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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 |
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.
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. |
[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. |
[3] |
E. Bombieri,
Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Debarre,
Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France, 110 (1982), 319-346.
|
[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.
|
[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. |
[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. |
[18] |
E. Horikawa,
Algebraic surfaces of general type with small $c_1^2$. Ⅱ, Invent. Math., 37 (1976), 121-155.
doi: 10.1007/BF01418966. |
[19] |
E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅲ, Invent. Math., 47, (1978), 209–248.
doi: 10.1007/BF01579212. |
[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. |
[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. |
[22] |
Y. Kawamata,
A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.
doi: 10.1007/BF01456407. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[28] |
S. Takayama,
Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.
doi: 10.1007/s00222-006-0503-2. |
[29] |
H. Tsuji,
Pluricanonical systems of projective varieties of general type. Ⅰ, Osaka J. Math., 43 (2006), 967-995.
|
[30] |
E. Viehweg,
Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.
doi: 10.1515/crll.1982.335.1. |
[31] |
G. Xiao, Surfaces Fibrées en Courbes de Genre Deux, Lecture Notes in Mathematics, 1137. Springer-Verlag, Berlin, 1985.
doi: 10.1007/BFb0075351. |
[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. |
[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. |
[3] |
E. Bombieri,
Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 171-219.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
O. Debarre,
Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France, 110 (1982), 319-346.
|
[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.
|
[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. |
[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. |
[18] |
E. Horikawa,
Algebraic surfaces of general type with small $c_1^2$. Ⅱ, Invent. Math., 37 (1976), 121-155.
doi: 10.1007/BF01418966. |
[19] |
E. Horikawa, Algebraic surfaces of general type with small $c_1^2$. Ⅲ, Invent. Math., 47, (1978), 209–248.
doi: 10.1007/BF01579212. |
[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. |
[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. |
[22] |
Y. Kawamata,
A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann., 261 (1982), 43-46.
doi: 10.1007/BF01456407. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[28] |
S. Takayama,
Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551-587.
doi: 10.1007/s00222-006-0503-2. |
[29] |
H. Tsuji,
Pluricanonical systems of projective varieties of general type. Ⅰ, Osaka J. Math., 43 (2006), 967-995.
|
[30] |
E. Viehweg,
Vanishing theorems, J. Reine Angew. Math., 335 (1982), 1-8.
doi: 10.1515/crll.1982.335.1. |
[31] |
G. Xiao, Surfaces Fibrées en Courbes de Genre Deux, Lecture Notes in Mathematics, 1137. Springer-Verlag, Berlin, 1985.
doi: 10.1007/BFb0075351. |
[32] |
G. Xiao, The Fibrations of Algebraic Surfaces, Modern Mathematics Series, Shanghai Scientific & Technical Publishers, 1991. Google Scholar |
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