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Weighted Projective Hypersurfaces with Extreme Invariants
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Dissertation
Weighted Projective Hypersurfaces with Extreme Invariants
2023
Overview
The goal of this dissertation is to study weighted projective hypersurfaces and their application to optimization problems in algebraic geometry. First, we generalize and strengthen several well-known results on the automorphisms of hypersurfaces due to Grothendieck-Lefschetz and Matsumura-Monsky to the weighted setting. Then, we construct special examples of weighted projective hypersurfaces with extreme properties. These are used to prove strong asymptotics on certain invariants from birational geometry as dimension increases. In particular, we show that the minimum volume of smooth varieties of general type approaches zero doubly exponentially with dimension; we also show that the index of mildly singular Calabi-Yau varieties can grow doubly exponentially with dimension. For several classes of varieties, we conjecture the optimal bounds on volume or index in every dimension; these conjectures are supported by low-dimensional evidence.
Publisher
ProQuest Dissertations & Theses
Subject
ISBN
9798379617080
