Moduli spaces of branchvarieties

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. I will also explain a recent theorem about connected components of this moduli space. (Based in part on joint works with Knutson.)