Spaces of Rational Curves in Projective Varieties

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The projects described in this proposal aim to contribute towards understanding moduli spaces of rational curves on complete intersections. The study of rational curves on smooth complete intersections is fundamental to a broad spectrum of important problems about Fano varieties, rationally connected varieties, and diophantine geometry. Despite some progress over the past few years, some of the basic properties of these spaces are still unknown. In the proposed research, some open questions on the dimension, irreducibility, Kodaira dimension, and several other aspects of the geometry of spaces of rational curves on complete intersections in projective space and other homogeneous varieties are investigated. Algebraic varieties are common zeros of collections of polynomial equations. An important approach to study the geometry of algebraic varieties is to study parameter spaces of rational curves contained in them. These parameter spaces are themselves varieties with rich geometry, and in the case of hypersurfaces, the study of them has broad applications in higher dimensional algebraic geometry, modern enumerative geometry, and questions inspired by mirror symmetry.