Finitely related algebras in congruence modular varieties have few subpowers

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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On multiplicative systems defined by generators and relations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028772 ◽

1953 ◽

Vol 49 (4) ◽

pp. 579-589 ◽

Cited By ~ 11

Author(s):

Trevor Evans

Keyword(s):

Automorphism Group ◽

Finite Number ◽

Cyclic Group ◽

Complete Information ◽

Isomorphism Problem ◽

Infinite Cyclic Group ◽

Generators And Relations ◽

Decision Method ◽

Multiplicative Systems ◽

Finitely Related

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.

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Chern numbers of Hilbert modular varieties.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1981.324.192 ◽

1981 ◽

Vol 1981 (324) ◽

pp. 192-210

Keyword(s):

Hilbert Modular ◽

Chern Numbers ◽

Hilbert Modular Varieties ◽

Modular Varieties

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A finitely generated, infinitely related group with trivial multiplicator

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046955 ◽

1971 ◽

Vol 5 (1) ◽

pp. 131-136 ◽

Cited By ~ 14

Author(s):

Gilbert Baumslag

Keyword(s):

Metabelian Group ◽

Finitely Generated ◽

Finitely Generated Group ◽

Related Group ◽

Finitely Related

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

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