  Motivated by questions in interpolation theory and on linear systems of
rational varieties, one is interested in upper bounds for the
Castelnuovo-Mumford regularity of arbitrary subschemes of fat points. An
optimal upper bound, named after Segre, was conjectured by Trung and,
independently, by Fattabi and Lorenzini. It is shown that this conjecture is
true. Furthermore, an alternate regularity bound is established that improves
the Segre bound in some cases. Among the arguments is a new partition result
for matroids.
