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Geometric Genus of Plane Projective Curves
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.
Note: The primary decomposition is done over
Q.
To obtain the points, we have to extend the field.
Analysis of Singularities |