- Author(s): Boocher, Adam Lee
- Advisor(s): Eisenbud, David
- et al.
One way to obtain geometric information about a homogeneous ideal is to pass to a monomial ideal via a flat degeneration. Flatness is strong enough to ensure this degeneration preserves the Hilbert function, which allows us to make geometric statements about the original ideal. Although it is by no means trivial, full analysis of monomial ideals is aided by a wealth of interactions with combinatorics, topology, and commutative algebra. However, since flatness only goes so far, finer invariants than the Hilbert function cannot typically be detected via this technique.
One finer invariant is the minimal free resolution. Originally introduced by Hilbert, free resolutions encode algebraic relations among the generators of an ideal. Numerically, the data of a free resolution are the graded Betti numbers which detect surprising geometric information. In recent years there has been much study devoted to the relationship between the modules occurring in a free resolution (collectively called syzygies) and geometric invariants.
Flatness is not strong enough to guarantee that the free resolution will be preserved upon degeneration. In fact, in some sense, the expected behavior is that the resolution will become more poorly behaved. This dissertation studies situations in which flat degenerations preserve more than they ought, and how these superflat degenerations allow us to better understand the resolution of our original ideal. It contains a brief introduction followed by three self-contained chapters.
In Chapter 2 we study ideals associated to sparse-generic matrices, those whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. Here we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over the integers and that the projective dimension depends only on the number of columns of the matrix that are identically zero. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Groebner basis.
Chapter 3 presents joint work with Elina Robeva and initiates a systematic study of ideals minimally generated by a universal Groebner basis. We call such an ideal robust. We show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible generalizations to higher degree, providing a tight classification for determinantal ideals, and a counterexample to a natural extension for Lawrence ideals. We close with a discussion of robustness of higher Betti numbers.
Chapter 4 is joint work with Federico Ardila concerning the closure of linear spaces in a product of projective lines. Let L be an linear space in affine space. We study the closure of L in a product of projective lines and show that the degree, defining equations, graded Betti numbers, and universal Groebner basis of its defining ideal are all combinatorially determined by the linear matroid associated to L. We explicitly compute these invariants.