Complete Moduli Spaces of Branchvarieties

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