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# RationalMapping * RationalMapping -- compose rational maps between projective varieties

• Operator: *

## 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