  We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on
$\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional
Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient
criteria for smoothability and smoothness of points of the Gorenstein locus. In
particular we prove that this locus is irreducible when $d\leq 13$ and find its
components when $d = 14$. The proof is relatively self-contained and it does
not rely on a computer algebra system. As a by--product, we give equations of
the fourth secant variety to the $d$-th Veronese reembedding of $\mathbb{P}^n$
for $d\geq 4$.
