An intrinsic construction of the principal component of the Hilbert scheme

Abstract. The principal component of the Hilbert scheme of n points on a scheme X is the closure of the open subset parame-terizing n distinct points. In this article we construct the princi-pal component as a certain blow-up of the symmetric product of X. Our construction is based on a local explicit analysis of étale families from where the appropriate universal property, needed to identify the principal component with the blow-up, is derived