arXiv 1901.025412019-01math.AG

开曲面上曲线对数正则曲线的次数的显式界

An explicit bound for the log-canonical degree of curves on open surfaces

Pietro Sabatino

备注：: 24 pages, to appear in Publ. RIMS Kyoto Univ

摘要：

设$X$，$D$分别是$X$上的光滑射影曲面和简单法交因子.设$κ(X, K_X + D)\ge 0$，$C$是$X$上的不可约曲线，其支集不包含在$D$中，$α$是$ [ 0, 1 ]$中的有理数.在Miyaoka之后，我们定义了一个轨道丛$\mathcal{E}_α$作为$X$的Galois覆盖上的对数微分的一个合适的自由子层。利用$\mathcal{E}_α$证明了$(X, D+αC)$对的一个Bogomolov-Miyaoka-Yau不等式.此外，假设$K_X+D$是大的，并且nef和$(K_X+D)^2 $大于$e_{X\setminus D}$，即开放曲面$X\setminus D$的拓扑欧拉数。作为不等式的结果，通过改变$α$，我们推导出一个明确的功能$(K_X+D)\cdot C)$的不变量：$(K_X+D)^2$，$e_{X\setminus D}$和$e_{C \setminus D} $，即拓扑欧拉数的规范化$C$减去点的集合论反像的$D$。最后，我们证明了在这类曲面上，$- e_{C\setminus D}$有界的曲线构成一个有界曲线族，特别是在$X$上只有有限条曲线$C$使得$- e_{C\setminus D}\le 0$ .

英文摘要：

Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $κ(X, K_X + D)\ge 0$, let $C$ be an irreducible curve on $X$ whose support is not contained in $D$ and $α$ a rational number in $ [ 0, 1 ]$. Following Miyaoka, we define an orbibundle $\mathcal{E}_α$ as a suitable free subsheaf of log differentials on a Galois cover of $X$. Making use of $\mathcal{E}_α$ we prove a Bogomolov-Miyaoka-Yau inequality for the couple $(X, D+αC)$. Suppose moreover that $K_X+D$ is big and nef and $(K_X+D)^2 $ is greater than $e_{X\setminus D}$, namely the topological Euler number of the open surface $X\setminus D$. As a consequence of the inequality, by varying $α$, we deduce a bound for $(K_X+D)\cdot C)$ by an explicit function of the invariants: $(K_X+D)^2$, $e_{X\setminus D}$ and $e_{C \setminus D} $, namely the topological Euler number of the normalization of $C$ minus the points in the set theoretic counterimage of $D$. We finally deduce that on such surfaces curves with $- e_{C\setminus D}$ bounded form a bounded family, in particular there are only a finite number of curves $C$ on $X$ such that $- e_{C\setminus D}\le 0$.

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英文摘要：

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