Subvarieties of Hypersurfaces and Local Positivity of Ample Line Bundles

徐9700491这个项目是关于代数几何的研究。根据镜像对称原理，物理学家几年前提出了一个公式，用来预测一般五次三次曲线上不同次数的有理曲线的数目。最近，Givental证明了物理学家提出的数字与一般五次三次J-全纯曲线上不同次数的J-全纯曲线的数目一致。主要研究者将致力于Clemens关于一般五次三次有理曲线在每一次上的有限性的猜想。他还致力于通过研究Seshadri常数来理解光滑射影簇上充裕线丛的局部正性，Seshadri常数与Nagata关于平面代数曲线的奇性与曲线的次数之间的关系的猜想有关。此外，主要研究者还将研究一般超曲面的补的代数双曲性和光滑射影曲面上的代数曲线的计数。最后，他将研究Fano流形的量子上同调代数，看看Fano超曲面的量子上同调代数是否在Dubrovin意义下是半单的。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

Xu 9700491 This project is concerned with research in algebraic geometry. Based on the mirror symmetry principle, physicists proposed several years ago a formula to predict the numbers of rational curves of various degrees on a general quintic threefold. Recently, Givental showed that the numbers suggested by the physicists agree with the numbers of J-holomorphic curves of genus 0 of various degrees on a general quintic threefold for a generic almost complex structure J. The principal investigator will work on Clemens' conjecture about the finiteness of rational curves on a general quintic threefold in each degree. He is also working towards an understanding of the local positivity of ample line bundles on a smooth projective variety through the study of Seshadri constants, which is related to a conjecture of Nagata on a relation between the singularity of a plane algebraic curve and the degree of the curve. In addition, the principal investigator will study the algebraic hyperbolicity of the complement of a general hypersurface and the enumeration of algebraic curves on smooth projective surfaces. Finally, he will work on the quantum cohomology algebras of Fano manifolds, to see whether the quantum cohomology algebras of Fano hypersurfaces are semi-simple in the sense of Dubrovin. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.