As we know there are a lot of principle of connectedness in algebraic geometry. Here is a useful and interesting one:

Suppose T is an integral curve over k. X-->T is a flat family of closed subvarieties in $P_k^n$. If there is a non-empty open subset U in T such that at every closed point t in U, the fiber X_t is connected. Then show every fiber X_t is connected for any t in T.

In consideration of uppercontinuous property, this says that if the parameter space is a curve, then if $h^0(X_t, O_{X_t})$ is locally constant on some open set, then it's locally constant **everywhere**!(If we further require k is algebraically closed here).

I thing this property is interesting and useful, but I can't prove it, and every reference I can find traces back to Hartshorne's exercise III.11.4. If anyone can give me a proof, I would be very grateful!