Isolated singularities, inverse systems and the punctual Hilbert scheme

We study of the punctual Hilbert scheme from an algorithmic point of view. We first present algorithms, which allow to compute the inverse system of an isolated point. We define the punctual Hilbert scheme as a subvariety of a Grassmannian variety and provide explicit equations defining it. Then we localised our study to the algebraic variety Hilb_B of inverse systems, which admits a given dual (monomial) basis B. Building on the algorithm to compute the inverse system and exploiting combinatorial properties of the monomial basis B, we derive a minimal set of equations defining Hilb_B. Using this effective presentation, we prove new properties of transversality of Hilb_B , give new dimension formula for its irreducible components and prove that they are birational.