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August  2021, 29(3): 2293-2323. doi: 10.3934/era.2020117

On projective threefolds of general type with small positive geometric genus

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  August 2021 Early access  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.

Citation: Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, 2021, 29 (3) : 2293-2323. 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. BirkarP. CasciniC. 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. BirkarP. CasciniC. 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

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