When is a variety a quotient of a smooth variety by a finite group?

10/25/2011 - 12:15pm
10/25/2011 - 1:10pm
Anton Geraschenko (California Institute of Technology)

If a variety X is a quotient of a smooth variety by a finite group, it has quotient singularities -- that is, it is locally a quotient by a finite group. In this talk, we will see that the converse is true if X is quasi-projective and is known to be a quotient by a torus (e.g. X a simplicial toric variety). Though the proof is stack-theoretic, the construction of a smooth variety U and finite group G so that X=U/G can usually be made explicit purely scheme-theoretically. To illustrate the construction, I'll produce a smooth variety U with an action of G=Z/2 so that U/G is the blow-up of P(1,1,2) at a smooth point. This example is interesting because even though U/G is toric, U cannot be taken to be toric. This is joint work with Matthew Satriano.

Millikan 134 (Pomona College)