arXiv 1607.012762016-07-05math.AGmath.NT

二次型的二次解

Quadratic solutions of quadratic forms

János Kollár

摘要：

我们研究齐次二次方程$q(x_0,\dots, x_n)=0$的解决方案，定义在一个字段$K$，其中$x_i$本身是齐次多项式的某种程度$d$ $r+1$变量。等价地，我们正在研究从射影$r$ -空间$P^r$到定义在域$K$上的二次超曲面PDMATHAMT的有理映射。如果$d$是偶数，则$P^1$到二次$Q$的映射空间对$Q$是稳定双有理的;如果$d$是奇数，则对$Q$中直线的正交格拉斯曼运动是稳定双有理的.大部分的文件是致力于获得类似的描述的空间参数化映射的$P^2$二次曲面，由2次多项式。最有趣的情况是当有5个不可约分量时的4维二次曲面。这些方法大多是经典的，涉及Veronese曲面，它的方程和投影。在实际情况下，这些结果提供了Kummer和Darboux开始的项目的最后几步，以描述所有包含至少2个通过每个点的圆的曲面。

英文摘要：

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational maps from projective $r$-space $P^r$ to a quadric hypersurface $Q$, defined over a field $K$. The space of maps of $P^1$ to a quadric $Q$ is stably birational to $Q$ if $d$ is even and to the orthogonal Grassmannian of lines in $Q$ if $d$ is odd. Most of the paper is devoted to obtaining similar descriptions for the spaces parametrizing maps of $P^2$ to quadrics, given by degree 2 polynomials. The most interesting case is 4-dimensional quadrics when there are 5 irreducible components. The methods are mostly classical, involving the Veronese surface, its equations and projections. In the real case, these results provide some of the last steps of a project, started by Kummer and Darboux, to describe all surfaces that contain at least 2 circles through every point.

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