The division algorithm and the hilbert scheme
David Allen Bayer · Harvard University eBooks · 1982
In this thesis, a division algorithm is studied, following work of Macaulay, Hironaka, Buchberger, and others, which generalizes row reduction and the euclidean algorithm, in the same way that elimination theory generalizes the determinant and the resultant. The main result is a cohomological interpretation of the complexity of this algorithm, for a fixed number of variables. This follows from a new result on the vanishing of coherent sheaf cohomology, which generalizes previous work by Gotzmann, and Macaulay. The Hilbert scheme offers a setting in which results about this algorithm can be understood; this relationship is described. The theory of the division algorithm is related to the problem of manipulating objects in algebraic geometry by computer. The problem of computing coherent sheaf cohomology is considered, as a guiding example. Finally, explicit equations are given for the Hilbert scheme.