Complete moduli spaces of branchvarieties

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.