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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.
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14G35 Modular and Shimura varieties
14J29 Surfaces of general type
Full Text: DOI arXiv
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