Computable Hilbert Schemes

TL;DR

This thesis develops new algorithmic methods to study Hilbert schemes, including equations, properties of Borel-fixed ideals, and connectedness proofs, advancing computational algebraic geometry.

Contribution

It introduces novel algorithms for computing Borel-fixed ideals, new equations for Hilbert schemes, and a proof of connectedness for certain Hilbert schemes.

Findings

New lower-degree equations for Hilbert schemes

Algorithm for all saturated Borel-fixed ideals with given Hilbert polynomial

Connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in P^3

Abstract

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert polynomial assigned and we investigate their combinatorial properties. In Chapter 3 we show a new type of flat deformations of Borel-fixed ideals which lead us to give a new proof of the connectedness of the Hilbert scheme. In Chapter 4 we construct families of ideals that generalize the notion of family of ideals sharing the same initial ideal with respect to a fixed term ordering. Some of these families correspond to open subsets of the Hilbert scheme and can be used to a local study of the Hilbert scheme. In Chapter 6 we deal with the problem of the connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in the projective 3-space. We show that one of the Hilbert scheme considered a "good" candidate to be non-connected, is instead connected. Moreover there are three appendices that present and explain how to use the implementations of the algorithms proposed.