The Asymptotic Behavior of Generic Initial Systems.

Consider a homogeneous ideal I contained inside of a polynomial ring over a field of characteristic zero with the reverse lexicographic order. The family of ideals formed by taking the generic initial ideals of the powers, or symbolic powers, of I is called the generic initial system, or the symbolic generic initial system, of I. The asymptotic behavior of such families is nicely captured by their limiting shapes. We study the generic initial systems of two classes of ideals. The first types of ideals that we investigate are complete intersections. In particular, we explicitly describe the limiting shape of the generic initial system of any complete intersection and show that it only depends on the type of the complete intersection. We also give algorithms that explicitly compute the minimal generators of each of the ideals in the generic initial system of any 2-complete intersection. The second class of ideals that we explore are those corresponding to sets of points in the projective plane. We describe the limiting shape of the symbolic generic initial systems of ideals corresponding to several point arrangements including points in general position, points on a conic, and small numbers of points. The first chapter provides a more elementary introduction to basic algebraic geometry, generic initial ideals, and limiting shapes accessible to senior undergraduate math majors and non-algebraists.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/98028/1/mayess_1.pd