American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras
  • ERA Home
  • This Issue
  • Next Article
    Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
doi: 10.3934/era.2020117

On projective threefolds of general type with small positive geometric genus

Meng Chen 1, , Yong Hu 2,3, and  Matteo Penegini 2,3,,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2. 

School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
3. 

Università degli Studi di Genova, DIMA Dipartimento di Matematica, I-16146 Genova, Italy

* Corresponding author: Matteo Penegini

Received  April 2020 Revised  October 2020 Published  November 2020

Fund Project: The first author was supported by National Natural Science Foundation of China (#12071078, #11731004) and Program of Shanghai Subject Chief Scientist (#16XD1400400). The second author is supported by a KIAS Individual Grant (MP062501) at Korea Institute for Advanced Study. The third author was partially supported by PRIN 2015 "Geometry of Algebraic Varieties" and by GNSAGA of INdAM

In this paper we study the pluricanonical maps of minimal projective 3-folds of general type with geometric genus $ 1 $, $ 2 $ and $ 3 $. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $ \varphi_{16} $, $ \varphi_{6} $ and $ \varphi_{5} $ respectively.

Keywords: 3-folds of general type, birational geometry, singularities of the Minimal Model Program.
Mathematics Subject Classification: Primary: 14J30, 14E05; Secondary: 14E30, 14B05.
Citation: Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, doi: 10.3934/era.2020117
References:
[1]

W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, 1984. doi: 10.1007/978-3-642-96754-2.  Google Scholar

[2]

C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar

[3]

E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.   Google Scholar

[4]

M. Chen, Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Cambridge Phil. Soc., 131 (2001), 241-264.  doi: 10.1017/S030500410100531X.  Google Scholar

[5]

M. Chen, Canonical stability of 3-folds of general type with $p_g\geq 3$, Int. J. Math., 14 (2003), 515-528.  doi: 10.1142/S0129167X03001934.  Google Scholar

[6]

M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann., 337 (2007), 887-908.  doi: 10.1007/s00208-006-0060-4.  Google Scholar

[7]

M. Chen, Some birationality criteria on 3-folds with $p_g>1$, Sci. China Math., 57 (2014), 2215-2234.  doi: 10.1007/s11425-014-4890-3.  Google Scholar

[8]

M. Chen, On minimal 3-folds of general type with maximal pluricanonical section index, Asian J. Math., 22 (2018), 257–268. arXiv: 1604.04828. doi: 10.4310/AJM.2018.v22.n2.a3.  Google Scholar

[9]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅰ, Ann. Sci. Éc. Norm. Supér., 43 (2010), 365-394.  doi: 10.24033/asens.2124.  Google Scholar

[10]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅱ, J. Differ. Geom., 86 (2010), 237-271.  doi: 10.4310/jdg/1299766788.  Google Scholar

[11]

J. A. Chen and M. Chen, Explicit birational geometry for 3-folds and 4-folds of general type, Ⅲ, Compos. Math., 151 (2015), 1041-1082.  doi: 10.1112/S0010437X14007817.  Google Scholar

[12]

M. Chen and D.-Q. Zhang, Characterization of the 4-canonical birationality of algebraic threefolds, Math. Z., 258 (2008), 565-585.  doi: 10.1007/s00209-007-0186-4.  Google Scholar

[13]

M. Chen and Q. Zhang, Characterization of the 4-canonical birationality of algebraic threefolds, Ⅱ, Math. Z., 283 (2016), 659-677.  doi: 10.1007/s00209-016-1616-y.  Google Scholar

[14]

O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France, 110 (1982), 319-346.   Google Scholar

[15]

A. R. Fletcher, Contributions to Riemann-Roch on projective $3$-folds with only canonical singularities and applications, Proceedings of Symposia in Pure Mathematics, 46 (1987), 221-231.   Google Scholar

[16]

C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1-25.  doi: 10.1007/s00222-006-0504-1.  Google Scholar

[17]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$ Ⅰ, Ann. of Math., 104 (1976), 357-387.  doi: 10.2307/1971050.  Google Scholar

[18]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅱ, Invent. Math., 37 (1976), 121-155.  doi: 10.1007/BF01418966.  Google Scholar

[19]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅲ, Invent. Math., 47, (1978), 209–248. doi: 10.1007/BF01579212.  Google Scholar

[20]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅳ, Invent. Math., 50, (1978/79), 103–128. doi: 10.1007/BF01390285.  Google Scholar

[21]

A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 281 (2000), 101–173.  Google Scholar

[22]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.  doi: 10.1007/BF01456407.  Google Scholar

[23]

Y. Kawamata, On the extension problem of pluricanonical forms, Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., Amer. Math. Soc., Providence, RI, 241 (1999), 193–207. doi: 10.1090/conm/241/03636.  Google Scholar

[24]

Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Algebraic Geometry, Sendai, (1985), 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. doi: 10.2969/aspm/01010283.  Google Scholar

[25] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[26]

M. Reid, Young person's guide to canonical singularities, Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345–414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.  Google Scholar

[27]

Y.-T. Siu, Finite generation of canonical ring by analytic method, Sci. China Ser. A, 51 (2008), 481-502.  doi: 10.1007/s11425-008-0073-4.  Google Scholar

[28]

S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.  doi: 10.1007/s00222-006-0503-2.  Google Scholar

[29]

H. Tsuji, Pluricanonical systems of projective varieties of general type. Ⅰ, Osaka J. Math., 43 (2006), 967-995.   Google Scholar

[30]

E. Viehweg, Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.  doi: 10.1515/crll.1982.335.1.  Google Scholar

[31]

