g = geometricGenus R
g = geometricGenus I
The geometric genus of a plane curve C0 is the genus of the normalization of C0








Every hyperelliptic curve of genus g can be represented as a plane curve of degree g+2 with a gfold ordinary singularity, and thus conductor equal to the (g1)st power of the maximal ideal. As of 1/20/2024, Macaulay2 crashes on computing the conductor when g >= 6, but knowing the conductor one can go much farther:
We make a general hyperelliptic curve of genus g with singularity at q'.
Example g = 20 S = ZZ/101[a,b,c] q' = ideal(a,b); Text Example I = q'^g C = S/(ideal random(g+2, I)); p = sub(p', C); q = sub(q', C); geometricGenus (C, Conductor => q'^(g1))
The object geometricGenus is a method function with options.