arXiv subjects

Roy Skjelnes

Publications and source records attributed to Roy Skjelnes.

7 recordsLinked to original sources

arXiv2026

Classifying Smooth Quot Schemes

The Quot scheme $\operatorname{Quot}^{q}(\mathcal{O}_{\mathbb{P}^n}^{r})$ parametrizes the quotients of the trivial vector bundle of rank $r$ on $n$-dimensional projective space that have Hilbert polynomial $q$ and are flat over a base scheme. We identify numerical conditions on the polynomial $q$ that completely determine when this Quot scheme is smooth and irreducible. Our approach also uncovers further geometric features of the projective scheme $\operatorname{Quot}^{q} (\mathcal{O}_{\mathbb{P}^n}^{r})$ including the smoothness of the lexicographic point.

math.AG↗

arXiv2022

Smooth Hilbert schemes: their classification and geometry

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.

math.AG↗

arXiv2021

The space of twisted cubics

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space.

math.AG↗

arXiv2008

Recovering the good component of the Hilbert scheme

In the Hilbert scheme of points on a scheme X there is an open subset parameterizing distinct points. The closure of that open set is by definition the good component. When X is flat over the base, we show that a certain blow-up of the symmetric product of X is the good component. The center of the blow-up we describe by giving generators for its defining ideal. In the non-flat case we obtain similar result by replacing the symmetric product with the divided power product. For smooth surfaces X the good component equals the Hilbert scheme of points.

math.AG↗

arXiv2008

The space of generically étale families

We construct a space $G^n_X$ and a rank $n$, generically etale family of closed subspaces in a separated ambient space $X$. The constructed pair satisfies a universal property of generically etale families of closed subspaces in $X$. This universal property is derived directly from the construction and does in particular not use the Hilbert scheme. The constructed space $G^n_X$ is by its universal property canonically identified with a closed subspace of the Hilbert scheme.

math.AG↗

arXiv2007

Non-effective Deformations of Grothendieck's Hilbert Functor

Let X be a scheme that does not satisfy the valuative criterion of separatedness. We show that the Hilbert functor parametrizing closed families of X that are flat, finite and of rank one is not represented by a scheme or an algebraic space.

math.AG↗

arXiv2005

An elementary, explicit, proof of the existence of Hilbert schemes of points

We give a short, elementary and explicit proof of the existence of Hilbert schemes of points on affine schemes. As a direct consequence we obtain the existence of the Hilbert scheme of points on any projective scheme, not necessarily of finite type, over any base scheme.

math.AG↗