Question

This is a somewhat vague question which came up MSRI a few days ago: Suppose I have a family of curves over a one dimensional base, given in a computationally explicit way. For example, maybe I have a family F_t(x,y,z) of homogenous polynomials, whose coefficients are polynomials in t, and which cut out smooth curves for t \neq 0.

Is there an algorithmically practical way to write down the limit of this family in \overline{M}_g?

Answer 1

If your curves are in P^n (specifically in P^2 - as in your example), I think there is something you can do: project your curves from a general P^{n-2} to P^1. This means that you are now looking for a limit in a Hurwitz scheme. This can be broken into two problems:

• looking for the limit on the underlying M_{0,n}

• tracing the ramification structure.

Here is an example: find the limit of F+t Q^2 where F is a plane quartic, and Q is a plane quadric.

Project from your favorite random point. You can verify that the limit of the ramification points on the family are

• the eight intersection points of F and Q

• twice on each of ramification points of the projection of p from Q.

From here you can continue in a variety of ways (e.g. you have a pencil of g^1_4 s on the limit curve which break through a map from the limit curve to a plane conic, which has 8 ramification points)

Answer 2

This answers a variant of your question.

The analogous problem in number theory -- computing a regular proper minimal model of a curve over Q at a prime p -- is hard. For genus 1 this is Tate's algorithm, for genus 2 this is done in a paper of Qing Liu and requires a lot of work. For higher genus (even hyperelliptic) curves I believe this is still open and probably too hard to do.

However, this is a much harder problem than semistable reduction (for instance in the genus 1 case the primes dividing the denominator of the j-invariant are exactly the primes of potentially semistable reduction).