An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces
Pietro Sabatino
Università di Roma Tor Vergata, Italy
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Abstract
Let , be a smooth projective surface and a simple normal crossing divisor on , respectively. Suppose , let be an irreducible curve on whose support is not contained in and a rational number in . Following Miyaoka, we define an orbibundle as a suitable free subsheaf of log differentials on a Galois cover of . Making use of we prove a Bogomolov–Miyaoka–Yau inequality for the couple . Suppose moreover that is big and nef and is greater than , namely the topological Euler number of the open surface . As a consequence of the inequality, by varying , we deduce a bound for by an explicit function of the invariants: , and , namely the topological Euler number of the normalization of minus the points in the set-theoretic counterimage of . We finally deduce that on such surfaces curves, with bounded, form a bounded family, in particular there are only a finite number of curves on such that .
Cite this article
Pietro Sabatino, An Explicit Bound for the Log-Canonical Degree of Curves on Open Surfaces. Publ. Res. Inst. Math. Sci. 58 (2022), no. 4, pp. 817–853
DOI 10.4171/PRIMS/58-4-6