论文标题
扭转点和同时在Del Pezzo表面上的特殊曲线
Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one
论文作者
论文摘要
Del Pezzo表面1 $ 1 $ S $ 1的抗态基点的爆炸导致了仅具有不可纤维的理性椭圆表面$ \ Mathscr {e} $。 $ \ Mathscr {e} $的最小高度的部分与$ S $上的$ 240 $ excial曲线相应。研究这些曲线的配置时会出现一个自然的问题：如果$ s $上的一个点包含在“许多”特殊曲线中，则它是$ \ mathscr {e} $上的纤维上的扭矩？在2005年，库瓦塔（Kuwata）证明了在del pezzo表面上的类似问题$ 2 $，那里有56个特殊曲线，如果“许多”等于$ 4 $或更多，则答案是肯定的。在本文中，我们证明，对于1度的Del Pezzo表面，如果“许多”等于$ 9 $或更多，则答案是肯定的。此外，我们给出反例，其中\ textsl {non} - torsion点位于$ 7 $ gract的曲线的交集。我们为8个相交的特殊曲线的仍然开放的情况给出了部分结果。
The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $S$. A natural question arises when studying the configuration of these curves: if a point on $S$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.
