The ubiquity of smooth Hilbert schemes
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Mathematische Zeitschrift
The ubiquity of smooth Hilbert schemes Andrew P. Staal1 Received: 16 February 2018 / Accepted: 16 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction Hilbert schemes parametrizing closed subschemes with a fixed Hilbert polynomial in projective space are fundamental moduli spaces. However, with the exception of Hilbert schemes parametrizing hypersurfaces [10, Example 2.3] and points in the plane [12], the geometric features of typical Hilbert schemes are still poorly understood. Techniques for producing pathological Hilbert schemes are known, generating Hilbert schemes with many irreducible components [11,20], with generically nonreduced components [27], and with arbitrary singularity types [35]. This raises the questions: What should we expect from a random Hilbert scheme? Can we understand the geography of Hilbert schemes? Our answer is that the set of nonempty Hilbert schemes forms a collection of trees and a discrete probability space, and that irreducible, nonsingular Hilbert schemes are unexpectedly common. Let Hilbp (Pn ) be the Hilbert scheme parametrizing closed subschemes of PnK with Hilbert polynomial p, where K is a field. Macaulay classified Hilbert polynomials of homogeneous ideals in [22]. Any such admissible Hilbert polynomial p(t) has a unique combinatorial expression of the form rj=1 t+b jb− j+1 , for integers b1 ≥ b2 ≥ · · · ≥ br ≥ 0. Our main j result is the following theorem. Theorem 1.1 The lexicographic ideal is the unique saturated strongly stable ideal of codimension c with Hilbert polynomial p if and only if at least one of the following holds: (i) br > 0,
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Andrew P. Staal [email protected] Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada
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(ii) c ≥ 2 and r ≤ 2, (iii) c = 1 and b1 = br , or (iv) c = 1 and r − s ≤ 2, where b1 = b2 = · · · = bs > bs+1 ≥ · · · ≥ br . If K is algebraically closed, then the lexicographic ideal is the unique saturated Borel-fixed ideal of codimension c with Hilbert polynomial p if and only if at least one of (i)–(iv) holds. Strongly stable ideals, including lexicographic ideals, are Borel-fixed. In characteristic 0, the converse also holds. Many fundamental properties of Hilbert schemes have been understood through these ideals. Hartshorne and later Peeva–Stillman found rational curves linking B
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