Equivariant extensions of $\mathbb{G}_a$-torsors over punctured surfaces

Abstract

Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X → X//Ga restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms π : X → X//Ga are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A 2 when equipped with their reduced structures.

Rees algebras of additive group actions

“…In this subsection, we present an application of Rees algebras to the construction of affine extensions of G a -torsors over punctured surfaces. The following notion was introduced in [11,18]. Definition 32.…”

“…The last section is devoted to a selection of examples which illustrate the interplay between relative and absolute Rees algebras. We also present an application of Rees algebras to the construction and classification of affine extensions of G a -torsors over punctured surfaces [11,18] Given a scheme or an algebraic space S, we denote by G a,S = S × Z G a,Z = Spec(O S [t]) the additive group scheme over S. We denote by m : G a,S × S G a,S → G a,S and e : S → G a,S its group law and neutral section respectively. By an affine S-scheme f : X → S, we mean the relative spectrum of a quasi-coherent sheaf A = f * O X of O S -algebras.…”