Smooth Hilbert schemes: their classification and geometry

TL;DR

This paper classifies smooth Hilbert schemes in projective space, providing numerical criteria for smoothness and relating them to generalized partial flag varieties, thereby advancing understanding of their structure and geometry.

Contribution

It offers a complete classification of smooth Hilbert schemes and links them to generalized partial flag varieties, revealing their geometric structure.

Findings

Numerical conditions for smoothness of Hilbert schemes

Classification of smooth Hilbert schemes in projective space

Identification of these schemes as generalized partial flag varieties

Abstract

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.