Page 127 - Geometric Modeling and Algebraic Geometry

128 T. Beck and J. Schicho
Up to isomorphism the resolution of a complete curve is unique (which follows
from [16, II.4.4, Cor. 2]). We can effectively deal with the divisors Div(C) by iden-

tifying prime divisors on C with discrete valuations of K(C) over K (and K(C) is

isomorphic to K(C) because of birationality). We never have to construct the curve

C explicitly.

Above we have seen how a point P ∈ C induces a discrete valuation ν P of

K(C). On the other hand let ν be such a discrete valuation. Since C is complete

ν has a center P ν ∈ C. This way P → ν P and ν → P ν constitute a one-one

correspondence between points of C and discrete valuations of K(C).

Also C is complete and so we let Center(ν) denote the center of a valuation ν
in C. Note that with the above notation Center(ν)= π(P ν ). For any subset M ⊂ C
we define V M to be the set of all discrete valuations ν of K(C) s.t. Center(ν) ∈ M.
7.4 Rational curves

It is well-known that a curve is parametrizable if and only if it has genus zero. In this
section we will show how to compute the genus in our setting. Afterwards we give
the general idea of a curve parametrization algorithm.

7.4.1 The genus of a curve
To each point Q ∈ C one can associate its delta invariant δ Q . It is a measure of
singularity, which is defined as the length of the quotient of the integral closure of
the local ring by the local ring at Q (see [9, exercise IV.1.8]). For instance, if Q is an
ordinary singularity of multiplicity µ, i.e. a self-intersection point where µ branches
meet transversally, then δ Q = µ(µ−1) . In particular δ Q =0 for Q smooth.
2
If Π ⊂ R is a lattice polygon we denote by #(Π):= |Π ∩ Z | the number of
2
2
◦
lattice points in Π. We also write Π for the polygon spanned by the interior points.
In the toric situation the genus can be computed as follows:
Theorem 4. The genus of C is equal to the number of interior lattice points of Π(f)
minus the sum of the delta invariants of all points on C:

◦
genus(C)=#(Π(f) ) − δ Q
Q∈C
Proof. See [1, Proposition 9].
The sum actually ranges over the singular points of C only.

Remark 5. Assume that Π(f) is an isosceles triangle with vertices (0, 0), (n, 0) and
(0,n) and that all singularities of the curve are ordinary. Then the number of interior
points is equal to (n−1)(n−2) and we recover the well-known genus formula for plain
2
curves.