Electronic Research Archive

2021, Issue 3: 2293-2323. doi: 10.3934/era.2020117

On projective threefolds of general type with small positive geometric genus

• Received: 01 April 2020 Revised: 01 October 2020 Published: 24 November 2020
• Primary: 14J30, 14E05; Secondary: 14E30, 14B05

• 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[J]. Electronic Research Archive, 2021, 29(3): 2293-2323. doi: 10.3934/era.2020117

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• 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.

 [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] Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. (2010) 23: 405-468. [3] Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. (1973) 42: 171-219. [4] Canonical stability in terms of singularity index for algebraic threefolds. Math. Proc. Cambridge Phil. Soc. (2001) 131: 241-264. [5] Canonical stability of 3-folds of general type with $p_g\geq 3$. Int. J. Math. (2003) 14: 515-528. [6] A sharp lower bound for the canonical volume of 3-folds of general type. Math. Ann. (2007) 337: 887-908. [7] Some birationality criteria on 3-folds with $p_g>1$. Sci. China Math. (2014) 57: 2215-2234. [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] Explicit birational geometry of threefolds of general type, Ⅰ. Ann. Sci. Éc. Norm. Supér. (2010) 43: 365-394. [10] Explicit birational geometry of threefolds of general type, Ⅱ. J. Differ. Geom. (2010) 86: 237-271. [11] Explicit birational geometry for 3-folds and 4-folds of general type, Ⅲ. Compos. Math. (2015) 151: 1041-1082. [12] Characterization of the 4-canonical birationality of algebraic threefolds. Math. Z. (2008) 258: 565-585. [13] Characterization of the 4-canonical birationality of algebraic threefolds, Ⅱ. Math. Z. (2016) 283: 659-677. [14] Inégalités numériques pour les surfaces de type général. Bull. Soc. Math. France (1982) 110: 319-346. [15] Contributions to Riemann-Roch on projective $3$-folds with only canonical singularities and applications. Proceedings of Symposia in Pure Mathematics (1987) 46: 221-231. [16] Boundedness of pluricanonical maps of varieties of general type. Invent. Math. (2006) 166: 1-25. [17] Algebraic surfaces of general type with small $c_1^2$ Ⅰ. Ann. of Math. (1976) 104: 357-387. [18] Algebraic surfaces of general type with small $c_1^2$. Ⅱ. Invent. Math. (1976) 37: 121-155. [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] A generalization of Kodaira-Ramanujam's vanishing theorem. Math. Ann. (1982) 261: 43-46. [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] (1998) Birational Geometry of Algebraic Varieties.Cambridge Tracts in Mathematics, 134. Cambridge University Press. [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] Finite generation of canonical ring by analytic method. Sci. China Ser. A (2008) 51: 481-502. [28] Pluricanonical systems on algebraic varieties of general type. Invent. Math. (2006) 165: 551-587. [29] Pluricanonical systems of projective varieties of general type. Ⅰ. Osaka J. Math. (2006) 43: 967-995. [30] Vanishing theorems. J. Reine Angew. Math. (1982) 335: 1-8. [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.
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