论文标题
与一般的超曲面和应用高度接触的线锥
Cones of lines having high contact with general hypersurfaces and applications
论文作者
论文摘要
给定一个光滑的超出表面$ x \ subset \ mathbb {p}^{n+1} $ of度量$ d \ geqslant 2 $，我们研究锥体$ v^h_p \ subset \ subset \ mathbb {p}^{n+1} $ swept of table of confacts unccters of constance $ h \ geqslant $ p $ p $ p $ p $ p $ p point $ p point $ p point $ p $ p point $ p point $ p。特别是，我们证明，如果$ x $是一般的，那么对于任何$ p \ in x $和$ 2 \ leqslant h \ leqslant \ min \ min \ {n+1，d \} $，圆锥$ v^h_p $的尺寸正好$ n+2-h $。此外，当$ x $是$ d \ geqslant 2n+2 $的非常普遍的高度表面时，我们描述了锥体$ v^h_p $与$ k $的非理性程度之间的关系 - $ x $的尺寸subvarieties $ x $的$ x $。作为一个应用程序，我们以$ k $的最小非理性性给出了一些界限 - $ x $的尺寸亚属性通过$ x $的一般点，我们证明了$ x $的连接gonations of $ x $ $ d- \ left \ lfloor \ frac {\ sqrt {16n+25} -3} -3} {2} \ right \ rfloor \ rfloor \ leqslant \ conngon（x）\ leqslant d- \ left \ lfloor \ frac {\ sqrt {8n+1} +1} {2} {2} \ right \ rfloor $。
Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant 2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we prove that if $X$ is general, then for any $p\in X$ and $2 \leqslant h\leqslant \min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when $X$ is a very general hypersurface of degree $d\geqslant 2n+2$, we describe the relation between the cones $V^h_p$ and the degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$. As an application, we give some bounds on the least degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$, and we prove that the connecting gonality of $X$ satisfies $d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$.
