| Classify |
| Coding |
| Deformations |
| Equidim Part |
| Existence |
| Finite Groups |
| Flatness |
| Genus |
| Hilbert Series |
| Membership |
| Monodromy |
| Normalization |
| Primdec |
| Puiseux |
| Plane Curves |
| Solving |
| Space Curves |
| Spectrum |
Let C be a plane projective curve. We compute the geometric genus by a local analysis of the singularities.
Choose an affine covering of P2, and assume that in such an open affine set the curve C is defined by f=0.
We know:
the ideal Sing(f):= < f, fx, fy > defines the singular locus
the ideal Sing(Sing(f)):=< f, fx, fy, det(Hess(f)) > defines the non-nodal locus
the ideal S:= Sing(Sing(Sing(f))) defines the non-nodal-cuspidal locus
delta(C,x)=1 in nodal or cuspidal singularities, so we have just to count them.
the singular points different from cusps and nodes are obtained by a primary decomposition of S.