### Background

We are given a curve with integer coefficients. I want to make a suggestion in another question (Computationally bounding a curve's genus from below?) into a deterministic algorithm for finding the genus of a plane curve.

The suggestion is: reduce modulo a random prime and find all singular points there. If the prime was of good reduction, then these are the reductions of all of the algebraic singular points, and you can compute the genus easily from here.

### Question

What is an effective bound on the largest prime of bad reduction?

### What I imagine

Say $C$ is given by $\sum a_{ij} x^i y^j$, then I imagine a bound similar to: $\displaystyle\sum_{\sigma} \prod a_{i\sigma^{-1} (i)}$

Qas well as modulo your prime, correct? If this is so, then by using various resultants you should be able to produce an integer all of whose divisors correspond to "bad primes". I suspect that the bound coming from this description will be bigger than the one you suggested. – damiano Mar 31 '10 at 18:24Sover Spec(Z) defined by $f=df/dx=df/dy=0$ will have some primary decomposition. Some of the primes in the decomposition ofSwill dominate Spec(Z), some will not. The primes of Spec(Z) of "good reduction" should be the primes above which there is no isolated component ofS. I am not sure how computationally feasible this approach is. – damiano Mar 31 '10 at 18:33