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D.5.1.8 rncItProjEven
Procedure from library paraplanecurves.lib (see paraplanecurves_lib).
- Usage:
- rncItProjEven(I); I ideal
- Assume:
- I is a homogeneous ideal in the basering with n+1 variables
defining a rational normal curve C in PP^n with n even.
- Note:
- The procedure will fail or give a wrong output if I is not the
ideal of a rational normal curve. It will test whether n is odd.
- Return:
- ring with an ideal CONIC defining a conic C2 in PP^2.
In addition, an ideal PHI in the basering defining an isomorphic
projection of C to C2 will be exported.
Note that the entries of PHI should be considered as
representatives of elements in R/I, where R is the basering.
- Theory:
- We iterate the procedure rncAntiCanonicalMap to obtain PHI.
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 = mapToRatNormCurve(f,adj);
==> //'mapToRatNorm' created a ring together with an ideal RNC.
==> // Supposing you typed, say, def RPn = mapToRatNorm(f,AI);
==> // you may access the ideal by typing
==> // setring RPn; RNC;
setring(Rn);
RNC;
==> RNC[1]=y(5)*y(6)-y(4)*y(7)
==> RNC[2]=y(4)*y(6)-y(3)*y(7)
==> RNC[3]=y(2)*y(6)-y(1)*y(7)
==> RNC[4]=y(4)*y(5)-y(2)*y(7)
==> RNC[5]=y(3)*y(5)-y(1)*y(7)
==> RNC[6]=y(1)*y(5)-y(7)^2
==> RNC[7]=y(4)^2-y(1)*y(7)
==> RNC[8]=y(3)*y(4)-y(1)*y(6)
==> RNC[9]=y(2)*y(4)-y(1)*y(5)
==> RNC[10]=y(1)*y(4)-y(6)*y(7)
==> RNC[11]=y(2)*y(3)-y(6)*y(7)
==> RNC[12]=y(1)*y(3)-y(6)^2
==> RNC[13]=y(2)^2-y(5)*y(7)
==> RNC[14]=y(1)*y(2)-y(4)*y(7)
==> RNC[15]=y(1)^2-y(3)*y(7)
==> RNC[16]=y(1)*y(6)^2-y(3)^2*y(7)
==> RNC[17]=y(6)^4-y(3)^3*y(7)
def Rc = rncItProjEven(RNC);
PHI;
==> PHI[1]=-y(7)
==> PHI[2]=-y(2)
==> PHI[3]=-y(5)
setring Rc;
CONIC;
==> y(2)^2-y(1)*y(3)
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See also:
rncItProjOdd.
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