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

研究人员将致力于两个与算术几何相关的项目。第一个项目是研究有理数代数族中有理点的存在性，并建立有理数域的丢番图子集库，这里所说的丢番图集是指一族中对应的簇有一个有理点的有理参数值的集合。特别是，研究人员将探索纤维是Chatlet表面或更复杂的圆锥束的家族，对于这些家族，Brauer-Manin障碍会产生有趣的不透明带。第一个项目的长期目标是利用丢番图集构造一个整数模型，因为这将证明Mazur关于有理点拓扑的一个猜想是错误的，同时证明了判定一个多元多项式方程是否有有理解的问题是不可判定的。第二个项目是研究可数代数闭域上的子簇的可数并，特别是证明在自然发生的情况下，正如各种构造所要求的那样，在可数并之外存在一个闭点。例如，非正则簇中有理曲线的并，交换簇的模空间轨迹与雅可比同源，簇的基中纤维的Picard数跳跃处的轨迹，以及因簇的自同态迭代而产生的子簇的并。算术几何位于数论和代数几何的交叉处：像代数几何一样，它研究多变量多项式方程的解，但它是在数论限制下进行的，解的坐标是整数(像-37的整数)或有理数(像-3/5的分数)，或者可能是与传统的实数或复数系统不同的其他数系的元素。从古希腊时代起，这些问题就因为它们的内在兴趣而被研究，并在20世纪发现了在密码学和纠错码中意想不到的应用。研究人员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的解。这笔拨款所涵盖的研究将研究方程式族中的模式，希望推导出否定的答案，同时也证明在更大的系统中满足无限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.