On differentiable compactifications of the hyperbolic space

The group of direct isometries of the hyperbolic space This isometric action admits many differentiable compactifications into an action on the closed n-dimensional ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the classifications of real analytic and smooth compactifications.     

Torsion-Free Sheaves and Moduli of Generalized Spin Curves

This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or r-spin curves, are pairs (X,L) with X a smooth curve and L a line bundle whose rth tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over Z[1/r], all using torsion-free sheaves, are constructed. All three yield algebraic stacks, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when r=2.     

Means, homomorphisms, and compactifications of weighted semitopological semigroups

We consider some almost periodic type function algebras on a weighted semitopological semigroup, and using the set of multiplicative means on each of these algebras, we define their corresponding weighted semigroup compactifications. This will constitute an effective tool for investigating the properties of the function algebras concerned. We also show that these compactifications do not retain all the nice properties of the ordinary semigroup compactifications unless we impose some restrictions on the weight functions.     

Conditions under which the least compactification of a regular continuous frame is perfect

We characterize those regular continuous frames for which the least compactification is a perfect compactification. Perfect compactifications are those compactifications of frames for which the right adjoint of the compactification map preserves disjoint binary joins. Essential to our characterization is the construction of the frame analog of the two-point compactification of a locally compact Hausdorff space, and the concept of remainder in a frame compactification. Indeed, one of the characterizations is that the remainder of the regular continuous frame in each of its compactifications is compact and connected.     

On the compactifications of contractible affine threefolds and the Zariski Cancellation Problem

In this article, we consider the compactifications of some kinds of contractible smooth affine threefolds and the characterization of the affine 3-space GIF1³ keeping the 3-dimensional Zariski Cancellation Problem in mind. We classify the compactifications of contractible smooth affine threefolds with a certain condition concerning the numerical property of boundary divisors with respect to compactifications and, then, we show that if an affine threefold X satisfies X×GIF1¹GIF1⁴ and this numerical condition, then X is isomorphic to the affine 3-space GIF1³.Mathematics Subject Classification (2000):14R10, 14E30     

Filters and semigroup compactification properties

Stone-Čech compactifications derived from a discrete semigroup S can be considered as the spectrum of the algebra ℬ(S) or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too. In the present paper, first we review the characterizations of semigroup compactifications in terms of filters and then extend some of the results in Papazyan (Semigroup Forum 41:329–338, 1990) concerning the Stone-Čech compactification to a semigroup compactification associated with a Hausdorff semitopological semigroup.     

A tropical view on Bruhat-Tits buildings and their compactifications

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.     

Compactification of the universal Picard over the moduli of stable curves

This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. Received January 5, 1998; in final form April 1, 1999 / Published online July 3, 2000     

A sharp cusp count for complex hyperbolic surfaces and related results

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in Di Cerbo and Di Cerbo (see [5]), to study the geometry of cusped complex hyperbolic surfaces and their compactifications.     

Compactification of Limit ?-Net Spaces

 Limit ?-net spaces are defined as convergence spaces whose convergence is expressed by using generalized nets, the so-called ?-nets (where ? is a construct). For limit ?-net spaces we study compactifications, especially those ones that are analogous to the Alexandrov and Čech-Stone compactifications known for topological spaces.     

Compactification of Limit ?-Net Spaces

 Limit ?-net spaces are defined as convergence spaces whose convergence is expressed by using generalized nets, the so-called ?-nets (where ? is a construct). For limit ?-net spaces we study compactifications, especially those ones that are analogous to the Alexandrov and Čech-Stone compactifications known for topological spaces. (Received 24 February 2000)     

Oids,finite sums and the structure of the Stone-Čech compactification of a discrete semigroup

The equivalence between oids and semigroups which satisfy a condition involving finite sums is established. Some of the already known results on the structure of Stone-Čech compactifications of discrete semigroups are obtained as immediate consequences. It is also shown that most commutative semigroups contain oids so that oid theory has applications to the Stone-Čech compactifications of many semigroups.     

On some parallelism between complete regularity and zero-dimensionality

We explore some parallelism between the categories CRFrm and 0DFrm of completely regular frames and zero-dimensional frames, respectively, with a view to establishing zero-dimensional analogues of C*-quotients. A lattice homomorphism between the cozero parts of two completely regular frames can be lifted to a frame homomorphism between the Stone-?ech compactifications of the frames involved [13]. Here we lift a lattice homomorphism ψ: BL → BM between the Boolean parts of two zero-dimensional frames to a frame homomorphism between their universal zero-dimensional compactifications, and then study some properties of the lift.     

Lattice measures and topologies

Summary We abstract Frink's notion of a normal base of a topological space to an arbitrary lattice, and replace the notion of filters on a base by zero-one measures on a lattice. This offers analytical simplification and clarijication, and extends to arbitrary measures as well. By putting a topology on the set of measures, we generalize the notion of Wallman-type compactifications, and we look at relations between the compactifications by examining the underlying lattices. Entrata in Redazione il 20 gennaio 1975.     

Moduli stacks of stable toric quasimaps

We construct new “virtually smooth” modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.     

On a characterization of analytic compactifications for ℂ* × ℂ*

Let M be a minimal compact surface, let Γ ⊂ M be a compact analytic sub-variety. Assume that X:= M \ Γ is Stein. Then we will show that X admits algebraic compactifications M _i (resp. non algebraic compactifications $ \mathbb{M}_i $ \mathbb{M}_i ) which are not birationally equivalent (resp. not bimeromorphically equivalent) iff X is biholomorphic to      

The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

In this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.     

Points of Bounded Height on Equivariant Compactifications of Vector Groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.     

Remarks on the Stone-?ech and Alexandroff compactifications of locales

Relations of strong inclusion are considered on pseudocomplemented distributive lattices to refine existing constructions of (Stone-?ech and Alexandroff) compactifications of frames.     

Products of Compactifications

A product operation of compactifications is defined and its different properties are studied. Some applications are considered.