[Paper Review] The space of generically étale families

David Rydh, Roy Skjelnes|Mar 12, 2007

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper constructs a space $\mathscr{G}^{n}_{X}$ parameterizing generically étale families of rank $n$ closed subspaces in a separated algebraic space $X\to S$, using explicit algebraic constructions involving symmetric tensors and blow-ups. The key contribution is a universal property derived directly from the construction, without relying on the Hilbert scheme, and it is shown that $\mathscr{G}^{n}_{X}$ is canonically isomorphic to the schematic closure of the open subspace of étale families inside the Hilbert scheme.

ABSTRACT

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.

Motivation & Objective

• To construct a space $\mathscr{G}^n_X$ that universally parameterizes generically étale families of closed subspaces of rank $n$ in a separated algebraic space $X\to S$.
• To establish the universal property of $\mathscr{G}^n_X$ directly from the construction, without using the Hilbert scheme.
• To show that $\mathscr{G}^n_X$ is canonically isomorphic to the schematic closure of the open subspace of étale families within the Hilbert scheme.
• To provide an independent, abstract global construction of the universal family over $\mathscr{G}^n_X$ using blow-ups and norm maps.

Proposed method

• Construct an algebra homomorphism $\mathscr{A} \to \mathscr{R}$ using symmetric tensors and the alternator map, defined via explicit identities in the tensor algebra.
• Define $\mathscr{R}$ as the localization of a quotient algebra at the non-vanishing locus of a symmetric tensor $\alpha^2(x)$, ensuring étale behavior.
• Prove that $\mathscr{R}$ is a free $\mathscr{A}$-module with basis formed by the images of $x_1,\dots,x_n$, using trace maps and universal properties.
• Extend the construction globally via blow-ups of canonical ideals in symmetric products, using the Grothendieck-Deligne norm map to ensure flatness.
• Show that the universal family $\mathscr{Z}_X \to \mathscr{G}^n_X$ is generically étale by analyzing the discriminant and its locus via blow-ups.
• Establish the universal property by verifying that any generically étale family over a base $T$ factors uniquely through $\mathscr{G}^n_X$.

Experimental results

Research questions

• RQ1 Can a universal space for generically étale families of closed subspaces be constructed without relying on the Hilbert scheme?
• RQ2 What algebraic structure underlies the universal family over $\mathscr{G}^n_X$, and how can it be described explicitly?
• RQ3 How does the construction of $\mathscr{G}^n_X$ relate to the schematic closure of the étale locus inside the Hilbert scheme?
• RQ4 What role do symmetric tensors and the alternator map play in controlling the étale and free module structure of the universal family?
• RQ5 Is the universal property of $\mathscr{G}^n_X$ derivable directly from the construction, without auxiliary moduli-theoretic tools?

Key findings

• The space $\mathscr{G}^n_X$ is constructed as a closed subspace of the Hilbert scheme, via blow-ups of symmetric products along canonical ideals.
• The universal family $\mathscr{Z}_X \to \mathscr{G}^n_X$ satisfies a universal property: any generically étale family of rank $n$ closed subspaces over a base $T$ factors uniquely through $\mathscr{G}^n_X$.
• The construction of $\mathscr{G}^n_X$ is independent of the Hilbert scheme; its universal property is derived directly from algebraic identities in symmetric tensors.
• The algebra extension $\mathscr{A} \to \mathscr{R}$ is étale and $\mathscr{R}$ is free of rank $n$ over $\mathscr{A}$, with the images of $x_1,\dots,x_n$ forming a basis.
• The space $\mathscr{G}^n_X$ is canonically isomorphic to the schematic closure of the open subspace $\mathscr{U}^n_X$ of étale families inside the Hilbert scheme $\mathscr{H}^n_X$, confirming its role as the Hilbert scheme compactification.
• The construction is compatible with étale base change and uses the Grothendieck-Deligne norm map to ensure flatness of the universal family.