Geodesic
Upper bounds on the genus of hyperelliptic Albanese fibrations
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e84

Voir la notice de l'article provenant de la source Cambridge University Press

Let S be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus g. We prove a quadratic upper bound on the genus g (i.e., $g\leq h\big (\chi (\mathcal {O}_S)\big )$, where h is a quadratic function). We also construct examples showing that the quadratic upper bounds cannot be improved to linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$. 
Ling, Songbo; Lü, Xin. Upper bounds on the genus of hyperelliptic Albanese fibrations. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e84. doi: 10.1017/fms.2025.43
@article{10_1017_fms_2025_43,
     author = {Ling, Songbo and L\"u, Xin},
     title = {Upper bounds on the genus of hyperelliptic {Albanese} fibrations},
     journal = {Forum of Mathematics, Sigma},
     pages = {e84},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.43},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.43/}
}
TY  - JOUR
AU  - Ling, Songbo
AU  - Lü, Xin
TI  - Upper bounds on the genus of hyperelliptic Albanese fibrations
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e84
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.43/
DO  - 10.1017/fms.2025.43
ID  - 10_1017_fms_2025_43
ER  - 
%0 Journal Article
%A Ling, Songbo
%A Lü, Xin
%T Upper bounds on the genus of hyperelliptic Albanese fibrations
%J Forum of Mathematics, Sigma
%D 2025
%P e84
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.43/
%R 10.1017/fms.2025.43
%F 10_1017_fms_2025_43

[1] Barth, W. P., Hulek, K., Peters, C. A. M. and Van De Ven, A., Compact Complex Surfaces, vol. 4, second edn. (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]) (Springer-Verlag, Berlin, 200). Google Scholar

[2] Catanese, F., ‘Fibred surfaces, varieties isogenous to a product and related moduli spaces’, Amer. J. Math. 122(1) (2000), 1–44. Google Scholar

[3] Catanese, F. and Ciliberto, C., ‘Surfaces with ’, in Problems in the Theory of Surfaces and Their Classification (Cortona, 1988) (Sympos. Math., XXXII) (Academic Press, London, 1991), 49–79. Google Scholar

[4] Catanese, F., Ciliberto, C. and Mendes Lopes, M., ‘On the classification of irregular surfaces of general type with nonbirational bicanonical map’, Trans. Amer. Math. Soc. 350(1) (1998), 275–308. Google Scholar

[5] Cornalba, M. and Harris, J., ‘Divisor classes associated to families of stable varieties, with applications to the moduli space of curves’, Ann. Sci. École Norm. Sup. (4) 21(3) (1988), 455–475. Google Scholar

[6] Cartwright, D. I., Koziarz, V. and Yeung, S.-K., ‘On the Cartwright-Steger surface’, J. Algebraic Geom. 26(4) (2017), 655–689. Google Scholar

[7] Debarre, O., ‘Inégalités numériques pour les surfaces de type général’, Bull. Soc. Math. France 110(3) (1982), 319–346. With an appendix by A. Beauville. Google Scholar

[8] Frapporti, D. and Pignatelli, R., ‘Mixed quasi-étale quotients with arbitrary singularities’, Glasg. Math. J. 57(1) (2015), 143–165. Google Scholar

[9] Gong, C., Guo, Z. and Lü, X., ‘A note on singularity indices of hyperelliptic fibrations’, Preprint, 2025. Google Scholar

[10] Hartshorne, R., Algebraic Geometry (Graduate Texts in Mathematics) no. 52 (Springer-Verlag, New York-Heidelberg, 1977). Google Scholar

[11] Hacon, C. D. and Pardini, R., ‘Surfaces with ’, Trans. Amer. Math. Soc. 354(7) (2002), 2631–2638. Google Scholar

[12] Horikawa, E., ‘On algebraic surfaces with pencils of curves of genus ’, in Complex Analysis and Algebraic Geometry (Iwanami Shoten Publishers, Tokyo, 1977), 79–90. Google Scholar

[13] Ishida, H., ‘Bounds for the relative Euler-Poincaré characteristic of certain hyperelliptic fibrations’, Manuscripta Math. 118(4) (2005), 467–483. Google Scholar

[14] Konno, K., ‘Even canonical surfaces with small . III’, Nagoya Math. J. 143 (1996), 1–11. Google Scholar

[15] Ling, S. and Lü, X., ‘Albanese fibrations of surfaces with low slope’, Math. Ann. 391 (2025), 4641–4671. Google Scholar

[16] Liu, X. and Tan, S., ‘Families of hyperelliptic curves with maximal slopes’, Sci. China Math. 56(9) (2013), 1743–1750. Google Scholar

[17] Lu, X. and Zuo, K., On the slope of hyperelliptic fibrations with positive relative irregularity. Trans. Amer. Math. Soc. 369(2) (2017), 909–934. Google Scholar

[18] Miyaoka, Y., ‘The maximal number of quotient singularities on surfaces with given numerical invariants’, Math. Ann. 268(2) (1984), 159–171. Google Scholar

[19] Penegini, M., ‘The classification of isotrivially fibred surfaces with ’, Collect. Math. 62(3) (2011), 239–274. With an appendix by Sönke Rollenske. Google Scholar

[20] Pirola, G. P., ‘Surfaces with ’, Manuscripta Math. 108(2) (2002), 163–170. Google Scholar

[21] Penegini, M. and Polizzi, F., ‘A note on surfaces with and an irrational fibration’, Adv. Geom. 17(1) (2017), 61–73. Google Scholar

[22] Reider, I., ‘Vector bundles of rank and linear systems on algebraic surfaces’, Ann. of Math . (2) 127(2) (1988), 309–316. Google Scholar

[23] Xiao, G., ‘Fibered algebraic surfaces with low slope’, Math. Ann. 276(3) (1987), 449–466. Google Scholar

[24] Xiao, G., ‘Irregularity of surfaces with a linear pencil’, Duke Math. J. 55(3) (1987), 597–602. Google Scholar

[25] Xiao, G., ‘ of elliptic and hyperelliptic surfaces’, Internat. J. Math. 2(5) (1991), 599–615. Google Scholar

[26] Xiao, G., The Fibrations of Algebraic Aurfaces (Chinese) (Shanghai Scientific and Technical Publishers, 1992). Google Scholar

[27] Zucconi, F., ‘Surfaces with and an irrational pencil’, Canad. J. Math. 55(3) (2003), 649–672. Google Scholar

Cité par Sources :