# Singular

#### D.5.4.8 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)