Imad Jaradat

Proper triangular a-actions on 4 are translations

We describe the structure of geometric quotients for proper locally triangulable additve group actions on locally trivial A^3-bundles over a noetherian normal base scheme X defined over a field of characteristic 0. In the case where dim X=1, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank 2 over X. As a consequence, every proper triangulable Ga-action on the affine four space A^4 over a field of characteristic 0 is a translation with geometric quotient isomorphic to A^3.

Locally trivial 𝔾a−actions on ℂ5 with singular algebraic quotients

ABSTRACT Simple examples are given of proper algebraic actions of the additive group of complex numbers on ℂ5 whose geometric quotients are, respectively, affine, strictly quasiaffine, and algebraic spaces which are not schemes. Moreover, a Zariski locally trivial action is given whose ring of invariant regular functions defines a singular factorial affine fourfold embedded in ℂ12. The geometric quotient for the action embeds as a strictly quasiaffine variety in the smooth locus of the algebraic quotient with complement isomorphic to the normal affine surface with the A2−singularity at the origin.

On graded P-compactly packed modules

Abstract Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce the concept of graded P-compactly packed modules and we give a number of results concerning such graded modules. In fact, our objective is to investigate graded P-compactly packed modules and examine in particular when graded R-modules are P-compactly packed. Finally, we introduce the concept of graded finitely P-compactly packed modules and give a number of its properties.