Abstract
In the current project proposal, we study important aspects of the global geometry of moduli spaces of translation surfaces. Translation surfaces can be seen as plane polygons with opposite edges identified. They are very simple and fundamental geometric objects yet the geometry of their moduli spaces remains at large a mystery. Moduli spaces of translation surfaces are universal geometric objects for plane polygons, they parameterize all isomorphism classes of translation surfaces of a given type. They were initially studied in Teichmüller dynamics and it is only in the last decade that they have been brought to the attention of algebraic geometry attracting substantial research focus. The proposal is divided into three interconnected working packages (WP). First, we study the divisor geometry, i.e., line bundles on them. The geometry of algebraic varieties is largely governed by their line bundles. Concretely, we aim at understanding their Picard groups (WP1). Secondly, we study their birational geometry, i.e., geometry up to birational equivalence (WP2), concretely the Kodaira dimension, and rational models for low-genus cases. Finally, we study their topology, concretely their cohomology groups (WP3). A key input is a recent breakthrough (2018 and 2019) where algebraic modular compactifications of these spaces were constructed. This opens the door to answering questions that before were out of reach. Each working package has intermediate and more ambitious goals.
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