Generic Distractions and Strata of Hilbert Schemes Defined by the Castelnuovo-Mumford Regularity

Consider the standard graded polynomial ring in $n$ variables over a field $k$ and fix the Hilbert function of a homogeneous ideal. In the nineties Bigatti, Hulett, and Pardue showed that the Hilbert scheme consisting of all the homogeneous ideals with such a Hilbert function contains an extremal point which simultaneously maximizes all the graded Betti numbers. Such a point is the unique lexsegment ideal associated to the fixed Hilbert function.

 For such a scheme, we consider the individual strata defined by all ideals with Castelnuovo-Mumford regularity bounded above by m. In 1997 Mall showed that when k is of characteristic 0 there exists an ideal in each nonempty strata with maximal possible Betti numbers among the ideals of the strata. In chapter 4 of this thesis we provide a new construction of Mall's ideal, extend the result to fields of any characteristic, and show that these ideals have other extremal properties. For example, Mall's ideals satisfy an equation similar to Green's hyperplane section theorem.

 The key technical component needed to extend the results of Mall is discussed in Chapter 3. This component is the construction of a new invariant called the distraction-generic initial ideal. Given a homogeneous ideal I C S we construct the associated distraction-generic initial ideal, D-gin_< (I), by iteratively computing initial ideals and general distractions. The result is a monomial ideal that is strongly stable in any characteristic and which has many properties analogous to the generic initial ideal of I.