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 just have to count them.
the singular points different from cusps and nodes are obtained by a primary decomposition of S.