D.4.5.2 reg_curve

Procedure from library mregular.lib (see mregular_lib).

Usage:

reg_curve (i[,e]); i ideal, e integer

Return:

an integer, the Castelnuovo-Mumford regularity of i-sat.

Assume:

i is a homogeneous ideal of the basering S=K[x(0)..x(n)] where the field K is infinite, and it defines a projective curve C in the projective n-space (dim(i)=2). We assume that K[x(n-1),x(n)] is a Noether normalization of S/i-sat.
e=0: (default)
Uses a random choice of an element of K when it is necessary. This is absolutely safe (if the element is bad, another random choice will be done until a good element is found).
e=1: Substitutes the random choice of an element of K by a simple transcendental field extension of K.

Note:

The output is the integer reg(C)=reg(i-sat).
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-sat the regularity of C is attained, and sometimes gives the value of the regularity of the Hilbert function of S/i-sat (otherwise, an upper bound is given).

Example:

LIB "mregular.lib";
ring s = 0,(x,y,z,t),dp;
// 1st example is Ex.2.5 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
ideal i  = x17y14-y31, x20y13, x60-y36z24-x20z20t20;
reg_curve(i);
→ 72
// 2nd example is Ex.2.9 in [Bermejo-Gimenez], Proc.Amer.Math.Soc. 128(5):
int k=43;
ideal j=x17y14-y31,x20y13,x60-y36z24-x20z20t20,y41*z^k-y40*z^(k+1);
reg_curve(j);
→ 93
// Additional information can be obtained as follows:
printlevel = 1;
reg_curve(j);
→ // Ideal i of S defining a projective curve C in P3:
→ //   - i is saturated: YES
→ //   - C is arithm. Cohen-Macaulay: NO
→ //   - reg(C) attained at the last step of a m.g.f.r. of i-sat: YES
→ //   - regularity of the Hilbert function of S/i-sat: 92
→ //   - time for computing reg(C): 0 sec.
→ // Castelnuovo-Mumford regularity of C:
→ 93