In this paper, we present an effective study of the punctual Hilbert scheme. First, we recall algorithms to compute the inverse system of an isolated singular point. These inverse systems are points of the punctual Hilbert scheme, that we define as a subvariety of a Grassmannian variety, providing explicit defining equations. We localise our study to the algebraic variety
H
i
l
b
B
\mathrm {Hilb}_{B}
of inverse systems, which admits a given dual (monomial) basis
B
B
. Building on the algorithm to compute inverse systems and exploiting combinatorial properties of the monomial basis
B
B
, we derive a set of equations defining
H
i
l
b
B
\mathrm {Hilb}_{B}
. Using this effective presentation, we prove new properties of transversality of
H
i
l
b
B
\mathrm {Hilb}_{B}
, give new dimension formula for its irreducible components and prove that they are birational.