Spaces of Rational Curves in Projective Varieties

本方案中所描述的项目旨在帮助理解完全交叉口上有理曲线的模空间。光滑完全交点上有理曲线的研究是Fano簇、有理连通簇和丢番图几何等一系列重要问题的基础。尽管在过去的几年里取得了一些进展，但这些空间的一些基本性质仍然未知。在所提出的研究中，研究了射影空间和其他齐次簇中完备交上有理曲线的空间几何的一些公开问题，包括维数、不可约性、Kodaira维数等。代数族是多项式方程集合的公共零点。研究代数簇几何的一个重要途径是研究其中包含的有理曲线的参数空间。这些参数空间本身就是具有丰富几何意义的变种，在超曲面的情况下，它们的研究在高维代数几何、现代计数几何和镜面对称问题中有着广泛的应用。

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.