Description
Given a rational map $f : X \to Y$, inverseOfMap computes the inverse of the induced map $X \to \overline{f(X)}$, provided it is birational." The target and source must be varieties; their defining ideals must be prime.
If AssumeDominant is set to true (default is false) then it assumes that the rational map of projective varieties is dominant, otherwise the function will compute the image by finding the kernel of $f$.
The Strategy option can be set to HybridStrategy (default), SimisStrategy, ReesStrategy, or SaturationStrategy. Note that SimisStrategy will never terminate for non-birational maps. If CheckBirational is set to false (default is true), then no check for birationality will be done. If it is set to true and the map is not birational, then an error will be thrown if you are not using SimisStrategy. The option HybridLimit controls HybridStrategy. Larger values of HybridLimit (the default value is 15) will mean that SimisStrategy is executed longer, smaller values will mean that ReesStrategy will be switched to sooner.
i1 : R = ZZ/7[x,y,z];
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i2 : S = ZZ/7[a,b,c];
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i3 : h = map(R, S, {y*z, x*z, x*y});
o3 : RingMap R <-- S
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i4 : inverseOfMap (h, Verbosity=>0)
o4 = Proj S - - - > Proj R {-b*c, -a*c, -a*b}
o4 : RationalMapping
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Notice that removal of the leading minus signs would not change the projective map. Next let us compute the inverse of the blowup of $P^2$ at a point.
i5 : P5 = QQ[a..f];
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i6 : M = matrix{{a,b,c},{d,e,f}};
2 3
o6 : Matrix P5 <-- P5
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i7 : blowUpSubvar = P5/(minors(2, M)+ideal(b - d));
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i8 : h = map(blowUpSubvar, QQ[x,y,z],{a, b, c});
o8 : RingMap blowUpSubvar <-- QQ[x..z]
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i9 : g = inverseOfMap(h, Verbosity=>0)
4 3 3 3 2 2 2
o9 = Proj(QQ[x..z]) - - - > Proj blowUpSubvar {-x , -x y, -x z, -x y, -x y , -x y*z}
o9 : RationalMapping
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i10 : baseLocusOfMap(g)
o10 = ideal (y, x)
o10 : Ideal of QQ[x..z]
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i11 : baseLocusOfMap(h)
o11 = ideal 1
o11 : Ideal of blowUpSubvar
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The next example is a birational map on $\mathbb{P}^4$.
i12 : Q=QQ[x,y,z,t,u];
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i13 : phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
o13 : RingMap Q <-- Q
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i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
-- used 0.434825s (cpu); 0.336316s (thread); 0s (gc)
125 124 120 5 124 100 25 104 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 28 90 7 32 85 8 36 80 9 40 75 10 44 70 11 48 65 12 52 60 13 56 55 14 60 50 15 64 45 16 68 40 17 72 35 18 76 30 19 80 25 20 84 20 21 88 15 22 92 10 23 96 5 24 100 25 24 100 28 95 32 90 2 36 85 3 40 80 4 44 75 5 48 70 6 52 65 7 56 60 8 60 55 9 64 50 10 68 45 11 72 40 12 76 35 13 80 30 14 84 25 15 88 20 16 92 15 17 96 10 18 100 5 19 104 20 48 75 2 52 70 2 56 65 2 2 60 60 3 2 64 55 4 2 68 50 5 2 72 45 6 2 76 40 7 2 80 35 8 2 84 30 9 2 88 25 10 2 92 20 11 2 96 15 12 2 100 10 13 2 104 5 14 2 108 15 2 72 50 3 76 45 3 80 40 2 3 84 35 3 3 88 30 4 3 92 25 5 3 96 20 6 3 100 15 7 3 104 10 8 3 108 5 9 3 112 10 3 96 25 4 100 20 4 104 15 2 4 108 10 3 4 112 5 4 4 116 5 4 120 5 124
o14 = Proj Q - - - > Proj Q {x , x y, - x y + x z, x y - 5x y z + 10x y z - 10x y z + 5x y z - x z + x t, - y + 25x y z - 300x y z + 2300x y z - 12650x y z + 53130x y z - 177100x y z + 480700x y z - 1081575x y z + 2042975x y z - 3268760x y z + 4457400x y z - 5200300x y z + 5200300x y z - 4457400x y z + 3268760x y z - 2042975x y z + 1081575x y z - 480700x y z + 177100x y z - 53130x y z + 12650x y z - 2300x y z + 300x y z - 25x y z + x z - 5x y t + 100x y z*t - 950x y z t + 5700x y z t - 24225x y z t + 77520x y z t - 193800x y z t + 387600x y z t - 629850x y z t + 839800x y z t - 923780x y z t + 839800x y z t - 629850x y z t + 387600x y z t - 193800x y z t + 77520x y z t - 24225x y z t + 5700x y z t - 950x y z t + 100x y z t - 5x z t - 10x y t + 150x y z*t - 1050x y z t + 4550x y z t - 13650x y z t + 30030x y z t - 50050x y z t + 64350x y z t - 64350x y z t + 50050x y z t - 30030x y z t + 13650x y z t - 4550x y z t + 1050x y z t - 150x y z t + 10x z t - 10x y t + 100x y z*t - 450x y z t + 1200x y z t - 2100x y z t + 2520x y z t - 2100x y z t + 1200x y z t - 450x y z t + 100x y z t - 10x z t - 5x y t + 25x y z*t - 50x y z t + 50x y z t - 25x y z t + 5x z t - x t + x u}
o14 : RationalMapping
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Finally, we do an example of plane Cremona maps whose source is not minimally embedded.
i15 : R=QQ[x,y,z,t]/(z-2*t);
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i16 : F = {y*z*(x-z)*(x-2*y), x*z*(y-z)*(x-2*y),y*x*(y-z)*(x-z)};
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i17 : S = QQ[u,v,w];
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i18 : ident = rationalMapping map(S, S)
o18 = Proj S - - - > Proj S {u, v, w}
o18 : RationalMapping
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i19 : h = rationalMapping(R, S, F);
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i20 : g = inverseOfMap(h, Verbosity=>0)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
o20 = Proj S - - - > Proj R {- 2u v + 8u v*w - 6u*v w - 8u w + 12u*v*w - 4v w , - 2u v + 6u v*w - 4u*v w - 4u w + 6u*v*w - 2v w , - 2u v + 6u v*w - 6u*v w - 4u w + 8u*v*w - 4v w , - u v + 3u v*w - 3u*v w - 2u w + 4u*v*w - 2v w }
o20 : RationalMapping
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i21 : h*g == ident
o21 = true
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