Abstract: Noncommutative algebra (=quivers) can be used to solve both explicit and non-explicit problems in algebraic geometry, and these lectures will try to explain some of the features of both approaches. I want to use these notes to give a gentle (!) introduction to the subject, and will try and make them as self-contained as possible. Since I want to eventually end up doing non-toric geometry, throughout I shall never adopt the language of toric geometry, even if the example I am considering is toric. First some motivation: From a noncommutative perspective we would like to take a singularity X = Spec R and produce a NC ring A from which we can extract resolution(s) of X . We can then ask whether the NC ring has some geometrical meaning, and if so whether this gives information about A. We can also ask what A says about X and its resolutions. From a more geometric perspective we may already have some resolution Y of X and would like produce other resolutions, for example by flopping certain curves. We may also want to describe the derived category of Y . This can sometimes be done using noncommutative algebra. In practice however things are not quite as simple as this, since most of the time a specific problem will be a mixture of the two above problems. Sometimes it is easier to solve the problem using the geometry, sometimes it is easier using quivers. Thus geometry can give us results in noncommutative algebra and noncommutative algebra can give us results in geometry; it is the process of playing the two sides off each other which gives us the strongest results. Today I’m going to define quivers and tell you how to think of them, then following King [King1] I’ll talk about their moduli space(s) of finite dimensional representations. Time permitting I’ll then show how to calculate the moduli spaces in some easy examples.
Publication Year: 2009
Publication Date: 2009-01-01
Language: en
Type: article
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