The article summarizes progress by many persons; it is accessible to nonexperts. My goal is
to point to what is available, and give some feeling for the subject. When X is a nonsingular …

Given a smooth complex threefold X, we define the virtual motive Hilb^n(X)_vir of the Hilbert
scheme of n points on X. In the case when X is Calabi–Yau, Hilb^n(X)_vir gives a motivic …

L'objet principal de ce travail est le théorème de division également baptisé «technique des
escaliers». Cette technique permet d'associer à chaque idéal de séries entières …

We review the open problems in the theory of deformations of zero-dimensional objects,
such as algebras, modules or tensors. We list both the well-known ones and some new ones …

We present counterexamples to four conjectures which appeared in the literature in
commutative algebra and algebraic geometry. The four questions to be studied are largely …

The moduli space of rank-n commutative algebras equipped with an ordered basis is an
affine scheme 3n of finite type over Z, with geometrically connected fibers. It is smooth if and …

L'objet de ce travail est l'étude des déformations plates de" points" ou genne d'espaces
analytiques (non réduits!) de dimension zéro, dans un espace analytique complexe X. En …

We study a natural Hodge module on the Hilbert scheme of four points on affine three-
space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We …

Let I be a homogeneous ideal of a polynomial ring over a field, v (I) the number of elements
of any minimal basis of I, e= e (I) the multiplicity or degree of R/I, h= h (I) the height or …

In this paper we provide a new method to certify that a nearby polynomial system has a
singular isolated root and we compute its multiplicity structure. More precisely, given a …