G. Xiao, Surfaces Fibrées en Courbes de Genre Deux, Lecture Notes in Mathematics, 1137. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075351.  Google Scholar

[32]

G. Xiao, The Fibrations of Algebraic Surfaces, Modern Mathematics Series, Shanghai Scientific & Technical Publishers, 1991. Google Scholar

show all references

References:
[1]

W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, 1984. doi: 10.1007/978-3-642-96754-2.  Google Scholar

[2]

C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468.  doi: 10.1090/S0894-0347-09-00649-3.  Google Scholar

[3]

E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.   Google Scholar

[4]

M. Chen, Canonical stability in terms of singularity index for algebraic threefolds, Math. Proc. Cambridge Phil. Soc., 131 (2001), 241-264.  doi: 10.1017/S030500410100531X.  Google Scholar

[5]

M. Chen, Canonical stability of 3-folds of general type with $p_g\geq 3$, Int. J. Math., 14 (2003), 515-528.  doi: 10.1142/S0129167X03001934.  Google Scholar

[6]

M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann., 337 (2007), 887-908.  doi: 10.1007/s00208-006-0060-4.  Google Scholar

[7]

M. Chen, Some birationality criteria on 3-folds with $p_g>1$, Sci. China Math., 57 (2014), 2215-2234.  doi: 10.1007/s11425-014-4890-3.  Google Scholar

[8]

M. Chen, On minimal 3-folds of general type with maximal pluricanonical section index, Asian J. Math., 22 (2018), 257–268. arXiv: 1604.04828. doi: 10.4310/AJM.2018.v22.n2.a3.  Google Scholar

[9]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅰ, Ann. Sci. Éc. Norm. Supér., 43 (2010), 365-394.  doi: 10.24033/asens.2124.  Google Scholar

[10]

J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, Ⅱ, J. Differ. Geom., 86 (2010), 237-271.  doi: 10.4310/jdg/1299766788.  Google Scholar

[11]

J. A. Chen and M. Chen, Explicit birational geometry for 3-folds and 4-folds of general type, Ⅲ, Compos. Math., 151 (2015), 1041-1082.  doi: 10.1112/S0010437X14007817.  Google Scholar

[12]

M. Chen and D.-Q. Zhang, Characterization of the 4-canonical birationality of algebraic threefolds, Math. Z., 258 (2008), 565-585.  doi: 10.1007/s00209-007-0186-4.  Google Scholar

[13]

M. Chen and Q. Zhang, Characterization of the 4-canonical birationality of algebraic threefolds, Ⅱ, Math. Z., 283 (2016), 659-677.  doi: 10.1007/s00209-016-1616-y.  Google Scholar

[14]

O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France, 110 (1982), 319-346.   Google Scholar

[15]

A. R. Fletcher, Contributions to Riemann-Roch on projective $3$-folds with only canonical singularities and applications, Proceedings of Symposia in Pure Mathematics, 46 (1987), 221-231.   Google Scholar

[16]

C. D. Hacon and J. McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1-25.  doi: 10.1007/s00222-006-0504-1.  Google Scholar

[17]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$ Ⅰ, Ann. of Math., 104 (1976), 357-387.  doi: 10.2307/1971050.  Google Scholar

[18]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅱ, Invent. Math., 37 (1976), 121-155.  doi: 10.1007/BF01418966.  Google Scholar

[19]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅲ, Invent. Math., 47, (1978), 209–248. doi: 10.1007/BF01579212.  Google Scholar

[20]

E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅳ, Invent. Math., 50, (1978/79), 103–128. doi: 10.1007/BF01390285.  Google Scholar

[21]

A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 281 (2000), 101–173.  Google Scholar

[22]

Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.  doi: 10.1007/BF01456407.  Google Scholar

[23]

Y. Kawamata, On the extension problem of pluricanonical forms, Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., Amer. Math. Soc., Providence, RI, 241 (1999), 193–207. doi: 10.1090/conm/241/03636.  Google Scholar

[24]

Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Algebraic Geometry, Sendai, (1985), 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. doi: 10.2969/aspm/01010283.  Google Scholar

[25] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9780511662560.  Google Scholar
[26]

M. Reid, Young person's guide to canonical singularities, Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345–414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.  Google Scholar

[27]

Y.-T. Siu, Finite generation of canonical ring by analytic method, Sci. China Ser. A, 51 (2008), 481-502.  doi: 10.1007/s11425-008-0073-4.  Google Scholar

[28]

S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.  doi: 10.1007/s00222-006-0503-2.  Google Scholar

[29]

H. Tsuji, Pluricanonical systems of projective varieties of general type. Ⅰ, Osaka J. Math., 43 (2006), 967-995.   Google Scholar

[30]

E. Viehweg, Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.  doi: 10.1515/crll.1982.335.1.  Google Scholar

[31]

G. Xiao, Surfaces Fibrées en Courbes de Genre Deux, Lecture Notes in Mathematics, 1137. Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0075351.  Google Scholar

[32]

G. Xiao, The Fibrations of Algebraic Surfaces, Modern Mathematics Series, Shanghai Scientific & Technical Publishers, 1991. Google Scholar

[1]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090
[2]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315
[3]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[4]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[5]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[6]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027
[7]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[8]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[9]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
[10]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[11]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346
[12]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076
[13]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011
[14]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[15]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[16]

Jintai Ding, Zheng Zhang, Joshua Deaton. The singularity attack to the multivariate signature scheme HIMQ-3. Advances in Mathematics of Communications, 2021, 15 (1) : 65-72. doi: 10.3934/amc.2020043
[17]

Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351
[18]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[19]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[20]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences