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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Complete moduli spaces of branchvarieties

Valery Alexeev1 / Allen Knutson2

1Department of Mathematics, University of Georgia, Athens GA 30602, USA. e-mail:

2Department of Mathematics, UCSD, La Jolla CA 92093, USA. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 639, Pages 39–71, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2010.010, January 2010

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Published Online:


The space of subvarieties of ℙn with a fixed Hilbert polynomial is not complete. Grothendieck defined a completion by relaxing “variety” to “scheme”, giving the complete Hilbert scheme of subschemes of ℙn with fixed Hilbert polynomial.

We instead relax “sub” to “branch”, where a branchvariety of n is defined to be a reduced (though possibly reducible) scheme with a finite morphism ton. Our main theorems are that the moduli stack of branchvarieties of ℙn with fixed Hilbert polynomial and total degrees of i-dimensional components is a proper (complete and separated) Artin stack with finite diagonal, and has a coarse moduli space which is a proper algebraic space.

Families of branchvarieties have many more locally constant invariants than families of subschemes; for example, the number of connected components is a new invariant. In characteristic 0, one can extend this count to associate a ℤ-labeled rooted forest to any branchvariety.

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David Rydh
Communications in Algebra, 2011, Volume 39, Number 7, Page 2632

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