Smooth Hilbert schemes: their classification and geometry

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.

The fiber-full scheme

“…The list includes the Hilbert scheme of points and classical examples such as the Hilbert scheme of twisted cubics and the Hilbert scheme of skew lines (see Section 6). Furthermore, by using the recent classification of Skjelnes and Smith [46], we show in Proposition 7.5 that smooth Hilbert schemes coincide with a fiberfull scheme (i.e., cohomological data is constant for points in a smooth Hilbert scheme).…”

“…In this section, we study the fiber-full scheme as a subscheme of smooth Hilbert schemes, the latter were recently classified in [46]. Our main result states that if the Hilbert scheme is smooth, then it is equal to a fiber-full scheme.…”

Hilbert Schemes with Two Borel-fixed Points in Arbitrary Characteristic

“…Remark 4.9. Concurrent work by Skjelnes-Smith (initially inspired by our geography described in §2.2) has resulted in a full classification of smooth Hilbert schemes [SS20]; their cases (4), (5), (6) correspond to cases (i)(c), (i)(b), (i)(a) here. In fact, our results show that the difficulties associated with nonstandard Borel-fixed ideals in positive characteristic are avoided by the classification.…”

“…Note that when K has characteristic 0, the local deformation theory at these Borel-fixed ideals is worked out in [Ram19]. The Hilbert schemes described in Theorem 1.1(i) also appear in the recent classification of smooth Hilbert schemes [SS20] (see Remark 4.9).…”

Some Singular Lex-Segments

“…Based on our underlying "geography" of Hilbert schemes, further smooth examples are promptly found, leaving open the challenge of understanding whether all smooth Hilbert schemes fall into known classes. This is achieved by Skjelnes-Smith in [SS20], where our classification comprises two of the main classes of smooth Hilbert schemes.…”