  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 stabilizer, 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.
