# The Ubiquity of Smooth Hilbert Schemes

## TL;DR

This paper explores the structure and properties of Hilbert schemes, revealing regularities and probabilistic results about their irreducibility and smoothness within a combinatorial and geometric framework.

## Contribution

It classifies certain Hilbert schemes with unique Borel-fixed points and uncovers probabilistic regularities in their geometric properties.

## Key findings

- Hilbert schemes are irreducible with probability > 0.5 under natural distributions.
- Classification of Hilbert schemes with unique Borel-fixed points.
- Hilbert schemes form vertices of an infinite binary tree with regular geometric patterns.

## Abstract

We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials. These expressions naturally lead to an arrangement of nonempty Hilbert schemes as the vertices of an infinite full binary tree. Here we discover regularities in the geometry of Hilbert schemes. Specifically, under natural probability distributions on the tree, we prove that Hilbert schemes are irreducible and nonsingular with probability greater than $0.5$.