The Hilbert scheme of points and its link with border basis

Mariemi Alonso (1) , Jérome Brachat (2) , Bernard Mourrain (2)

1 Departamento de Álgebra [Madrid]
2 GALAAD - Geometry, algebra, algorithms

In this paper, we give new explicit representations of the Hilbert scheme of $\mu$ points in $\PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous ones and global. It involves equations of degree $2$. We show how these equations are deduced from the commutation relations characterizing border bases. Next, we consider infinitesimal perturbations of an input system of equations on this Hilbert scheme and describe its tangent space. We propose an effective criterion to test if it is a flat deformation, that is if the perturbed system remains on the Hilbert scheme of the initial equations. This criterion involves in particular formal reduction with respect to border bases.

Mots clés: algebra; geometry; effective algebraic computation; Hilbert scheme; deformation; flatness

Domaines: Calcul formel [cs.SC]; Géométrie algébrique [math.AG]; Algèbre commutative [math.AC]

Mariemi Alonso, Jérome Brachat, Bernard Mourrain. The Hilbert scheme of points and its link with border basis. 2010. ⟨inria-00433127v2⟩