  Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq
\mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of
$x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the
number of equations and syzygies of subschemes parametrized by the Hilbert
scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of
points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq
\mathbb{P}^n$.
