Base Loci of Linear Series and Diophantine Approximation

从最抽象的理论物理概念到数百万人每天使用的密码系统（通常没有意识到这一点），代数几何和数论在21世纪的生活中有着真正而重要的存在。本提案旨在发展代数几何的工具与数论应用的观点。最重要的是找到多项式方程的有理数或整数解的问题，其中最著名的例子是怀尔斯最近建立的费马大定理。在此建议下发展的新几何技术将应用于限制更一般多项式方程的有理解的数量。丢番图近似的许多情形在光滑射影变化上产生线性级数。通常会有使用消失定理和交集理论的论点，迫使级数移动，显示基本轨迹为空。另一方面，各种假设性的假设——例如，假设一条至少有两个属的曲线上有无限多个有理点，或者假设一个代数无理数有很好的有理数近似值——迫使所讨论的线性序列有一个非空的基轨迹。由此产生的矛盾表明假设假设是错误的。这条思路或推理已分别用于建立莫德尔猜想和罗斯定理，并将在本项目中应用于高维问题的研究，如schmidt子空间定理和Faltings关于阿贝尔变分的有理点的定理，长期目标是推广或加强这些定理。这些论点所需要的关键代数几何背景是对线性级数的基轨迹的数值性质的详细研究。特别地，一个关键的问题是，即使束不是正的，Seshadri常数是否控制着线束的全局截面的存在性。

From the most abstract concepts of theoretical physics to thecryptosystems used by millions of people every day (often without realizing it) , algebraic geometry and numbertheory have a genuine and vital presence in twenty-first century life. This proposal seeks to develop tools of algebraic geometry with a viewtoward number theoretic applications. Of central importance is the problem of finding rational or whole number solutions to polynomialequations, the most famous example of which is Fermat's Last Theoremrecently established by Wiles. The new geometric techniques developed under this proposal will be applied to limit the number of rational solutions of a more generalpolynomial equation.Many situations in diophantine approximation produce a linear series on asmooth projective variety. Often there are arguments employing vanishing theorems and intersection theory that force the series to move, showing the base locus to be empty. On the other hand, varous hypothetical assumptions-for example, assuming that there are infinitely many rational points on a curve of genus at least two, or assuming that there are good rational approximations of an algebraic irrational number-- force the linear series in question to have a non-empty base locus. The resulting contradiction shows the hypothetical assumption to be false. This line or reasoning has been used to establish Mordell's conjecture and Roth's theorem respectively, andwill be applied in this project to study higher dimensional problems, such as theSchmidt subspace theorem and Faltings' theorem on rational points ofsubvarieties of abelian varieties, with a long term goal of extending andor strengthening these theorems. The key algebro-geometric backgroundrequired in these arguments is a detailed study of numerical properties ofthe base locus of a linear series. In particular, a key question iswhether or not Seshadri constants control the existence of global sectionsof a line bundle even when the bundle is not positive.