D.4.6.3 genus

Procedure from library normal.lib (see normal_lib).

Usage:

genus(I) or genus(i,1); I a 1-dimensional ideal

Return:

an integer, the geometric genus p_g = p_a - delta of the projective curve defined by I, where p_a is the arithmetic genus.

Note:

delta is the sum of all local delta-invariants of the singularities, i.e. dim(R’/R), R’ the normalization of the local ring R of the singularity.
genus(i,1) uses the normalization to compute delta. Usually this is slow but sometimes not.

Example:

LIB "normal.lib";
ring r=0,(x,y),dp;
ideal i=y^9 - x^2*(x - 1)^9;
genus(i);
→ 0