Birational geometry and spaces of rational curves

这是一个复杂代数几何领域的研究项目，围绕以下两个问题：(i)证明存在一个非酉的Fano流形，（ii）在曲面的满射态射的几何一般纤维上寻找保证有理截面存在的唯一障碍是Brauer障碍的条件。虽然这些项目看起来很不同，但它们都以一种至关重要的方式涉及对变量上有理曲线空间的研究。在第一个问题中，可以通过证明在流形上的有理曲线空间上有“很少”的有理曲线来证明一个范诺流形不是酉的。在第二个问题中，假定几何型纤维上的有理曲线空间本身是有理连接的。在复杂代数几何中，在非代数闭域上定义的变量的概念是变量x，y,z，…的多项式方程序列的常见情况。它依赖于参数s t u…（它们本身可以满足一些多项式方程）。一个基本的问题是，对于每一个参数的选择，是否有如下形式的方程的解：x，y,z，…是参数s，t,u，…中多项式的商吗？这是有理点的问题。另一个问题，在多项式不依赖于参数的情况下，是x y z…可以写成一组自由变量的多项式，a,b,c，…使得a，b,c，…给出多项式的一个解，本质上每个多项式的解都是这样出现的。这就是非理性的问题。两者都是在代数几何和数论中应用的古老而困难的问题。本项目将有理曲线空间的新成果应用到这些项目中。

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.