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?



Algorithms for semistable reduction of families of curves
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)


