Limits of varieties: subschemes vs. branchvarieties -- Allen Knutson

Given a family of algebraic varieties indexed by a nonzero parameter $t$, it is usually desirable to define a ``limit'' at $t=0$. Grothendieck studied the case where the varieties are all embedded in a fixed projective space, and did give a well-defined limit; unfortunately to do so he had to invent schemes, whose geometry is often murky. I'll present an alternative limit, where we keep ``variety'' but give up ``sub''; the limit is still a variety (not just a scheme) with a map to projective space that is no longer necessarily an inclusion. Then I'll discuss an application to intersection theory on algebraic varieties. To create such a limit requires introducing certain Nth roots of $t$; consequently the corresponding moduli space is a stack, not a scheme. If time permits, I'll explain some relations to the Hilbert scheme and Chow variety.