Mathematics Colloquia and Seminars

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The reduced Groebner basis of a polynomial ideal I (with respect to a fixed term order) gives a one-parameter (flat) deformation of the (complex) variety V(I) defined by I to a union of coordinate subspaces. This deformation carries a lot of the information about V(I) over to this simple variety of coordinate subspaces: for instance, the dimension, the degree, the Hilbert function etc. are preserved throughout. When this deformation is done algebraically by computing the so-called initial ideal of I, other "embedded" coordinate subspaces emerge. This joint work with Rekha Thomas investigates the isolated and embedded components of the initial ideals (of the defining ideals) of toric varieties. We show that these associated coordinate subspaces come in saturated chains: this puts a strong condition on when a given monomial ideal could be the initial ideal of a toric ideal. Moreover, we prove a tight upperbound on the length of these chains that depends only on the codimension of the toric variety. The techniques are combinatorial-geometric, and use a characterization of a toric initial ideal in terms of a set of polytopes obtained from the toric variety (and the underlying term order). This characterization helps to explain the multiplicites of the associated components as well.