×

A remark on the intersection of plane curves. (English) Zbl 1443.14035

Kuchment, Peter (ed.) et al., Functional analysis and geometry. Selim Grigorievich Krein centennial. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 733, 109-128 (2019).
Summary: Let \(D\) be a very general curve of degree \(d=2\ell -\varepsilon\) in \(\mathbb{P}^2\), with \(\varepsilon\in\{0,1\}\). Let \(\Gamma\subset\mathbb{P}^2\) be an integral curve of geometric genus \(g\) and degree \(m,\Gamma\neq D\), and let \(\nu:C\to\Gamma\) be the normalization. Let \(\delta\) be the degree of the reduction modulo 2 of the divisor \(\nu^*(D)\) of \(C\) (see §2.1). In this paper we prove the inequality \(4g+\delta\geqslant m(d-8+2\varepsilon)+5\). We compare this with similar inequalities due to G. Xu [Am. J. Math. 118, No. 3, 611–620 (1996; Zbl 0872.14023); J. Algebr. Geom. 7, No. 1, 1–13 (1998; Zbl 0954.14023)] and X. Chen [Math. Res. Lett. 7, No. 5–6, 631–641 (2000; Zbl 0990.14019); Commun. Contemp. Math. 6, No. 4, 513–559 (2004; Zbl 1083.14052)]. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.
For the entire collection see [Zbl 1420.35010].

MSC:

14H50 Plane and space curves
14J70 Hypersurfaces and algebraic geometry
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
14H30 Coverings of curves, fundamental group
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Araujo, Carolina; Koll\'{a}r, J\'{a}nos, Rational curves on varieties. Higher dimensional varieties and rational points, Budapest, 2001, Bolyai Soc. Math. Stud. 12, 13-68 (2003), Springer, Berlin · Zbl 1080.14521 · doi:10.1007/978-3-662-05123-8\_3
[2] Autissier, Pascal; Chambert-Loir, Antoine; Gasbarri, Carlo, On the canonical degrees of curves in varieties of general type, Geom. Funct. Anal., 22, 5, 1051-1061 (2012) · Zbl 1276.14051 · doi:10.1007/s00039-012-0188-1
[3] Ballico, E., Algebraic hyperbolicity of generic high degree hypersurfaces, Arch. Math. (Basel), 63, 3, 282-283 (1994) · Zbl 0803.14023 · doi:10.1007/BF01189831
[4] Ballico, Edoardo, Non-existence of smooth rational curves of degree \(d=13,14,15\) contained in a general quintic hypersurface of \(\mathbb{P}^4\) and in some quadric hypersurface, Note Mat., 36, 2, 77-88 (2016) · Zbl 1387.14109
[5] Ballico, Edoardo; Fontanari, Claudio, Finiteness of rational curves of degree 12 on a general quintic threefold, Pure Appl. Math. Q., 11, 4, 537-557 (2015) · Zbl 1362.14039 · doi:10.4310/PAMQ.2015.v11.n4.a1
[6] Beheshti, Roya, Hypersurfaces with too many rational curves, Math. Ann., 360, 3-4, 753-768 (2014) · Zbl 1304.14065 · doi:10.1007/s00208-014-1024-8
[7] G. Berczi. Towards the Green-Griffiths-Lang Conjecture via equivariant localization. arXiv:1509.03406 (2015), 26p. · Zbl 1420.32013
[8] Bernardara, M., Calabi-Yau complete intersections with infinitely many lines, Rend. Semin. Mat. Univ. Politec. Torino, 66, 2, 87-97 (2008) · Zbl 1183.14055
[9] Bogomolov, F. A., Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR, 236, 5, 1041-1044 (1977)
[10] Bonavero, Laurent; H\`“{o}ring, Andreas, Counting conics in complete intersections, Acta Math. Vietnam., 35, 1, 23-30 (2010) · Zbl 1204.14025
[11] Brotbek, Damian, Hyperbolicity related problems for complete intersection varieties, Compos. Math., 150, 3, 369-395 (2014) · Zbl 1386.32025 · doi:10.1112/S0010437X13007458
