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D.5.1.7 rncItProjOdd
Procedure from library paraplanecurves.lib (see paraplanecurves_lib).
- Usage:
- rncItProjOdd(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 odd.
- 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:
- ideal PHI defining an isomorphic projection of C to PP^1.
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 = -x7-10x5y2-10x4y3-3x3y4+8x2y5+7xy6+11y7+3x6+10x5y +30x4y2
+26x3y3-13x2y4-29xy5-33y6-3x5-20x4y-33x3y2-8x2y3+37xy4+33y5
+x4+10x3y+13x2y2-15xy3-11y4;
f = homog(f,z);
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(4)*y(5)-y(3)*y(6)
==> RNC[2]=y(2)*y(5)-y(1)*y(6)
==> RNC[3]=y(4)^2-y(2)*y(6)
==> RNC[4]=y(3)*y(4)-y(1)*y(6)
==> RNC[5]=11*y(1)*y(4)+7*y(2)*y(4)+8*y(1)*y(6)-3*y(2)*y(6)-10*y(3)*y(6)-10*y\
(4)*y(6)-y(6)^2
==> RNC[6]=y(3)^2-y(1)*y(5)
==> RNC[7]=11*y(2)*y(3)+7*y(2)*y(4)+8*y(1)*y(6)-3*y(2)*y(6)-10*y(3)*y(6)-10*y\
(4)*y(6)-y(6)^2
==> RNC[8]=11*y(1)*y(3)+7*y(1)*y(4)+8*y(1)*y(5)-10*y(3)*y(5)-3*y(1)*y(6)-10*y\
(3)*y(6)-y(5)*y(6)
==> RNC[9]=121*y(1)*y(2)+77*y(2)^2-89*y(2)*y(4)-174*y(1)*y(6)-86*y(2)*y(6)+80\
*y(3)*y(6)+69*y(4)*y(6)+8*y(6)^2
==> RNC[10]=1331*y(1)^2-539*y(2)^2+1246*y(2)*y(4)-1914*y(1)*y(5)+880*y(3)*y(5\
)+984*y(1)*y(6)+335*y(2)*y(6)-691*y(3)*y(6)-1373*y(4)*y(6)+88*y(5)*y(6)-1\
45*y(6)^2
rncItProjOdd(RNC);
==> _[1]=121*y(3)+77*y(4)
==> _[2]=-11*y(5)-7*y(6)
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See also:
rncItProjEven.
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