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Example text

Theorem. There is a natural isomorphism V. V. Shokurov 42 ))1 1 C{ 2 2 ... Fig. 11. A hyperelliptic projection HI(S,Z) ~ 1r(S)j [1r(S), 1r(S)] , where [1r(S) , 1r(S)] is the commutator subgroup of the fundamental group 1r(S) . Every path is homotopically equivalent to a path along the edges of a triangulation and can therefore be represented by a simplicial 1-chain. Correspondingly, a loop can be represented by a 1-cycle. This is how the isomorphism is defined (cf. Springer [1957]). In particular, the edges ai and bi of a standard development (see Fig.

6 for geometric genera. Example 2. On an elliptic curve

Definition 3. , if p is a pole of Ij otherwise. That multp I is well-defined, follows from the geometric interpretation of multiplicity. In the full inverse image 1- 1 (q) of a point q E 8 2 - I(p) close to f(p), one finds precisely multp I points close to p. Example 1. II} ~ II} has 0 as a single ramification point, with index n - l. Example 2. C ~ C has no ramification points. Example 3. If I(z) E M(Cp1) is a polynomial of degree d then ordoo 1= -d, and ordp J is equal to the multiplicity of p as a root of f if J(p) = o.

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