DMS Algebra Seminar

Speaker: Doug Leonard (Auburn University)

Title: Desingularization of function fields

Abstract: We’ll recast resolution of singularities as a purely commutative algebra topic, in terms of irreducible polynomials in d + 1 variables defining d-dimensional (affine) domains, all with a common field of fractions, L, also called a function field. We’ll then define d-dimensional valuations, having d explicit, independent parameters, and also a dependent unit, having a recursively computable formal series expansion in the parameters. This then gives us a purely algebraic objective, namely that of finding formal Laurent series expansions of the original variables used to define the function field in terms of the parameters of each valuation, rewritten in polynomial form by using the unit as well. This theory is independent of characteristic, unlike other approaches, and gives a complete overhaul of the theory in reasonably elementary terms, assuming commutative algebra at the level of valuations and local monomial orderings. The paper was posted pre-pandemic on arXiv. There is Macaulay 2 code written to implement this, though it needs to be put in package form. There is a file of examples, mostly from the literature, used to test it against other methods. (Both are included in the LaTeX file of the paper, so available from me by request.) Singular has a resolve.lib that can do the examples in char 0. It has the same general structure of a rooted tree, but based on blowups, exceptional divisors, and the like. In Magma the desingularization of surfaces is much harder to sort out and use. I am not aware of other code written to do this.

live-stream on Zoom via the following link: https://auburn.zoom.us/j/83583246175