Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties

For m C 2; consider the m×m determinantal variety of m×(m+1) matrices mod t2: this is the variety Zm;m+1 m;2 obtained by considering generic m× (m+ 1) matrices over the ring F[t]~(t2), and setting the coe cients of powers of t of all m×m minors to zero. The corresponding object mod t is of course the classical determinantal variety Zm;m+1 m of m× (m+ 1) matrices of rank less than m. We refer to the mod t2 variety Zm;m+1 m;2 as the tangent bundle of the classical variety Zm;m+1 m. In this thesis, we begin by providing a conjectured Groebner basis for the ideal Im;m+1 m;2 which de nes Zm;m+1 m;2 , and as well, conjectured lead terms of the Groebner basis. These conjectures were made based on explicit computations for the cases 2 B m B 6 that were done using the computer algebra system Singular ([8]). Since our conjectured lead terms are squarefree for all m, we can construct Im, the Stanley- Reisner simplicial complex attached to the ideal generated by our conjectured lead terms. This complex has an existence of its own, independent of the conjectures, and is an interesting object in its own right. The bulk of our thesis consists of the analysis of this simplicial complex. For all values of m, we describe the facets of Im in terms of their intersections with certain antidiagonals of a matrix of variables. Also for all values of m, we derive a formula that counts the number of facets of this complex. This number turns out to be the square of the degree of the classical variety Zm;m+1 m , which yields evidence that our conjectures about the Groebner basis and their lead terms are correct. In addition, for 2 B m B 6, we show that Im is shellable. Using standard results, we conclude from the shellability of Im that for 2 B m B 6, the coordinate ring of Zm;m+1 m;2 is Cohen-Macaulay, a property of great interest in algebraic geometry. Further, for 2 B m B 6, we compute the Hilbert function of the Stanley-Reisner ring of Im. The corresponding Hilbert series of Zm;m+1 m;2 pleasingly turns out to be the square of the Hilbert series of the classical variety Zm;m+1 m , giving further credibility to our conjectures.