Rational points on varieties in families, and countable unions of varieties over countable fields

研究者将从事两个与算术和几何有关的项目. 第一个项目是研究代数簇族中有理数点的存在性，并建立有理数域的丢番图子集库，其中丢番图集是指族中相应簇族具有有理数点的有理数参数值的集合。 特别是，调查员将探讨家庭whosefields是夏特莱表面或更复杂的圆锥束，为Brauer-Manin阻塞产生有趣的diophantinesets。 第一个项目的长期目标是利用丢番图集构造整数的模型，因为这将反驳Mazur关于有理点拓扑的一个猜想，同时证明判定多元多项式方程是否有有理解的问题的不可判定性。第二个项目是研究可数代数闭域上的子簇的可数并，例如代数数域，特别是证明在自然发生的情况下，存在一个封闭点以外的可数并，作为所需的各种建设。 例子包括在非uniruled品种的分数曲线的工会，模空间轨迹abelian品种isogenous到雅可比，轨迹在家庭的基础上的品种，皮卡德数的纤维跳跃，工会的子品种所产生的自同态迭代的品种。算术几何位于数论和代数几何的交叉：像代数几何一样，它研究多变量多项式方程的解，但它是在解的坐标为整数的条件下，（整数如-37）或有理数（分数如-3/5）或可能是其他一些数字系统的元素，这些数字系统与传统的真实的数字或复数系统不同。 从古希腊时代起，人们就对这些问题进行了研究，并在世纪发现了密码学和纠错码的应用。 研究者的研究重点不是这些应用，而是它们背后和周围的基本问题，例如是否有可能编写计算机程序来确定任意多变量多项式方程是否有有理数解的问题。 这项研究将研究方程族中的模式，希望能推导出否定的答案，同时也证明在一个更大的数字系统中满足无穷多个约束的解的存在性。

The investigator will work on two projects connected with arithmeticgeometry. The first project is to study the existence of rationalpoints in algebraic families of varieties, and to build a library ofdiophantine subsets of the field of rational numbers, where adiophantine set in this context means the set of rational parametervalues for which the corresponding variety in a family has a rationalpoint. In particular, the investigator will explore families whosefibers are Chatelet surfaces or more complicated conic bundles, forwhich the Brauer-Manin obstruction produces interesting diophantinesets. A long-term goal of the first project is to construct a modelof the integers using diophantine sets, because this would disprove aconjecture of Mazur regarding topology of rational points andsimultaneously prove the undecidability of the problem of decidingwhether a multivariable polynomial equation has a rational solution.The second project is to study countable unions of subvarieties over acountable algebraically closed field, such as the field of algebraicnumbers, and in particular to prove that in naturally occurringsituations, there exists a closed point outside the countable union,as required for various constructions. Examples include the union ofrational curves in a non-uniruled variety, the moduli space locus ofabelian varieties isogenous to a Jacobian, the locus in the base of afamily of varieties where the Picard number of the fiber jumps, andunions of subvarieties arising from iteration of endomorphisms ofvarieties.Arithmetic geometry lies at the intersection of number theory andalgebraic geometry: like algebraic geometry, it studies the solutionsto multivariable polynomial equations, but it does so under thenumber-theoretic restriction that the coordinates of the solutions beintegers (whole numbers like -37) or rational numbers (fractions like-3/5) or perhaps elements of some other number system different fromthe traditional systems of real numbers or complex numbers. Suchquestions were studied for their intrinsic interest since the time ofthe ancient Greeks, and in the 20th century they found unforeseenapplications to cryptography and error-correcting codes. Theinvestigator's research focuses not on these applications, but on thefundamental questions underlying and surrounding them, such as thequestion of whether it is possible to write a computer program todecide whether an arbitrary multivariable polynomial equation has asolution in rational numbers. The research covered by this grant willstudy patterns in families of equations in the hope of deducing anegative answer, while also proving the existence of solutionssatisfying infinitely many constraints in a larger number system.