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On finiteness of curves with high canonical degree on a surface. (English) Zbl 1360.14019
By exploiting a result of Y. Miyaoka [Publ. Res. Inst. Math. Sci. 44, No. 2, 403–417 (2008; Zbl 1162.14026)] this paper obtains a simple explicit bound on the canonical degree \(k_C=K_X.C\) of a curve \(C\neq{\mathbb P}^1\) of negative self-intersection on a smooth complex projective surface \(X\) of non-negative Kodaira dimension, in terms of the geometric genus \(g\) of \(C\) and the invariants of \(X\). In fact the only invariant needed is \(a=3c_2-K_X^2\). Then \[ k_C\leq 3(g-1)+\frac34 a+\frac14\sqrt{9a^2+24a(g-1)}. \] Moreover, for \(X\) of general type, there are finitely many curves with \(k_C\geq 3+\epsilon(g-1)\geq 0\). This (together with some refinements omitted here for brevity) yields several corollaries. One is that a Shimura surface contains only finitely many Shimura curves: this is known, but the proof here is more economical both mathematically and in workforce terms than the one in [T. Bauer et al., Duke Math. J. 162, No. 10, 1877–1894 (2013; Zbl 1272.14009)]. That paper is largely concerned with questions of bounded negativity (crudely, \(C^2\) should be bounded below) and leads the present authors to rephrase their results so as to obtain partial results of that type. In particular they address a conjecture of Nagata that there are no negative curves apart from \(-1\)-curves on \({\mathbb P}^2\) blown up in \( n\geq 10\) general points, using an extension of their result to some cases of negative Kodaira dimension, and a conjecture of Vojta that \(k_C\leq (4+\epsilon)(g-1)+B(\epsilon)\). In both cases they get interesting results, informative but weaker than what is conjectured, using relatively direct methods.
MSC:
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14G35 Modular and Shimura varieties
14J29 Surfaces of general type
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