The Homological Minimal Model Program

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One of the more startling discoveries of twentieth century mathematics was the realization that, in contrast to the theory of surfaces, and in contrast to many results and the prevailing wisdom, in dimension three geometric minimal models may be singular. Thus the `best' answers, indeed the spaces we are aiming for, may themselves have singularities. This project aims to understand these structures from a homological viewpoint, and will give new information even in the well-studied cases. Current derived category techniques focus on situations where the minimal models are smooth, whereas this project will push both our theoretical understanding and our computational ability deep into the singular setting.The first part of this project develops the homological algebra surrounding reconstruction algebras and maximal modification algebras, strengthening them and extending them in many new directions. These new techniques will then be applied to algebraic geometry, and will recover the existence of geometric minimal models in dimension three as a special case. The techniques, however, will give much more, and in the main body of this work we use this extra data to obtain information regarding contraction of curves and flops; this then allows us to run aspects of the minimal model program in an algorithmic manner.