We investigate the geography of Hilbert schemes that parametrize closed subschemes of projective space with a specified Hilbert polynomial. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials. We realize the set of all nonempty Hilbert schemes as a probability space and prove that Hilbert schemes are irreducible and nonsingular with probability greater than 0.5. Theorem 1.1. The lexicographic ideal is the unique saturated Borel ideal of codimension c with Hilbert polynomial p if and only if:Borel ideals generalize lexicographic ideals and in characteristic 0 define Borel-fixed points on Hilbert schemes. Some basic general properties of Hilbert schemes have been extracted from these ideals. Rational curves linking Borel-fixed points prove connectedness in [Har66, PS05]. The thesis [Bay82] uses them to give equations for Hilbert schemes and proposes studying their tangent cones. Further, [Ree95] studies their combinatorial properties to give general bounds for radii of Hilbert schemes, and [RS97] proves that lexicographic points are nonsingular. Theorem 1.1 specifies an explicit collection of wellbehaved Hilbert schemes, generalizes the main result of [Got89], and improves our understanding of the geography of Hilbert schemes.This collection of Hilbert schemes is ubiquitous. Our new interpretation of Macaulay's classification identifies an infinite binary tree H c whose vertices are the Hilbert schemes 1 2011-12, by Ontario Graduate Scholarships in 2012-15, and by Gregory G. Smith's NSERC Discovery Grant in 2015-16. THE TREE OF ADMISSIBLE HILBERT POLYNOMIALSWe identify a graph structure on the set of numerical polynomials determining nonempty Hilbert schemes of projective spaces. The pioneering work [Mac27] gives a combinatorial classification of these polynomials. We introduce two binary relations to equip this set with the structure of an infinite binary tree.
We answer an open problem posed by Iarrobino, Hilbert scheme of points: Overview of last ten years. Proceedings of Symposia in Pure Mathematics, 46 (American Mathematical Society, Providence, RI, 1987), 297–320: Is there a component of the punctual Hilbert scheme [Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert', in Séminaire Bourbaki, 6 (Societe Mathematique de France, Paris, 1995), 221, 249–276] $\operatorname {\mathrm {Hilb}}^d({\mathscr {O}}_{\mathbb {A}^n,p})$ with dimension less than $(n-1)(d-1)$ ? For each $n\geq 4$ , we construct an infinite class of elementary components in $\operatorname {\mathrm {Hilb}}^d(\mathbb {A}^n)$ producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring [Iarrobino and Kanev, Power sums, Gorenstein algebras, and determinantal loci (Springer-Verlag, Berlin, 1999), 221–226] of the form with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension $2$ .
We answer an open problem posed by Iarrobino in the '80s: is there a component of the punctual Hilbert scheme Hilb d (O A n ,p ) with dimension less than (n − 1)(d − 1)? We construct an infinite class of elementary components in Hilb d (A 4 ) producing such examples. Our techniques also allow us to construct an explicit example of a local Artinian ring with trivial negative tangents, vanishing nonnegative obstruction space, and socle-dimension 2.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.