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D.5.11.2 invertBirMap
Procedure from library paraplanecurves.lib (see paraplanecurves_lib).
- Usage:
- invertBirMap(phi, I); phi ideal, I ideal
- Assume:
- The ideal phi in the basering R represents a birational map of the
variety given by the ideal I in R to its image in projective space
P = PP^(size(phi)-1).
- Note:
- The procedure might fail or give a wrong output if phi does
not define a birational map.
- Return:
- ring, the coordinate ring of P, with an ideal named J and an ideal
named psi.
The ideal J defines the image of phi.
The ideal psi gives the inverse of phi.
Note that the entries of psi should be considered as representatives
of classes in the quotient ring R/J.
- Theory:
- We compute the ideal I(G) in R**S of the graph G of phi.
The ideal J is given by the intersection of I(G) with S.
The map psi is given by a relation mod J of those relations
in I(G) which are linear in the variables of R.
Example:
| | LIB "paraplanecurves.lib";
ring R = 0,(x,y,z),dp;
poly f = y^8-x^3*(z+x)^5;
ideal adj = adjointIdeal(f);
def Rn = invertBirMap(adj,ideal(f));
==> // 'invertBirMap' created a ring together with two ideals J and psi.
==> // Supposing you typed, say, def RPn = invertBirMap(phi,I);
==> // you may access the ideals by typing
==> // setring RPn; J; psi;
setring(Rn);
J;
==> J[1]=y(5)*y(6)-y(4)*y(7)
==> J[2]=y(4)*y(6)-y(3)*y(7)
==> J[3]=y(2)*y(6)-y(1)*y(7)
==> J[4]=y(4)*y(5)-y(2)*y(7)
==> J[5]=y(3)*y(5)-y(1)*y(7)
==> J[6]=y(1)*y(5)-y(7)^2
==> J[7]=y(4)^2-y(1)*y(7)
==> J[8]=y(3)*y(4)-y(1)*y(6)
==> J[9]=y(2)*y(4)-y(7)^2
==> J[10]=y(1)*y(4)-y(6)*y(7)
==> J[11]=y(2)*y(3)-y(6)*y(7)
==> J[12]=y(1)*y(3)-y(6)^2
==> J[13]=y(2)^2-y(5)*y(7)
==> J[14]=y(1)*y(2)-y(4)*y(7)
==> J[15]=y(1)^2-y(3)*y(7)
==> J[16]=y(1)*y(6)^2-y(3)^2*y(7)
==> J[17]=y(6)^4-y(3)^3*y(7)
psi;
==> psi[1]=-y(6)^2
==> psi[2]=-y(4)*y(7)
==> psi[3]=y(6)^2-y(5)*y(7)
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