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NSERC USRA Report – Summer 2013 Resolution of Singularities and Toric Varieties Shamil Asgarli

During the summer, I had the privilege and pleasure of studying under supervision of Professors Julia Gordon and Sujatha Ramdorai on the resolution of singularities and related topics in algebraic geometry. Very roughly speaking, the goal is to take some singular algebraic variety X (for example, a curve or surface), and to resolve (”smooth out”) its singular points, which in some sense are ”badly” behaved points, and at the end of the process to obtain a new variety X (called desingularization, or nonsingular model of X) which is birationally isomorphic to X away from the singular points. In order to give more precise formulation of the problem, one needs to introduce the concept of blow-up. In simplest terms, blow-up is the process of replacing a singular point with the projectivized tangent space. For example, one can blow-up the origin of the affine plane A as depicted in the diagram below.