  Given the Hilbert function $u$ of a closed subscheme of a projective space
over an infinite field $K$, let $m_u$ and $M_u$ be, respectively, the minimum
and the maximum among all the Castelnuovo-Mumford regularities of schemes with
Hilbert function $u$. I show that, for every integer $m$ such that $m_u \leq m
\leq M_u$, there exists a scheme with Hilbert function $u$ and
Castelnuovo-Mumford regularity $m$. As a consequence, the analogous algebraic
result for an O-sequence $f$ and homogeneous polynomial ideals over $K$ with
Hilbert function $f$ holds too.
  Although this result does not need any explicit computation, I also describe
how to compute a scheme with the above requested properties. Precisely, I give
a method to construct a strongly stable ideal defining such a scheme.

Commutative Algebra