论文标题
Néron-Severi组方案的扭转亚组的界限
Bounds on the Torsion Subgroups of Néron-Severi Group Schemes
论文作者
论文摘要
令$ x \ hookrightarrow \ mathbb {p}^r $为平滑的投射变体，该变体是由一个均质的多项式定义的，$ \ leq d $在代数封闭的字段上。令$ \ mathbf {pic} \，x $为$ x $的PICARD方案。令$ \ mathbf {pic}^0 x $为$ \ mathbf {pic} \，x $的身份组件。 $ x $的néron-Severi组方案由$ \ MathBf {ns} \，x =（\ MathBf {pic} \，x）/（\ Mathbf {pic}^0 x）_ {\ Mathrm {pic}^0 x）我们在有限组方案$（\ Mathbf {ns} \，x）_ {\ Mathrm {tor}} $中，以$ d $和$ r $ $表示，给出了一个明确的上限。作为推论，我们在有限组$π^1 _ {\ mathrm {et}}}（x，x_0）^{\ Mathrm {ab}} _ {\ Mathrm {tor}} $的顺序上给出了上限。我们还表明，$ x $的Néron--SeveriGroup的扭转子组$（\ Mathrm {ns} \，x）_ {\ Mathrm {tor}} $由$ x $组成的组小于或等于$（\ Mathrm {deg}} \，x -1）（\ Mathrm Mathrm eylets by by by by by by by by by by by by by by by by by by by by by Mathrm {\ mathrm {deg}
Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$. Let $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The Néron--Severi group scheme of $X$ is defined by $\mathbf{NS}\, X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$. We give an explicit upper bound on the order of the finite group scheme $(\mathbf{NS}\, X)_{\mathrm{tor}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the finite group $π^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that the torsion subgroup $(\mathrm{NS}\, X)_{\mathrm{tor}}$ of the Néron--Severi group of $X$ is generated by less than or equal to $(\mathrm{deg}\, X -1)(\mathrm{deg}\, X - 2)$ elements in various situations.
