三维有限射影空间中卵形面之特征及分类

This project studies ovoids in the projective three-dimensional space PG(3,q) coordinated by the finite field of order q. An ovoid of PG(3,q) (for q>2) is a set of q^2+1 points with no three on a line. The classical example of an ovoid is an elliptic quadric in PG(3,q). It is a set of q^2+1 points, with no three collinear, whose points satisfy a quadratic equation. All ovoid is elliptic quadrics when q is odd. When q is even,only one other type of ovoid is known, namely the Suzuki ovoids. No other types of ovoids have since been discovered, and has long been conjectured that every ovoid is either an elliptic quadric or a Suzuki ovoid. ..This is a famous long-standing problem which has received much international attention over the last 30 years. For q<32, the conjecture has been proven using characterisation theorems of ovoids by the ovals which can occur as plane sections. The classification of ovoids in PG(3, q) has so far proceeded a step behind the classification of ovals in the plane of the same order, so that all possible plane sections of an ovoid are known. ..This project involves innovative ways of studying characterisations of ovoids. We will use ovoidal fibrations and tangents of ovoid to look at the intersection properties of ovoids with each other, and so as to glean information about equivalence and structure. We will then concentrate our attention on the important special case where q is a square. There is a significant amount of structure, such as Baer subspaces, we can exploit in this case, making it feasible to solve the conjecture. The expected outcome is a deeper understanding of ovoids and fibrations, and significant progress towards the classification of ovoids.

在有限域上三维射影空间中，阶数q的卵形面是一个没有三点在同一直线上的基数q^2+1的点集。椭球面是卵形面的典型例子。当q是奇数时，已知所有卵形面是椭球面；当q是偶数时，铃木卵形面是现时唯一已知不是椭球面的卵形面，而q必为非平方数。..一个在过去30年一直受到了国际广泛关注的著名猜想为：所有卵形面皆是椭球面或铃木卵形面。当阶数是少于32的偶数时，猜想已被证实。现时已知卵形面的特征定理皆是用其自构群或截面描述，而已知的卵形面分类皆取决于射影平面的卵形线分类。..我们将用新方法来研究卵形面的特征和分类。我们计划透过研究射影空间的二阶曲线划分（又名纤维）及卵形面切线来寻找卵形面特征定理，而这些定理是由卵形面相交结构描述的。我们亦会尝试利用纤维及阶数为平方数的空间特性，去证明所有阶数为平方数的卵形面皆为椭球面。我们预期能更深入的了解卵形面及纤维，并在卵形面分类有显着进展。

在有限域GF(q)上的射影空间中的二次曲面是一个满足一条二次方程的点集。椭球面是二次曲面其中一个例子。椭球面有以下特性：没有三点在同一直线上，且基数为q^2+1。满足以上特性的点集被称为卵形面。在本项目研究中，我们使用新方法来研究卵形面的特征和分类。..本项目得到的重要结果的概述如下：(1) 我们研究了与曲面有特定相交的面集和点集，并对有相同组合相交特性的面集和点集进行分类。(2)通过直线几何和群组论，我们研究了卵形面纤维和双曲面纤维，并找出在五维空间、四维空间、及二维平面中与卵形面等阶的点集特性。这启发我们以不同维度射影空间的几何来为卵形面分类。(3)我们对曲面和贝尔子空间进行了研究，观察了与贝尔子空间息息相关的酉区组设计，考虑其关联图并以等周证明了一个酉区组设计的特征定理。此结果为分类阶为q^2的卵形面奠定了基础。