Birational geometry and spaces of rational curves

DMS-0353692 Jason M.Starr这是复代数几何领域的一项研究项目，围绕以下两个问题：(I)证明存在非单调的Fano流形，(Ii)找到曲面上满射的几何普通纤维上的条件，保证有理截面存在的唯一障碍是Brauer障碍。虽然这两个项目看起来很不同，但它们都涉及到对簇上有理曲线空间的关键研究。在第一个问题中，通过证明Fano流形上的有理曲线空间上存在“少数”有理曲线，可以证明Fano流形不是单旋的。第二个问题的条件是几何一般纤维上的有理曲线空间本身是有理连通的。在复代数几何中，定义在非代数闭域上的簇的概念是变量x，y，z，…的多项式序列的常见情况。这取决于参数S，t，u，..。(它们本身可能满足一些多项式方程)。一个基本的问题是，对于每个参数的选择，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。这就是理性观点的问题。另一个问题是，在多项式不依赖于参数的情况下，x，y，z，…可以写成一组自由变量的多项式，a，b，c，...这样每一个选择a，b，c，..。给出了多项式的解，基本上每个多项式的解都是这样产生的。这就是单调的问题。这两个问题都是在代数几何和数论中有应用的古老而困难的问题。这个项目将关于有理曲线空间的新结果应用到这些项目中的每一个。

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.