南方科技大学知识苑(SUSTech KC): Ovoids of PG(3,q)

题名	Ovoids of PG(3,q)
其他题名	PG(3,q)中的卵形体
姓名	胡栓阳
姓名拼音	HU Shuanyang
学号	12232853
学位类型	硕士
学位专业	070101 基础数学
学科门类/专业学位类别	07 理学
导师	QING XIANG
导师单位	数学系
论文答辩日期	2024-05-21
论文提交日期	2024-06-22
学位授予单位	南方科技大学
学位授予地点	深圳
摘要	In this paper, we review some known examples of ovoids in 𝑃𝐺(3, 𝑞), consisting of the two known infinite families, elliptic quadrics and Tits ovoids. We introduce the characterization of ovoids in 𝑃𝐺(3, 𝑞) with 𝑞 odd, which was proved by Barlotti and independently by Panella. We survey the best known classification results by using the secant plane sections, which was proved by Brown. Brown proved that if every secant plane section of a given ovoid in 𝑃𝐺(3, 𝑞) with 𝑞 even is a conic, then the ovoid must be an elliptic quadric. The theorem is proved by applying the generalized quadrangles 𝑇2(𝒞) (𝒞 a conic) constructed by Tits, 𝑊(𝑞) and the isomorphism between them to show that every secant plane section of the ovoid admitting a conic secant plane section must be a conic. Such result follows from a well-known theorem of Barlotti. We introduce another version of classification problem of ovoids in 𝑃𝐺(3, 𝑞) with 𝑞 even. Due to the work of Segre and Thas, the problem can be transformed into classifying the ovoids of the generalized quadrangle 𝑊(𝑞) and we can apply some results about strongly regular graphs to the collinearity graph of 𝑊(𝑞). Finally, we discuss some properties of the collinearity graph and analyze the necessary conclusions of the method.
关键词	Ovoid Conic Secant Plane Section Generalized Quadrangle Elliptic Quadric
语种	英语
培养类别	独立培养
入学年份	2022
学位授予年份	2024-06
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[40] PENTTILA T, PRAEGER C E. Ovoids and translation ovals[J/OL]. J. London Math. Soc. (2),1997, 56(3): 607-624. https://doi.org/10.1112/S002461079800564X. [41] O’KEEFE C M, PENTTILA T. Ovoids with a pencil of translation ovals[J/OL]. Geom. Dedicata, 1996, 62(1): 19-34. https://doi.org/10.1007/BF00239999. [42] O’KEEFE C M, PENTTILA T. Ovals in translation hyperovals and ovoids[J/OL]. European J.Combin., 1997, 18(6): 667-683. https://doi.org/10.1006/eujc.1996.0127. [43] BROUWER A E, VAN MALDEGHEM H. Encyclopedia of Mathematics and its Applications:Vol. 182 Strongly regular graphs[M/OL]. Cambridge University Press, Cambridge, 2022:xvii+462. https://doi.org/10.1017/9781009057226.
所在学位评定分委会	数学
国内图书分类号	O157.3
来源库	人工提交
成果类型	学位论文
条目标识符	http://sustech.caswiz.com/handle/2SGJ60CL/765842
专题	南方科技大学理学院_数学系
推荐引用方式 GB/T 7714	Hu SY. Ovoids of PG(3,q)[D]. 深圳. 南方科技大学,2024.

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