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D.15.10.3 curveDeligneNumber
Procedure from library curveInv.lib (see curveInv_lib).
- Usage:
- curveDeligneNumber(I); I ideal
- Assume:
- I is a radical ideal, dim(R/I) = 1
- Return:
- the Deligne number of R/I
- Remarks:
- The Deligne number e satisfies by definition: e = 3delta - m.
So the algorithm splits the computation into two parts: one part computes the delta
invariant, the other part the colength of derivations m.
- Note:
- the optional parameter can be used if the normalization has already
been computed. If a list L contains the output of the procedure
normal (with options prim, wd and usering if the ring has a mixed ordering),
apply curveDeligneNumber(I,L)
Example:
| | LIB "curveInv.lib";
//////////////////////////////
// Deligne number of curves //
//////////////////////////////
// Example 1:
ring R = 0,(x,y,z),ds;
ideal I = x2-y4z,z3y2+xy2;
I = std(radical(I));
curveDeligneNumber(I);
==> 30
// Example 2:
ring S = 0,(x,y),ds;
ideal I = (x+y)*(x2-y3);
curveDeligneNumber(I);
==> 5
// Example 3:
ideal J = (x2-y3)*(x2+y2)*(x-y);
curveDeligneNumber(J);
==> 15
// Let us also compute the milnor number of this complete intersection:
milnor(J);
==> 17
// We see that the Milnor number is bigger than the Deligne number. Hence, this
// curve cannot be quasi homogeneous. This can also be verified by Saitos criterion:
reduce(J[1],std(jacob(J[1])));
==> -1/5y6+19/50y7
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See also:
curveColengthDerivations;
curveDeltaInv.
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