  The purpose of this paper is to study the Zariski tangent space of the
punctual Hilbert scheme parametrizing subschemes of a smooth surface which are
supported at a single point. We give a lower bound on the dimension of the
tangent space in general and show the bound is sharp for subschemes of the
affine plane cut out by monomials. Furthermore for monomial subschemes we give
an explicit combinatorial formula for the dimension of the tangent space.
