On surfaces of general type with $q=5$

Lopes, Margarida Mendes; Pardini, Rita; Pirola, Gian Pietro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 999-1007.

We prove that a complex surface $S$ with irregularity $q\left(S\right)=5$ that has no irrational pencil of genus $>1$ has geometric genus $p_g\left(S\right)\ge 8$. As a consequence, we are able to classify minimal surfaces $S$ of general type with $q\left(S\right)=5$ and $p_g\left(S\right)<8$. 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 $q=3,4$).

Classification : 14J29

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

[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 | Zbl

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

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

[5] A. Beauville, L’inégalité $p_g\ge 2q-4$ pour les surfaces de type général, appendix to [10]. | Numdam | Zbl

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

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

[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 | Zbl

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

[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. | EuDML | Numdam | MR | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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