Suppose f:*X*→Y is a morphism of schemes. We can define a function on the underlying topological space Y by sending y∈Y to the dimension of the fiber of f over y. When is this function upper semi-continuous?
I have a "concrete" application in mind. If an algebraic group G acts on a scheme X, I'm pretty sure the dimensions of the stabilizers is upper semi-continuous (i.e. it can jump up on closed subschemes), but I don't know a proof. The stabilizers of points are the fibers of the map Stab→X in the following cartesean square:
Stab ---> GxX | | | cart | v v X ----> XxX
The map GxX→XxX is given by (g,x)→(gx,x), and the map X→XxX is the diagonal x→(x,x). So it would be nice to have a condition satisfied by GxX→XxX which guarantees upper semi-continuity of the fiber dimension.