Quantitative geometric and arithmetic results on projective surfaces

Liao, Hungzen

doi:10.1090/proc/13401

Quantitative geometric and arithmetic results on projective surfaces
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Proc. Amer. Math. Soc. 145 (2017), 2495-2504 Request permission

Abstract:

In this paper, we improve Ru’s defect relation and the height inequality in the case when $X$ is a normal projective surface and $D_j$, $1 \leq j \leq q$, are big and asymptotic free divisors without irreducible common components on $X$. As a consequence, we derive a sharp result in the qualitative statement.

References

Pascal Autissier, Géométries, points entiers et courbes entières, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 2, 221–239 (French, with English and French summaries). MR 2518077, DOI 10.24033/asens.2094
H. Cartan, Sur les zéros des combinaisions linéarires de $p$ fonctions holomorpes données, Mathematica (Cluj) 7 (1933), 5–29.
P. Corvaja and U. Zannier, On integral points on surfaces, Ann. of Math. (2) 160 (2004), no. 2, 705–726. MR 2123936, DOI 10.4007/annals.2004.160.705
Pietro Corvaja and Umberto Zannier, On a general Thue’s equation, Amer. J. Math. 126 (2004), no. 5, 1033–1055. MR 2089081
Jan-Hendrik Evertse and Roberto G. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Not. 25 (2002), 1295–1330. MR 1903776, DOI 10.1155/S107379280210804X
Jan-Hendrik Evertse and Roberto G. Ferretti, A generalization of the Subspace Theorem with polynomials of higher degree, Diophantine approximation, Dev. Math., vol. 16, SpringerWienNewYork, Vienna, 2008, pp. 175–198. MR 2487693, DOI 10.1007/978-3-211-74280-8_{9}
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
Aaron Levin, Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170 (2009), no. 2, 609–655. MR 2552103, DOI 10.4007/annals.2009.170.609
Aaron Levin, On the Schmidt subspace theorem for algebraic points, Duke Math. J. 163 (2014), no. 15, 2841–2885. MR 3285859, DOI 10.1215/00127094-2827017
Hungzen Liao and Min Ru, A note on the Second Main Theorem for holomorphic curves into algebraic varieties, Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 4, 671–684. MR 3309947
Min Ru, On a general form of the second main theorem, Trans. Amer. Math. Soc. 349 (1997), no. 12, 5093–5105. MR 1407711, DOI 10.1090/S0002-9947-97-01913-2
Min Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), no. 1, 215–226. MR 2033568
Min Ru, Holomorphic curves into algebraic varieties, Ann. of Math. (2) 169 (2009), no. 1, 255–267. MR 2480605, DOI 10.4007/annals.2009.169.255
M. Ru, A defect relation for holomorphic curves intersecting general divisors in projective varieties, J. of Geometric Analysis, first online: 15 October 2015.
M. Ru, A general Diophantine inequality, Functiones et Approximatio, to appear.
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Paul Vojta, On Cartan’s theorem and Cartan’s conjecture, Amer. J. Math. 119 (1997), no. 1, 1–17. MR 1428056
P. Vojta, Diophantine approximation and Nevanlinna theory, CIME notes, page 231, 2007.