[12] Brotbek, Damian, On the hyperbolicity of general hypersurfaces, Publ. Math. Inst. Hautes \'{E}tudes Sci., 126, 1-34 (2017) · Zbl 1458.32022 · doi:10.1007/s10240-017-0090-3
[13] Browning, Tim; Vishe, Pankaj, Rational curves on smooth hypersurfaces of low degree, Algebra Number Theory, 11, 7, 1657-1675 (2017) · Zbl 1442.14094 · doi:10.2140/ant.2017.11.1657
[14] J. Cao and A. Hoering. Rational curves on compact Kaehler manifolds. Avalaible at: http://math.unice.fr/ hoering/articles/a27-rat-curves.pdf
[15] Chang, M.-C.; Ran, Z., Divisors on some generic hypersurfaces, J. Differential Geom., 38, 3, 671-678 (1993) · Zbl 0807.14032
[16] Chen, Jungkai Alfred, On genera of smooth curves in higher-dimensional varieties, Proc. Amer. Math. Soc., 125, 8, 2221-2225 (1997) · Zbl 0883.14013 · doi:10.1090/S0002-9939-97-03908-7
[17] Chen, Xi, On the intersection of two plane curves, Math. Res. Lett., 7, 5-6, 631-641 (2000) · Zbl 0990.14019 · doi:10.4310/MRL.2000.v7.n5.a9
[18] Chen, Xi, On algebraic hyperbolicity of log varieties, Commun. Contemp. Math., 6, 4, 513-559 (2004) · Zbl 1083.14052 · doi:10.1142/S0219199704001422
[19] Chen, Xi; Zhu, Yi, \( \mathbb{A}^1\)-curves on affine complete intersections, Selecta Math. (N.S.), 24, 4, 3823-3834 (2018) · Zbl 1400.14018 · doi:10.1007/s00029-018-0421-3
[20] Chiantini, Luca; Ciliberto, Ciro, A few remarks on the lifting problem, Ast\'{e}risque, 218, 95-109 (1993) · Zbl 0813.14043
[21] Chiantini, Luca; Lopez, Angelo Felice, Focal loci of families and the genus of curves on surfaces, Proc. Amer. Math. Soc., 127, 12, 3451-3459 (1999) · Zbl 0929.14018 · doi:10.1090/S0002-9939-99-05407-6
[22] Chiantini, Luca; Lopez, Angelo Felice; Ran, Ziv, Subvarieties of generic hypersurfaces in any variety, Math. Proc. Cambridge Philos. Soc., 130, 2, 259-268 (2001) · Zbl 1068.14508 · doi:10.1017/S0305004100004904
[23] Ciliberto, Ciro; Flamini, Flaminio; Zaidenberg, Mikhail, Genera of curves on a very general surface in \(\mathbb{P}^3\), Int. Math. Res. Not. IMRN, 22, 12177-12205 (2015) · Zbl 1346.14081
[24] Ciliberto, C.; Flamini, F.; Zaidenberg, M., Gaps for geometric genera, Arch. Math. (Basel), 106, 6, 531-541 (2016) · Zbl 1344.14030 · doi:10.1007/s00013-016-0908-0
[25] Ciliberto, C.; Zaidenberg, M., Scrolls and hyperbolicity, Internat. J. Math., 24, 4, 1350026, 25 pp. (2013) · Zbl 1270.14027 · doi:10.1142/S0129167X13500262
[26] Clemens, Herbert, Curves on generic hypersurfaces, Ann. Sci. \'{E}cole Norm. Sup. (4), 19, 4, 629-636 (1986) · Zbl 0611.14024
[27] Clemens, Herbert, Curves on higher-dimensional complex projective manifolds. Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Berkeley, Calif., 1986, 634-640 (1987), Amer. Math. Soc., Providence, RI · Zbl 0682.14024
[28] Clemens, Herbert; Ran, Ziv, Erratum to: “Twisted genus bounds for subvarieties of generic hypersurfaces” [Amer. J. Math. {\bf126} (2004), no. 1, 89-120; MR2033565], Amer. J. Math., 127, 1, 241-242 (2005) · Zbl 1073.14542
[29] Coates, Tom; Givental, Alexander, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2), 165, 1, 15-53 (2007) · Zbl 1189.14063 · doi:10.4007/annals.2007.165.15
[30] Coskun, Izzet; Starr, Jason, Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. IMRN, 24, 4626-4641 (2009) · Zbl 1200.14051 · doi:10.1093/imrn/rnp102
[31] Coskun, Izzet; Riedl, Eric, Normal bundles of rational curves in projective space, Math. Z., 288, 3-4, 803-827 (2018) · Zbl 1391.14067 · doi:10.1007/s00209-017-1914-z
[32] Cotterill, Ethan, Rational curves of degree 10 on a general quintic threefold, Comm. Algebra, 33, 6, 1833-1872 (2005) · Zbl 1079.14048 · doi:10.1081/AGB-200063325
[33] Cotterill, Ethan, Rational curves of degree 11 on a general quintic 3-fold, Q. J. Math., 63, 3, 539-568 (2012) · Zbl 1260.14046 · doi:10.1093/qmath/har001
[34] Cotterill, Ethan, Rational curves of degree 16 on a general heptic fourfold, J. Pure Appl. Algebra, 218, 1, 121-129 (2014) · Zbl 1285.13034 · doi:10.1016/j.jpaa.2013.04.017
[35] Cox, David A.; Katz, Sheldon, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, xxii+469 pp. (1999), American Mathematical Society, Providence, RI · Zbl 0951.14026 · doi:10.1090/surv/068
[36] D’Almeida, Jean, Courbes rationnelles de degr\'{e} 11 sur une hypersurface quintique g\'{e}n\'{e}rale de \(\mathbf{P}^4\), Bull. Sci. Math., 136, 8, 899-903 (2012) · Zbl 1262.14030 · doi:10.1016/j.bulsci.2012.06.001
[37] O. Debarre. Rational curves on hypersurfaces. Lecture notes for the II Latin American School of Algebraic Geometry and Applications, 1-12 June 2015. Cabo Frio, Brazil. January 4, 2016, 48p.
[38] Demailly, Jean-Pierre, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math. 62, 285-360 (1997), Amer. Math. Soc., Providence, RI · Zbl 0919.32014 · doi:10.1090/pspum/062.2/1492539
[39] J. P. Demailly. Towards the Green-Griffiths-Lang Conjecture. Baklouti, Ali (ed.) et al., Analysis and geometry. MIMS-GGTM, Tunis, Tunisia, March 24-27, 2014. Proceedings of the international conference. In honour of Mohammed Salah Baouendi. Springer Proceedings in Mathematics and Statistics 127, 141-159 (2015). · Zbl 1327.14048
[40] J. P. Demailly. Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. arXiv:1801.04765 (2018), 76p. · Zbl 1436.32086
[41] Y. Deng. Effectivity in the hyperbolicity-related problems. arXiv:1606.03831 (2016), 13p.
[42] Y. Deng. On the Diverio-Trapani conjecture. arXiv:1703.07560 (2017), 31p. To appear in Ann. Sci. Ecole Norm. Sup. · Zbl 1447.32042
[43] Diverio, Simone; Rousseau, Erwan, Hyperbolicity of projective hypersurfaces, IMPA Monographs 5, xiv+89 pp. (2016), Springer, [Cham] · Zbl 1357.32001 · doi:10.1007/978-3-319-32315-2
[44] Ein, Lawrence, Subvarieties of generic complete intersections, Invent. Math., 94, 1, 163-169 (1988) · Zbl 0701.14002 · doi:10.1007/BF01394349
[45] Ein, Lawrence, Subvarieties of generic complete intersections. II, Math. Ann., 289, 3, 465-471 (1991) · Zbl 0746.14019 · doi:10.1007/BF01446583
[46] Eklund, David, Curves on Heisenberg invariant quartic surfaces in projective 3-space, Eur. J. Math., 4, 3, 931-952 (2018) · Zbl 1423.14223 · doi:10.1007/s40879-018-0216-2
[47] Ferrarese, Giorgio; Romagnoli, Daniela, Elliptic curves on the general hypersurface of degree \(7\) of \({\bf P}^4\), Nederl. Akad. Wetensch. Indag. Math., 50, 3, 249-252 (1988) · Zbl 0669.14008
[48] Furukawa, Katsuhisa, Rational curves on hypersurfaces, J. Reine Angew. Math., 665, 157-188 (2012) · Zbl 1246.14040 · doi:10.1515/CRELLE.2011.114
[49] K. Furukawa. Dimension of the space of conics on Fano hypersurfaces. arXiv:1702.08890 (2017), 14p.
[50] Green, Mark; Griffiths, Phillip, Two applications of algebraic geometry to entire holomorphic mappings. The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), 41-74 (1980), Springer, New York-Berlin · Zbl 0508.32010
[51] Hana, Gert Monstad; Johnsen, Trygve, Rational curves on a general heptic fourfold, Bull. Belg. Math. Soc. Simon Stevin, 16, 5, Linear systems and subschemes, 861-885 (2009) · Zbl 1183.14059
[52] Harris, Joe; Roth, Mike; Starr, Jason, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math., 571, 73-106 (2004) · Zbl 1052.14027 · doi:10.1515/crll.2004.045
[53] Harris, Joe; Starr, Jason, Rational curves on hypersurfaces of low degree. II, Compos. Math., 141, 1, 35-92 (2005) · Zbl 1083.14003 · doi:10.1112/S0010437X04001253
[54] Ikeda, Atsushi, Subvarieties of generic hypersurfaces in a nonsingular projective toric variety, Math. Z., 263, 4, 923-937 (2009) · Zbl 1243.14045 · doi:10.1007/s00209-008-0446-y
[55] Johnsen, Trygve; Kleiman, Steven L., Rational curves of degree at most \(9\) on a general quintic threefold, Comm. Algebra, 24, 8, 2721-2753 (1996) · Zbl 0860.14038 · doi:10.1080/02560049608542652
[56] Johnsen, Trygve; Kleiman, Steven L., Toward Clemens’ conjecture in degrees between \(10\) and \(24\), Serdica Math. J., 23, 2, 131-142 (1997) · Zbl 0942.14005
[57] Katz, Sheldon, On the finiteness of rational curves on quintic threefolds, Compositio Math., 60, 2, 151-162 (1986) · Zbl 0606.14039
[58] Knutsen, Andreas Leopold, On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds, Trans. Amer. Math. Soc., 364, 10, 5243-5264 (2012) · Zbl 1330.14014 · doi:10.1090/S0002-9947-2012-05461-4
[59] Knutsen, Andreas Leopold, Smooth, isolated curves in families of Calabi-Yau threefolds in homogeneous spaces, J. Korean Math. Soc., 50, 5, 1033-1050 (2013) · Zbl 1278.14054 · doi:10.4134/JKMS.2013.50.5.1033
[60] Kobayashi, Shoshichi, Hyperbolic manifolds and holomorphic mappings, xii+148 pp. (2005), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1084.32018 · doi:10.1142/5936
[61] Koll\'{a}r, J\'{a}nos, Rational curves on algebraic varieties, 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] 32, viii+320 pp. (1996), Springer-Verlag, Berlin · Zbl 0877.14012 · doi:10.1007/978-3-662-03276-3
[62] Kontsevich, Maxim, Enumeration of rational curves via torus actions. The moduli space of curves, Texel Island, 1994, Progr. Math. 129, 335-368 (1995), Birkh\`“{a}user Boston, Boston, MA · Zbl 0885.14028 · doi:10.1007/978-1-4612-4264-2\_12
[63] Lang, Serge, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.), 14, 2, 159-205 (1986) · Zbl 0602.14019 · doi:10.1090/S0273-0979-1986-15426-1
[64] Libgober, A.; Teitelbaum, J., Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations, Internat. Math. Res. Notices, 1, 29-39 (1993) · Zbl 0789.14005 · doi:10.1155/S1073792893000030
[65] Liu, Yuchen, Hyperbolicity of cyclic covers and complements, Trans. Amer. Math. Soc., 370, 8, 5341-5357 (2018) · Zbl 1394.32021 · doi:10.1090/tran/7097
[66] Liu, Yuchen, Construction of hyperbolic Horikawa surfaces, Ann. Inst. Fourier (Grenoble), 68, 2, 541-561 (2018) · Zbl 1435.32030
[67] Lu, Steven Shin-Yi; Miyaoka, Yoichi, Bounding codimension-one subvarieties and a general inequality between Chern numbers, Amer. J. Math., 119, 3, 487-502 (1997) · Zbl 0890.14028
[68] Lu, Steven Shin-Yi; Miyaoka, Yoichi, Bounding curves in algebraic surfaces by genus and Chern numbers, Math. Res. Lett., 2, 6, 663-676 (1995) · Zbl 0870.14020 · doi:10.4310/MRL.1995.v2.n6.a1
[69] Miyaoka, Yoichi, The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem, Publ. Res. Inst. Math. Sci., 44, 2, 403-417 (2008) · Zbl 1162.14026 · doi:10.2977/prims/1210167331
[70] Miyaoka, Yoichi, Counting lines and conics on a surface, Publ. Res. Inst. Math. Sci., 45, 3, 919-923 (2009) · Zbl 1185.14050 · doi:10.2977/prims/1249478969
[71] Nijsse, Pieter G. J., Clemens’ conjecture for octic and nonic curves, Indag. Math. (N.S.), 6, 2, 213-221 (1995) · Zbl 0910.14012 · doi:10.1016/0019-3577(95)91244-P
[72] Pacienza, Gianluca, Rational curves on general projective hypersurfaces, J. Algebraic Geom., 12, 2, 245-267 (2003) · Zbl 1054.14057 · doi:10.1090/S1056-3911-02-00328-4
[73] Pacienza, Gianluca, Subvarieties of general type on a general projective hypersurface, Trans. Amer. Math. Soc., 356, 7, 2649-2661 (2004) · Zbl 1056.14060 · doi:10.1090/S0002-9947-03-03250-1
[74] Riedl, Eric, Rational Curves on Hypersurfaces, 73 pp. (2015), ProQuest LLC, Ann Arbor, MI
[75] Roulleau, Xavier; Rousseau, Erwan, On the hyperbolicity of surfaces of general type with small \(c^2_1\), J. Lond. Math. Soc. (2), 87, 2, 453-477 (2013) · Zbl 1276.14053 · doi:10.1112/jlms/jds053
[76] Segre, B., The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford Ser., 14, 86-96 (1943) · Zbl 0063.06860 · doi:10.1093/qmath/os-14.1.86
[77] Shin, Dongsoo, Rational curves on general hypersurfaces of degree 7 in \(\mathbb{P}^5\), Osaka J. Math., 44, 1, 1-10 (2007) · Zbl 1120.14033
[78] Shin, Dongsoo, Conics on a general hypersurface in complex projective spaces, Bull. Korean Math. Soc., 50, 6, 2071-2077 (2013) · Zbl 1296.14025 · doi:10.4134/BKMS.2013.50.6.2071
[79] Siu, Yum-Tong, Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., 202, 3, 1069-1166 (2015) · Zbl 1333.32020 · doi:10.1007/s00222-015-0584-x
[80] B. Teissier. Resolution simultanee : I - Familles de courbes. Seminaire sur les singularites des surfaces (1976-1977), 1-10.
[81] Voisin, Claire, On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom., 44, 1, 200-213 (1996) · Zbl 0883.14022
[82] Voisin, Claire, A correction: “On a conjecture of Clemens on rational curves on hypersurfaces” [J. Differential Geom. {\bf44} (1996), no. 1, 200-213; MR1420353 (97j:14047)], J. Differential Geom., 49, 3, 601-611 (1998) · Zbl 0994.14026
[83] Voisin, Claire, On some problems of Kobayashi and Lang; algebraic approaches. Current Developments in Mathematics, 2003, 53-125 (2003), Int. Press, Somerville, MA · Zbl 1215.32014
[84] B. Wang. Genus of curves in generic hypersurfaces. arXiv:1110.0185v3 (2011), 14p.
[85] Wang, Bin, First order deformations of pairs and non-existence of rational curves, Rocky Mountain J. Math., 46, 2, 663-678 (2016) · Zbl 1356.14032 · doi:10.1216/RMJ-2016-46-2-663
[86] Wang, Lih-Chung, A remark on divisors of Calabi-Yau hypersurfaces, Asian J. Math., 4, 2, 369-372 (2000) · Zbl 0972.14032 · doi:10.4310/AJM.2000.v4.n2.a7
[87] Wang, Lih-Chung, Divisors of generic hypersurfaces of general type, Taiwanese J. Math., 6, 4, 507-513 (2002) · Zbl 1031.14023 · doi:10.11650/twjm/1500407474
[88] Xu, Geng, On the complement of a generic curve in the projective plane, Amer. J. Math., 118, 3, 611-620 (1996) · Zbl 0872.14023
[89] Xu, Geng, On the intersection of rational curves with cubic plane curves, J. Algebraic Geom., 7, 1, 1-13 (1998) · Zbl 0954.14023
[90] Xu, Geng, Subvarieties of general hypersurfaces in projective space, J. Differential Geom., 39, 1, 139-172 (1994) · Zbl 0823.14030
[91] Xu, Geng, Divisors on hypersurfaces, Math. Z., 219, 4, 581-589 (1995) · Zbl 0858.14023 · doi:10.1007/BF02572382
[92] Xu, Geng, Divisors on generic complete intersections in projective space, Trans. Amer. Math. Soc., 348, 7, 2725-2736 (1996) · Zbl 0871.14037 · doi:10.1090/S0002-9947-96-01613-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.