Infinitesimal Automorphisms of Algebraic Varieties

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The purpose of this project is to study the automorphism scheme of smooth projective varieties in positive characteristic. In this context, infinitesimal automorphisms, i.e. subgroup schemes of the automorphism scheme which are contained in an infinitesimal neighborhood of the identity, play a fundamental role. This is in stark contrast to the situation in characteristic 0 where such non-reduced group schemes do not even exist. The first part of this project will be the explicit calculation of infinitesimal automorphisms in the case of non-ruled surfaces of special type and the construction of varieties of higher dimension, such as Calabi-Yau threefolds, using the results of these computations. In the second part, I want to study the scheme structure of the group scheme of automorphisms of surfaces and varieties of general type along with their canonical models from a more abstract perspective and work towards giving a bound on the length of this automorphism scheme.