,1},R)), apply(4,i -> random({0,1},R)), apply(3,i -> random({1,0},R))} o2 = {{- x0 x1 + x0 x1 , x0 x1 + x0 x1 + x0 x1 + x0 x1 }, {- x1 - x1 - 1 0 0 1 0 0 1 0 0 2 1 2 0 1 ------------------------------------------------------------------------ x1 , - x1 - x1 , - x1 + x1 - x1 , x1 - x1 - x1 }, {-x0 , - x0 - 2 0 2 0 1 2 0 1 2 0 0 ------------------------------------------------------------------------ x0 , x0 + x0 }} 1 0 1 o2 : List i3 : H = Hom(PP_(ZZ/3)^{1,2},PP_(ZZ/3)^{1,3,2}) o3 = H o3 : Hom(PP^1 x PP^2,PP^1 x PP^3 x PP^2) i4 : f = H F; o4 : MultirationalMap (rational map from PP^1 x PP^2 to PP^1 x PP^3 x PP^2) i5 : show f o5 = -- multi-rational map -- ZZ ZZ source: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 , x1 ]) 3 0 1 3 0 1 2 ZZ ZZ ZZ target: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 , x1 , x1 ]) x Proj(--[x2 , x2 , x2 ]) 3 0 1 3 0 1 2 3 3 0 1 2 -- rational map 1/3 -- map 1/3, one of its representatives: { - x0 x1 + x0 x1 , 1 0 0 1 x0 x1 + x0 x1 + x0 x1 + x0 x1 0 0 1 0 0 2 1 2 } -- rational map 2/3 -- map 2/3, one of its representatives: { - x1 - x1 - x1 , 0 1 2 - x1 - x1 , 0 2 - x1 + x1 - x1 , 0 1 2 x1 - x1 - x1 0 1 2 } -- rational map 3/3 -- map 3/3, one of its representatives: { -x0 , 0 - x0 - x0 , 0 1 x0 + x0 0 1 }

The following equality is satisfied for every rational map f.

 i6 : assert( f == (Hom(source f,target f)) entries f )

Here it is shown how to make a dominant rational map.

 i7 : H' = Hom(PP_(ZZ/3)^{1,2},Dominant); o7 : Hom(PP^1 x PP^2,*,Dominant) i8 : H' F o8 = multi-rational map consisting of 3 rational maps source variety: PP^1 x PP^2 target variety: threefold in PP^1 x PP^3 x PP^2 cut out by 3 hypersurfaces of multi-degrees (0,0,1)^1 (0,1,0)^1 (1,1,1)^1 dominance: true o8 : MultirationalMap (dominant rational map from PP^1 x PP^2 to threefold in PP^1 x PP^3 x PP^2) i9 : assert(image oo == target oo)

## Ways to use this method:

• RAT List -- define a multi-rational map