We give lower bounds for the slope of higher dimensional fibrations over curves under conditions of GIT-semistability of the fibres, using a generalization of a method of Cornalba and Harris. With the same method we establish a sharp lower bound for the slope of trigonal fibrations of even genus and general Maroni invariant; this result in particular proves a conjecture due to Harris and Stankova-Frenkel.
@article{ASNSP_2009_5_8_4_647_0,
author = {Barja, Miguel \'Angel and Stoppino, Lidia},
title = {Slopes of trigonal fibred surfaces and of higher dimensional fibrations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {4},
year = {2009},
pages = {647-658},
zbl = {1204.14017},
mrnumber = {2647907},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2009_5_8_4_647_0}
}
Barja, Miguel Ángel; Stoppino, Lidia. Slopes of trigonal fibred surfaces and of higher dimensional fibrations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 4, pp. 647-658. http://www.numdam.org/item/ASNSP_2009_5_8_4_647_0/
[1] and , Global and local properties of pencils of algebraic curves, In: “Algebraic Geometry 2000”, Azumino (Hotaka), Adv. Stud. Pure Math., Vol. 36, Math. Soc. Japan, Tokyo, 2002, 1–49. | MR 1971511 | Zbl 1088.14010
[2] , “On the Slope and Geography of Fibred Surfaces and Threefolds”, Ph. D. Thesis, Univesity of Barcelona, 1998.
[3] , On the slope of fibred threefolds, Internat. J. Math. 11 (2000), 461–491. | MR 1768169 | Zbl 1100.14519
[4] and , Linear stability of projected canonical curves with applications to the slope of fibred surfaces, J. Math. Soc. Japan 60 (2008), 171–192. | MR 2392007 | Zbl 1135.14016
[5] and , On the slope of fibred surfaces, Nagoya Math. J. 164 (2001), 103–131. | MR 1869097 | Zbl 1075.14502
[6] , Numerical bounds of canonical varieties, Osaka J. Math. (3) 37 (2000), 701–718. | MR 1789444 | Zbl 1077.14540
[7] and , Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. École Norm. Sup. (4) 21 (1988), 455–475. | Numdam | MR 974412 | Zbl 0674.14006
[8] , “Intersection Theory”, second edition, Springer-Verlag, 1998. | MR 1644323
[9] and , On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton, Invent. Math. (1) 67 (1982), 23–88. | MR 664324 | Zbl 0506.14016
[10] , “Algebraic Geometry”, GTM, Vol. 52, Springer-Verlag, New York-Heidelberg, 1977. | MR 463157 | Zbl 0367.14001
[11] , On deformations of quintic surfaces, Invent. Math. 31 (1975), 43–85. | MR 1573789 | Zbl 0317.14018
[12] , and , Introduction to the minimal model problem. In: “Algebraic Geometry”, Sendai, 1985, Adv. Stud. Pure Math. 10, North Holland, Amsterdam, (1987), 283–360. | MR 946243
[13] , Instability in invariant theory, Ann. of Math. 108 (1978), 299–316. | MR 506989 | Zbl 0406.14031
[14] and , Log canonical thresholds of semistable plane curves, Math. Proc. Cambridge Philos. Soc. (2) 137 (2004), 273–280. | MR 2090618 | Zbl 1055.14032
[15] , Clifford index and the slope of fibered surfaces, J. Algebraic Geom. (2) 8 (1999), 207–220. | MR 1675150 | Zbl 0979.14004
[16] , A lower bound of the slope of trigonal fibrations, Internat. J. Math. (1) 7 (1996), 19–27. | MR 1369903 | Zbl 0880.14009
[17] , Chow stability criterion in terms of log canonical threshold, J. Korean Math. Soc. (2) 45 (2008), 467–477. | MR 2389549 | Zbl 1144.14300
[18] , Le serie lineari speciali sulle curve trigonali, Ann. Mat. Pura Appl. (4) 25 (1946), 341–354. | MR 24182 | Zbl 0061.35407
[19] and , Line bundles and syzygies of trigonal curves, Abh. Math. Sem. Univ. Hamburg 56 (1986), 169–189. | MR 882414 | Zbl 0628.14029
[20] , Projective stability of ruled surfaces, Invent. Math. (3) 56 (1980), 269–304. | MR 561975 | Zbl 0423.14005
[21] , Stability of projective varieties, L’Enseignement Math. (2) 23 (1977), 39–110. | MR 450272 | Zbl 0363.14003
[22] , Some inequalities for minimal fibrations of surfaces of general type over curves, J. Math. Soc. Japan (4) 44 (1992), 643–666. | MR 1180441 | Zbl 0783.14021
[23] , Chapters on algebraic surfaces, In: “Complex Algebraic Geometry” (Park City, UT, 1993), Amer. Math. Soc., Providence, RI, Vol. 3, 1997, 3–159. | MR 1442522 | Zbl 0910.14016
[24] , On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157–175. | MR 337983 | Zbl 0315.14010
[25] , Moduli of trigonal curves, J. Algebraic Geom. (4) 9 (2000), 607–662. | MR 1775309 | Zbl 1001.14007
[26] , Slope inequalities for fibered surfaces via GIT, Osaka Math. J. 45 (2008), 1027–1041. | MR 2493968 | Zbl 1160.14013
[27] , On the invariants of base changes of pencils of curves I, Manuscripta Math. 84 (1994), 225–244. | MR 1291119 | Zbl 0835.14012
[28] , On the invariants of base changes of pencils of curves II, Math. Z. 222 (1996), 655–676. | MR 1406272 | Zbl 0864.14016
[29] , On the slopes of the moduli spaces of curves, Internat. J. Math. 9 (1998), 119–127. | MR 1612259 | Zbl 0930.14017
[30] , The -energy on hypersurfaces and stability, Comm. Anal. Geom. (2) 2 (1994), 239-265. | MR 1312688 | Zbl 0846.32019
[31] , “Quasi-projective Moduli for Polarized Manifolds”, Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 30, 1995. | MR 1368632 | Zbl 0844.14004
[32] , Fibred algebraic surfaces with low slope, Math. Ann. 276 (1987), 449–466. | MR 875340 | Zbl 0596.14028
