Birational geometry and spaces of rational curves

DMS-0353692Jason M. Starr这是复杂代数几何领域的一个研究项目，围绕以下两个问题：（i）证明存在非无理数的法诺流形，以及（ii）找到曲面满射态射的几何通用纤维上的条件，保证有理截面存在的唯一障碍是布劳尔障碍。尽管这些项目看起来截然不同，但它们都在关键方面涉及对品种有理曲线空间的研究。 在第一个问题中，可以通过证明流形上的有理曲线空间上存在“少数”有理曲线来证明法诺流形不是无理的。 在第二个问题中，推测地，条件是几何通用纤维上的有理曲线空间本身有理连接。 在复杂代数几何中，在非代数闭域上定义的簇的概念是变量 x,y,z,... 中的一系列多项式方程的常见情况，这些方程取决于参数 s,t,u,...（它们本身可能满足某些多项式方程）。 一个基本问题是，对于每个参数选择，是否存在以下形式的方程的解：x,y,z,... 是参数 s,t,u,... 中多项式的商，这是有理点问题。 另一个问题是，在多项式不依赖于参数的情况下，x,y,z,... 是否可以写成一组自由变量 a,b,c,... 中的多项式，使得 a,b,c,... 的每个选择都给出多项式的解，并且基本上多项式的每个解都以这种方式出现。 这就是非理性的问题。 两者都是代数几何和数论应用中的古老难题。 该项目将有关有理曲线空间的新结果应用于每个项目。

DMS-0353692Jason M. StarrThis is a research project in the field of complex algebraic geometry around the following 2 problems: (i) proving there exists a Fano manifold that is not unirational, and (ii) finding conditions on the geometric generic fiber of a surjective morphism to a surface guaranteeing that the only obstruction to existence of a rational section is the Brauer obstruction. Although these projects seem quite different, they both involve the study of spaces of rational curves on varieties in a crucial way. In the first problem, one can prove that a Fano manifold is not unirational by proving there are "few" rational curves on the space of rational curves on the manifold. In the second problem, conjecturally, the condition is that the spaces of rational curves on the geometric generic fiber are themselves rationally connected. In complex algebraic geometry, the notion of a variety defined over a non-algebraically closed field is the common situation of a sequence of polynomial equations in variables x,y,z,... that depend on parameters s,t,u,... (which themselves may satisfy some polynomial equations). A basic question is whether for each choice of parameters there is a solution of the equation of the form: x,y,z,... are quotients of polynomials in the parameters s,t,u,... This is the problem of rational points. Another question, in the case where the polynomials don't depend on parameters, is whether x,y,z,... can be written as polynomials in a set of free variables, a,b,c,... such that every choice of a,b,c,... gives a solution of the polynomials, and essentially every solution of the polynomials arises in this way. This is the problem of unirationality. Both are old, difficult problems with applications in algebraic geometry and number theory. This project applies new results about spaces of rational curves to each of these projects.