A research on higher dimensional dual hyperovals in projective spaces

我们研究了高维对偶超卵圆，即构造了一些新的对偶超卵圆，并研究了它们的性质。推广了Yoshiara的构造，给出了PG （n, q）中q偶的对偶超卵圆的构造，并研究了n=2d+1.2时对偶超卵圆的同构类。（关于Buratti-Del Fra对偶超卵圆的研究。）(1)我们在$G （3d- 1,2）中实现了原构造于PG （d (d+3)/ 2,2）中的Buratti-Del Fra的对偶超卵圆。(2)我们也给出了Hybrechts的对偶超卵圆和Buratti-Del Fra的对偶超卵圆的表征。在$PG (d (d+3)/ 2,2)$上构造了一个新的对偶超卵圆族，并研究了它们的自同构群。本文研究了PG （2d, 2）中的d维对偶超卵圆。(1)从特征2的仿射平移平面构造对偶超卵圆，并证明了我们构造的一些对偶超卵圆与Yoshiara的对偶超卵圆不同构。(2)证明了如果近场不同构，则由特征2的近场构造的对偶超卵圆是不同构的。由此证明，对于某些d，在PG （2d, 2）中存在许多非同构对偶超卵圆。推广了具有q奇数的PG （n, q）上对偶超卵圆的定义，并给出了一些例子。

We investigate on higher dimensional dual hyperovals, that is, construct some new dual hyperovals and study their properties, as follows.1. We generalize the construction of Yoshiara e give a construction of dual hyperovals in PG (n, q) with q even, and study on the isomorphism classes in case n=2d+1.2. (Study on Buratti-Del Fra's dual hyperovals.)(1) We realize Buratti-Del Fra's dual hyperovals in $G (3d-1, 2), which are originally constructed in PG (d (d+3)/ 2, 2). (2) We also give a characterization of Hybrechts's dual hyperovals and Buratti-Del Fra's dual hyperovals.3. We constructed a new family of dual hyperovals in $PG (d (d+3)/2, 2)$, and study on the automorphism groups.4. We study on d-dimensional dual hyperovals in PG (2d, 2).(1) We construct dual hyperovals from affine translation planes of characteristic 2, and prove that some dual hyperovals constructed by us are not isomorphic to the Yoshiara's dual hyperovals.(2) We prove that dual hyperovals constructed from Near field of characteristic 2 (by the author) are not isomorphic if Near fields are not isomorphic. Hence we prove that, for some d, there are a lot of non-isomorphic dual hyperovals in PG (2d, 2).5. We try to generalize the definition of dual hyperovals in PG (n, q) with q odd, and give some examples.