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- W4288640492 abstract "Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $kappa (X, K_X + D)ge 0$, let $C$ be an irreducible curve on $X$ whose support is not contained in $D$ and $alpha$ a rational number in $ [ 0, 1 ]$. Following Miyaoka, we define an orbibundle $mathcal{E}_alpha$ as a suitable free subsheaf of log differentials on a Galois cover of $X$. Making use of $mathcal{E}_alpha$ we prove a Bogomolov-Miyaoka-Yau inequality for the couple $(X, D+alpha C)$. Suppose moreover that $K_X+D$ is big and nef and $(K_X+D)^2 $ is greater than $e_{Xsetminus D}$, namely the topological Euler number of the open surface $Xsetminus D$. As a consequence of the inequality, by varying $alpha$, we deduce a bound for $(K_X+D)cdot C)$ by an explicit function of the invariants: $(K_X+D)^2$, $e_{Xsetminus D}$ 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 $- e_{Csetminus D}$ bounded form a bounded family, in particular there are only a finite number of curves $C$ on $X$ such that $- e_{Csetminus D}le 0$." @default.
- W4288640492 created "2022-07-30" @default.
- W4288640492 creator A5080234571 @default.
- W4288640492 date "2019-01-08" @default.
- W4288640492 modified "2023-10-17" @default.
- W4288640492 title "An explicit bound for the log-canonical degree of curves on open surfaces" @default.
- W4288640492 doi "https://doi.org/10.48550/arxiv.1901.02541" @default.
- W4288640492 hasPublicationYear "2019" @default.
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