## Abstract

The space of subvarieties of ℙ^{n} 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 ℙ^{n} with fixed Hilbert polynomial.

We instead relax “sub” to “branch”, where a *branchvariety of *ℙ^{n} is defined to be a *reduced *(though possibly reducible) scheme *with a finite morphism to* ℙ^{n}. Our main theorems are that the moduli stack of branchvarieties of ℙ^{n} 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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