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- W2909414309 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.
- W2909414309 created "2019-01-25" @default.
- W2909414309 creator A5080234571 @default.
- W2909414309 date "2022-11-03" @default.
- W2909414309 modified "2023-09-24" @default.
- W2909414309 title "An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces" @default.
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- W2909414309 doi "https://doi.org/10.4171/prims/58-4-6" @default.
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