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# MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map

## Synopsis

• Operator: ^**
• Usage:
Phi^** Y
Phi^* Y
• Inputs:
• Phi,
• Y, , a subvariety of the ambient multi-projective space of the target of Phi
• Outputs:
• , the (closure of the) inverse image of Y via Phi

## Description

 i1 : ZZ/300007[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2}, g = rationalMap {x_1^2-x_0*x_2, x_0*x_3, x_1*x_3, x_2*x_3, x_3^2}; i2 : Phi = last graph rationalMap {f,g}; o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) i3 : Y = projectiveVariety ideal(random({1,1},ring target Phi), random({1,1},ring target Phi)); o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 i4 : time X = Phi^* Y; -- used 5.22588s (cpu); 4.14472s (thread); 0s (gc) o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension 2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 ) i5 : dim X, degree X, degrees X o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1), ------------------------------------------------------------------------ ({0, 8, 0}, 1), ({1, 0, 1}, 7), ({1, 0, 2}, 4), ({1, 1, 0}, 2), ({1, 2, ------------------------------------------------------------------------ 0}, 1), ({1, 4, 0}, 1), ({2, 0, 1}, 2), ({2, 1, 0}, 2), ({3, 0, 1}, 2), ------------------------------------------------------------------------ ({3, 1, 0}, 1), ({4, 0, 0}, 4)}) o5 : Sequence