The Hilbert scheme of a pair of linear spaces

Ritvik Ramkumar

doi:10.1007/s00209-021-02788-6

Abstract

Let mathcal {H}_{a,b}^n denote the component of the Hilbert scheme whose general point parameterizes an a-plane union a b-plane meeting transversely in {mathbf {P}}^n. We show that mathcal {H}_{a,b}^n is smooth and isomorphic to successive blow ups of mathbf {Gr}(a,n) times mathbf {Gr}(b,n) or text {Sym}^2 mathbf {Gr}(a,n) along certain incidence correspondences. We classify the subschemes parameterized by mathcal {H}_{a,b}^n and show that this component has a unique Borel fixed point. We also study the birational geometry of this component. In particular, we describe the effective and nef cones of mathcal {H}_{a,b}^n and determine when the component is Fano. Moreover, we show that mathcal {H}_{a,b}^n is a Mori dream space for all values of a, b, n.

Highlights

The Hilbert scheme HilbP(t) Pn, which parameterizes closed subschemes of Pn with a fixed Hilbert polynomial P(t), introduced by Grothendieck [12], has attracted a lot of interest
Fogarty [10] proved that Hilbm P2 is smooth and Arcara, Bertram, Coskun and Huizenga [1] proved that its a Mori dream space and described the stable base decomposition of its effective cone in numerous cases
Piene and Schlessinger [23] showed that Hilb3t+1 P3 has two smooth components that meet transversely and described the points of the component corresponding to twisted cubics explicitly

Summary

Introduction

The Hilbert scheme HilbP(t) Pn, which parameterizes closed subschemes of Pn with a fixed Hilbert polynomial P(t), introduced by Grothendieck [12], has attracted a lot of interest. Let N1 denote the divisor class of the locus of generically non-reduced subschemes in Hnn−c,n−c. Let F denote the divisor class of the locus of subschemes whose linear span meets a fixed n−2c. Let N1 denote the divisor class of the locus of generically non-reduced subschemes in Hcn−1,c−1.

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Results

Conclusion