Degenerations of K3 surfaces in projective space

The purpose of this article is to study a certain kind of numerical K3 surfaces, the so-called K3 carpets. These are double structures on rational normal scrolls with trivial dualizing sheaf and irregularity . As is deduced from our study, K3 carpets can be obtained as degenerations of smooth K3 surfaces. We also study the Hilbert scheme near the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. Contrary to the case of canonical ribbons, not all K3 carpets are smooth points of the Hilbert scheme.

     

The Horrocks-Mumford bundle and the Ferrand construction

There is an analogue of the Ferrand construction for smooth surfaces in ₄: if the normal bundle of such a surface X has a suitable 1-subbundle, then a 2-vector bundle can be constructed, which has a section vanishing doubly on X. In this way the Horrocks-Mumford bundle is recovered from the quintic scroll.K. Stein, dem B ahnbrecher der Funktionen-theorie, gewidmet     

Expanded Degenerations and Pairs

We provide a universal approach to the moduli of Jun Li's expanded pairs and expanded degenerations [20 Li , J. ( 2001 ). Stable morphisms to singular schemes and relative stable morphisms . J. Differential Geom. 57 : 509 – 578 .[Web of Science ®] , [Google Scholar]]. This enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16 Graber , T. , Vakil , R. ( 2005 ). Relative virtual localization and vanishing of tautological classes on moduli spaces of curves . Duke Math. J. 130 : 1 – 37 .[Crossref], [Web of Science ®] , [Google Scholar]], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.     

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let ${\mathcal{T}}$ be an ideal triangulation of M. The deformation variety ${\mathfrak{D}(\mathcal{T})}$ , a subset of which parameterises (incomplete) hyperbolic structures obtained on M using ${\mathcal{T}}$ , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.     

Degenerations of cubic threefolds and matroids

We present a surprising connection between cubic threefolds and the well-known regular matroid by making use of intermediate Jacobians of cubic threefolds realized as Prym varieties. As a corollary we obtain a new proof of the nonrationality of generic cubic threefolds.

     

Degenerations and Stable Homomorphisms #

Let Λ be an algebra over an algebraically closed field. We compare the partial order ≤ _hom in the module category of Λ with a certain relation ≤ _stab in the stable module category of Λ. Both relations coincide if Λ is hereditary. Starting with any non-hereditary representation-finite algebra Λ, we construct a representation-finite algebra Λ′, obtained by a covering of the Auslander-Reiten quiver of Λ, such that for Λ′ both relations do not coincide.     

Degenerations of Submodules and Composition Series

Let M and N be modules over an artin algebra such that M degenerates to N. We show that any submodule of M degenerates to a submodule of N. This suggests that a composition series of M will in some sense degenerate to a composition series of N. We then study a subvariety of the module variety, consisting of those representations where all matrices are upper triangular. We show that these representations can be seen as representations of composition series, and that the orbit closures describe the above mentioned degeneration of composition series.     

Degenerations of Representations of Associative Algebras

The purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finite-dimensional representations of finitely generated, and especially, finite-dimensional algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether. Received: July 2007     

Degenerations of toric ideals and toric varieties

Toric degenerations of toric varieties and toric ideals are important both in theory and in applications. In this paper, we study the correspondence between degenerations of toric variety and of toric ideal when the weight admits a regular subdivision.