The Hilbert scheme of a pair of linear spaces

Ritvik Ramkumar

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arXiv · 1903.06377

The Hilbert scheme of a pair of linear spaces

Ritvik Ramkumar

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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 $a,b,n$.

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Ritvik Ramkumar. 2019-03-15. The Hilbert scheme of a pair of linear spaces. https://arxiv.org/abs/1903.06377

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