Build. Blocks
Comb. Appl.
HCA Proving
Arrangements
Branches
Classify
Coding
Deformations
Equidim Part
Existence
Finite Groups
Flatness
Genus
Hilbert Series
Membership
Nonnormal Locus
Normalization
Primdec
Puiseux
Plane Curves
Saturation
Solving
Space Curves
Spectrum
Geometric Genus of Projective Curves
Definition: Let C be a projective curve, and let
HC(t) = d(C)*t - pa(C) + 1
be its Hilbert polynomial, then

  • d(C) =: degree of the curve C

  • pa(C) =: arithmetic genus of the curve.

The geometric genus g(C) is the arithmetic genus of the normalization Cn of C:
g(C):=pa(Cn)

If we are able to compute the normalization, we can compute the geometric genus. But this is very time consuming.

We propose a procedure based on the following knowledge:

  • pa(C)=g(C)+delta(C),   where delta(C) is the sum over the local delta invariants in the singular points.

  • There exist a projection C-->D to a plane curve D with degree d(D)=d(C), such that Cn=Dn. Then

    g(C) = pa(Cn) = pa(Dn) = g(D).
    Almost every projection has this property.
Plane Curves
Sao Carlos, 08/02 http://www.singular.uni-kl.de