Speaker: Daniele Alessandrini (Strasbourg)

Title: On the tropicalization of the Hilbert scheme

Abstract: The Hilbert scheme is the parameter space of subschemes of a projective space with a fixed Hilbert polynomial. The tropicalization of the Hilbert scheme is a tropical variety, and it is natural to ask whether it is also a parameter space, i.e. if there is a natural correspondence between its points and the elements of a suitable set of tropical varieties. In a joint work with M. Nesci, we showed that such a correspondence exists. The proof passes through a bound on the degree of a tropical basis of an ideal in terms of its Hilbert polynomial. I will discuss examples showing that this correspondence is surjective but not, in general, injective.