Next: Computing -bases of adjoint Up: Computational Aspects Previous: Computing the adjunction divisor

### Computing the divisors

The intersection divisors can be computed by the following algorithm:

Input
, list of closed places of (defined by ), .

Output
Extended list of closed places and list of integers such that .
1. Affine Intersection. Let , and consider . Proceed as in Step 1 of the algorithm in Section 2.4.1 to obtain a set of (triangular) ideals corresponding to the set of closed points in . For each closed point in do the following:
• check for each closed place in whether lies above (that is, the first entry of the triple representing has to be ). If this is the case, compute the multiplicity , where is the primitive parametrization obtained from the sHNE (third entry) of .
• If there is no such in then compute the sHNE for the (smooth) germ , add the resulting place to the list and proceed as before to obtain .
2. Intersection at infinity. Let and compute a prime factorization of the polynomial ,

Each factor that appeared also in the prime factorization (4) of corresponds to a closed point in the intersection of with the plane curve defined by . For the corresponding closed places we compute

where is the primitive parametrization obtained from the sHNE of (cf. Step 4 of the algorithm in Section 2.4.1).

Finally, if and then we compute for the corresponding closed places the multiplicities

Next: Computing -bases of adjoint Up: Computational Aspects Previous: Computing the adjunction divisor
Christoph Lossen
2001-03-21