Affine Extensions of Principal Additive Bundles Over a Punctured Surface

Abstract

The aim of this article is to make a first step towards the classification of complex normal affine G a -threefolds X. We consider the case where the restriction of the quotient morphism π : X → S to π −1 (S * ), where S * denotes the complement of some regular closed point in S, is a principal G a -bundle. The variety SL 2 will be of special interest and a source of many examples. It has a natural right G a -action such that the quotient morphism SL 2 → A 2 restricts to a principal G a -bundle over the punctured plane A 2 * .

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

“…This observation combined with the following re-interpretation of an example constructed in [10] suggests that locally trivial A 1 -bundles over the blow-up of closed point o in the smooth locus of a surface S form a natural class of schemes in which to search for nontrivial G a -extension of G a -bundles over punctured surfaces.…”

“…In the present article, as a step towards the understanding of the structure of two-dimensional degenerate fibers, we consider a particular type of non equidimensional surjective G a -quotient A 1 -fibrations π : X → S which have the property that they restrict to G a -torsors 1 over the complement of a finite set of smooth points in S. These are simpler than the general case illustrated in the previous example since they do not admit additional degeneration of their fibers over curves in S passing through the given points. The local and global study of some classes of such fibrations was initiated by the second author [10]. He constructed in particular many examples of G a -quotient A 1fibrations on smooth affine threefolds X with image A 2 whose restrictions over the complement of the origin are isomorphic to the geometric quotient SL 2 → SL 2 /G a of SL 2 by the action of unitary upper triangular matrices.…”

“…where G a acts on SL 2 via left multiplication by upper triangular unipotent matrices, were constructed in [10,Section 5 and 6]. Various other extensions were obtained from these by performing suitable equivariant affine modifications.…”

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.…”