Abstract. In this manuscript we consider the well-established problem of TDOA-based source localization and propose a comprehensive analysis of its solutions for arbitrary sensor measurements and placements. More specifically, we define the TDOA map from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space. We then study the identifiability of the model, giving a complete analytical characterization of the image of this map and its invertibility. This analysis has been conducted in a completely mathematical fashion, using many different tools which make it valid for every sensor configuration. These results are the first step towards the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.

We analyze the Gorenstein locus of the Hilbert scheme of d points on P^n i.e., the\ud open subscheme parameterizing zero-dimensional Gorenstein subschemes of P^n\ud of degree d. We give new sufficient criteria for smoothability and smoothness of\ud points of the Gorenstein locus. In particular we prove that this locus is irreducible\ud when d <= 13 and find its components when d = 14.\ud The proof is relatively self-contained and it does not rely on a computer algebra\ud system. As a by-product, we give equations of the fourth secant variety to the\ud d-th Veronese reembedding of P^n for d >= 4

a b s t r a c tLet k be an algebraically closed field and let H ilb G d (P N k ) be the open locus of the Hilbert scheme Hilb d (P N k ) corresponding to Gorenstein subschemes. We prove that H ilb G d (P N k ) is irreducible for d ≤ 9. Moreover we also give a complete picture of its singular locus in the same range d ≤ 9. Such a description of the singularities gives some evidence to a conjecture on the nature of the singular points in Hilb G d (P N k ) that we state at the end of the paper.

a b s t r a c tLet k be an algebraically closed field of characteristic 0 and let Hilb G d (P N k ) be the open locus of the Hilbert scheme H ilb d (P N k ) corresponding to Gorenstein subschemes. We proved in a previous paper that Hilb G d (P N k ) is irreducible for d ≤ 9 and N ≥ 1. In the present paper we prove that Hilb G 10 (P N k ) is irreducible for each N ≥ 1, giving also a complete description of its singular locus.

No abstract