论文标题
$ Q $ -BIC Hypersurfaces的几何形状
Geometry of $q$-bic Hypersurfaces
论文作者
论文摘要
传统的代数几何不变剂失去了积极特征的一些效力。例如，尽管任意高度，线条可能涵盖平滑的投射性超曲面。本论文的目的是定义一类表现出这种经典特性的超曲面，并提供一种概念化这种现象的观点。 具体而言，本文提出了同名$ q $ bic的高度弯曲面之间的类比 - 特殊的高度曲面$ q+1 $，并带有$ q $地面田地特征的任何力量，这是相应的Fermat HyperSurface给出的熟悉的示例 - 以及低度高度的Hypersurface，尤其是Quadrics和Cubics和Cubics。这种类比是通过混凝土结果证实的，例如：$ q $ - BIC Hypersurfaces是双线性形式的各向同性矢量的模量空间；平滑的$ Q $ bic超曲面中包含的线性空间的Fano方案是光滑的，不可还原的，并且携带类似于正交的Grassmannian的结构； $ q $ bic三倍的中间雅各布纯粹是不可分割的，这与阿尔巴尼亚的平滑狂热线条表面相关。
Traditional algebraic geometric invariants lose some of their potency in positive characteristic. For instance, smooth projective hypersurfaces may be covered by lines despite being of arbitrarily high degree. The purpose of this dissertation is to define a class of hypersurfaces that exhibits such classically unexpected properties, and to offer a perspective with which to conceptualize such phenomena. Specifically, this dissertation proposes an analogy between the eponymous $q$-bic hypersurfaces -- special hypersurfaces of degree $q+1$, with $q$ any power of the ground field characteristic, a familiar example given by the corresponding Fermat hypersurface -- and low degree hypersurfaces, especially quadrics and cubics. This analogy is substantiated by concrete results such as: $q$-bic hypersurfaces are moduli spaces of isotropic vectors for a bilinear form; the Fano schemes of linear spaces contained in a smooth $q$-bic hypersurface are smooth, irreducible, and carry structures similar to orthogonal Grassmannian; and the intermediate Jacobian of a $q$-bic threefold is purely inseparably isogenous to the Albanese variety of its smooth Fano surface of lines.
