Compactified moduli of projective bundles. (English) Zbl 1195.14016

Lieblich, Max

Algebra Number Theory 3, No. 6, 653-695 (2009).

The article develops a method for compactifying stacks of \(\text{PGL}_n\)-torsors on algebraic spaces. A general theory of twisted objects in a stack is presented. The theory is then applied to show that the stack of twisted sheaves is a natural cover of a compactification of the stack of \(\text{PGL}_n\)-torsors. A more concrete description of this candidate compactification is also provided, using an explicit description by generalized Azumaya algebras. Finally, by specializing to the situation with twisted sheaves on surfaces the author establishes the main result of the article: Let \(X\) be a smooth projective surface over an algebraically closed field, and assume that the integer \(n\) is invertible in the base field. Then the stack of stable \(\text{PGL}_n\)-torsors on a \(X\) with cohomology class \(\alpha \in H^2(X, \mu_n)\), and where the associated adjoint bundle has sufficiently large second Chern class, is of finite type and is irreducible.

Reviewer: Roy Mikael Skjelnes (Stockholm)

MSC:

14D20	Algebraic moduli problems, moduli of vector bundles
14D15	Formal methods and deformations in algebraic geometry

Keywords:

Projective bundles; moduli of stable bundles; Skolem-Noether theorem; derived categories; rigidification