Toric Varieties in Hilbert Schemes

Given a field K, let R be the power series ring K[[x1, . . . , xn]]. There is a natural action of Aut(R) on the Hilbert scheme Hilb(R) parameterizing ideals in R of colength d. Recall from [10] that given a smooth variety X of dimension n and a monomial ideal I ⊂ R of finite colength, the space U(I) parametrizing subschemes of X isomorphic to Spec(R/I) sits naturally inside a suitable Hilbert scheme H . In [10] a flag bundle B on the tangent bundle of X and a space Y ⊂ B × U(I) such that Y is a fiber bundle over B with respect to the first projection map and an étale covering of U(I) with respect to the second is constructed. Moreover, Y has the property that its closure in B × H is also a fiber bundle over B. Via an isomorphism φ of R and the local ring of the point of X over which the fiber lies, the fiber is isomorphic to the closure of the G-orbit of I in H where G is the subgroup of Aut(R) fixing the flag over which the fiber lies via φ. The complexity of the fiber F (I) of Y over B and its closure is partially measured by the measuring sequence of I as defined in [10]. In the first level of complexity, the G equivariance of F̄ (I) forces its normalization to be a projective space. In the second level, the normalization is forced to be a toric variety. In the third level, we lump together all of the other cases. The following three sections correspond to these three levels of difficulty, each illustrating techniques for understanding F and F̄ (I). For simplicity, we restrict ourselves to the case when n = 2, thus freeing the variable n for later use. Moreover, we will use the variables x