The Resolution of Toric Singularities

Acknowledgements The greatest thanks must go to my supervisor Daniel Chan. Daniel was on leave for 6 months this year, yet he nonetheless committed a lot of time to helping me. His enthusiasm and intelligence is very inspiring. Thanks also to my dear parents, who have supported me all the way through university. Thanks to everyone in the honours room! I appreciated sharing Wirkunggeschicte with Ambros and Maike. Coffee with Doug was great. The complaining of Gina and the steadfastness of Michael will be much missed! Introduction Algebraic geometry is the study of the zeros of systems of polynomials. We denote these sets algebraic varieties. For example, hyperbolas, parabolas and hyperplanes are algebraic varieties. In algebraic geometry a phenomenon of particular interest is the singularity. Informally, singularities occur when the tangent space to a particular point in a variety is not well defined. Consider for example the curve y 2 = x 3 − x 2 in R 2. The curve crosses itself at 0, and as a consequence if we intersect any line through the origin with the curve we get a double root. A question of great interest to the algebraic geometer is this: given any algebraic variety X how may I resolve its singularities? Suppose Y is a nonsingular variety. A resolution of singularities is a map φ : X → Y with the following characteristics. The map φ = (φ 1 ,. .. , φ k) must have rational components, and in addition φ must be defined almost everywhere on X, such that the set of points where φ isn't defined forms an algebraic subvariety. The Japanese mathematician Heisuke Hironaka provided an answer to our question in 1964. Hironaka's celebrated theorem is that a resolution of singularities of X always exists, as long as the variety X is over a field of characteristic zero. He was awarded a Fields Medal for his efforts in 1970. However, although we know a resolution of singularities of any algebraic variety X must exist, it is often exceedingly difficult to determine. Moreover Hironaka's Theorem employs a lot of sophisticated mathematics which goes well beyond the scope of this thesis. The beauty of toric varieties is that we can reduce the problem of finding a resolution of singularities to relatively straightforward combinatorics. Toric varieties are algebraic varieties which are constructed in a special way, using convex polyhedral cones. (A cone is …