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RationalMapping * RationalMapping -- compose rational maps between projective varieties

Synopsis

Description

This allows one to compose two rational maps between projective varieties.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing
 
i2 : P2 = Proj(R)

o2 = P2

o2 : ProjectiveVariety
 
i3 : phi = rationalMapping (P2, P2, {y*z,x*z,x*y})

o3 = P2 - - - > P2   {y*z, x*z, x*y}

o3 : RationalMapping
 
i4 : ident = rationalMapping (P2, P2, {x,y,z})

o4 = P2 - - - > P2   {x, y, z}

o4 : RationalMapping
 
i5 : phi*phi == ident

o5 = true

Raising a map to the negative first power means computing the inverse birational map. Raising a map to the first power simply returns the map itself. In the next example we compute the blowup of a point on $P^2$ and its inverse.

i6 : P5ring = ZZ/103[a..f];
 
i7 : R = ZZ/103[x,y,z];
 
i8 : P2 = Proj R;
 
i9 : identP2 = rationalMapping(P2, P2, {x,y,z});
 
i10 : M = matrix{{a,b,c},{d,e,f}};

                   2           3
o10 : Matrix P5ring  <-- P5ring
 
i11 : blowUp = Proj(P5ring/(minors(2, M)+ideal(b - d)));
 
i12 : identBlowUp = rationalMapping(blowUp, blowUp, {a,b,c,d,e,f});
 
i13 : tau = rationalMapping(P2, blowUp,{a, b, c});
 
i14 : tauInverse = tau^-1;
 
i15 : tau*tauInverse == identP2 --a map composed with its inverse is the identity

o15 = true
 
i16 : tauInverse*tau == identBlowUp

o16 = true

Note that one can only raise maps to powers (with the exception of 1 and -1) if the source and target are the same. In that case, raising a map to a negative power means compose the inverse of a map with itself. We illustrate this with the quadratic transformation on $P^2$ that we started with (an transformation of order 2 in the Cremona group).

i17 : phi^3 == phi^-1

o17 = true
 
i18 : phi^-2 == ident

o18 = true
 
i19 : phi^1 == ident

o19 = false

See also

Ways to use this method: