On surfaces of general type with q=5
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, p. 999-1007

We prove that a complex surface S with irregularity q(S)=5 that has no irrational pencil of genus >1 has geometric genus p g (S)8. As a consequence, we are able to classify minimal surfaces S of general type with q(S)=5 and p g (S)<8. This result is a negative answer, for q=5, to the question asked in [13] of the existence of surfaces of general type with irregularity q that have no irrational pencil of genus >1 and with the lowest possible geometric genus p g =2q-3 (examples are known to exist only for q=3,4).

Published online : 2018-06-21
Classification:  14J29
@article{ASNSP_2012_5_11_4_999_0,
     author = {Lopes, Margarida Mendes and Pardini, Rita and Pirola, Gian Pietro},
     title = {On surfaces of general type with $q=5$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     pages = {999-1007},
     zbl = {1272.14030},
     mrnumber = {3060707},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_999_0}
}
Lopes, Margarida Mendes; Pardini, Rita; Pirola, Gian Pietro. On surfaces of general type with $q=5$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 999-1007. http://www.numdam.org/item/ASNSP_2012_5_11_4_999_0/

[1] M. A. Barja, Numerical bounds of canonical varieties, Osaka J. Math. 37 (2000), 701–718. | MR 1789444 | Zbl 1077.14540

[2] M. A. Barja, J. C. Naranjo and G. P. Pirola, On the topological index of irregular surfaces, J. Algebraic Geom. 16 (2007), 435–458. | MR 2306275 | Zbl 1127.14020

[3] W. Barth, C. Peters and A. Van de Ven, “Compact Complex Surfaces”, Ergebnisse der Mathematik, 3. Folge, Band 4, Springer, Berlin, 1984. | MR 749574 | Zbl 1036.14016

[4] A. Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), 121–140. | MR 553705 | Zbl 0403.14006

[5] A. Beauville, L’inégalité p g 2q-4 pour les surfaces de type général, appendix to [10]. | Numdam | Zbl 0543.14026

[6] A. Beauville, “Complex Algebraic Surfaces”, second edition, L.M.S Student Texts 34, Cambridge University Press, Cambridge, 1996. | MR 1406314 | Zbl 0849.14014

[7] W. Bruns and J. Herzog, “Cohen-Macaulay Rings”, revised edition, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1998. | MR 1251956 | Zbl 0909.13005

[8] F. Catanese, C. Ciliberto and M. Mendes Lopes, On the classification of irregular surfaces of general type with non birational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), 275–308. | MR 1422597 | Zbl 0889.14019

[9] A. Causin and G. P. Pirola, Hermitian matrices and cohomology of Kaehler varieties, Manuscripta Math. 121 (2006), 157–168. | MR 2264019 | Zbl 1107.32006

[10] O. Debarre, Inégalités numériques pour les surfaces de type général, with an appendix by A. Beauville, Bull. Soc. Math. France 110 (1982), 319–346. | Numdam | MR 688038 | Zbl 0543.14026

[11] C. D. Hacon and R. Pardini, Surfaces with p g =q=3, Trans. Amer. Math. Soc. 354 (2002), 2631–2638. | MR 1895196 | Zbl 1009.14004

[12] R. Lazarsfeld and M. Popa, Derivative complex BGG correspondence and numerical inequalities for compact Khaeler manifolds, Invent. Math. 182 (2010), 605–633. | MR 2737707 | Zbl 1209.32011

[13] M. Mendes Lopes and R. Pardini, On surfaces with p g =2q-3, Adv. Geom. 10 (2010), 549–555. | MR 2660426 | Zbl 1200.14073

[14] M. Mendes Lopes and R. Pardini, The geography of irregular surfaces, In: “Current Developments in Algebraic Geometry”, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press 59 (2012), 349–378. | MR 2931875 | Zbl 1255.14028

[15] M. Mendes Lopes and R. Pardini, Severi type inequalities for surfaces with ample canonical class, Comment. Math. Helv. 86 (2011), 401–414. | MR 2775134 | Zbl 1210.14040

[16] S. Mukai, “Curves and Grassmannians”, Algebraic geometry and related topics (Inchon, 1992), 19–40, Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993. | MR 1285374 | Zbl 0846.14030

[17] G. Pareschi and M. Popa, Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality, Duke Math. J. 150 (2009), 269–28. | MR 2569614 | Zbl 1206.14067

[18] G. P. Pirola, Algebraic surfaces with p g =q=3 and no irrational pencils, Manuscripta Math. 108 (2002), 163–170. | MR 1918584 | Zbl 0997.14009

[19] C. Schoen, A family of surfaces constructed from genus 2 curves, Internat. J. Math. 18 (2007), 585–612. | MR 2331080 | Zbl 1118.14042

[20] G. Xiao, Irregularity of surfaces with a linear pencil, Duke Math. J. 55 (1987), 596–602. | MR 904942 | Zbl 0651.14021