D.4.5.3 reg_moncurve

Procedure from library mregular.lib (see mregular_lib).

Usage:

reg_moncurve (a0,...,an) ; ai integers with a0=0 < a1 < ... < an=:d

Return:

an integer, the Castelnuovo-Mumford regularity of the projective monomial curve C in Pn parametrically defined by:
x(0)=t^d , x(1)=s^(a1)t^(d-a1), ... , x(n)=s^d.

Assume:

a0=0 < a1 < ... < an are integers and the base field is infinite.

Note:

The defining ideal I(C) in S is determined using elimination. The procedure reg_curve is improved in this case since one knows beforehand that the dimension is 2, that the variables are in Noether position, that I(C) is prime.
If printlevel > 0 (default = 0) additional information is displayed. In particular, says if C is arithmetically Cohen-Macaulay or not, determines in which step of a minimal graded free resolution of I(C) the regularity is attained, and sometimes gives the value of the regularity of the Hilbert function of S/I(C) (otherwise, an upper bound is given).

Example:

LIB "mregular.lib";
// The 1st example is the twisted cubic:
reg_moncurve(0,1,2,3);
→ 2
// The 2nd. example is the non arithm. Cohen-Macaulay monomial curve in P4
// parametrized by: x(0)-s6,x(1)-s5t,x(2)-s3t3,x(3)-st5,x(4)-t6:
reg_moncurve(0,1,3,5,6);
→ 3
// Additional information can be obtained as follows:
printlevel = 1;
reg_moncurve(0,1,3,5,6);
→ // Sequence of integers defining a monomial curve C in P4:
→ //   - time for computing ideal I(C) of S (elimination): 0 sec.
→ //   - C is arithm. Cohen-Macaulay: NO
→ //   - reg(C) attained at the last step of a m.g.f.r. of I(C): YES
→ //   - reg(C) attained at the second last step of a m.g.f.r. of I(C): YES
→ //   - regularity of the Hilbert function of S/I(C): 2
→ //   - time for computing reg(C): 0 sec.
→ // Castelnuovo-Mumford regularity of C:
→ 3