Explicit birational geometry of threefolds of general type, I

Chen, Jungkai A.; Chen, Meng

doi:10.24033/asens.2124

Explicit birational geometry of threefolds of general type, I
[Géométrie birationnelle explicite des variétés de type général de dimension 3, I]

Chen, Jungkai A. ; Chen, Meng

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 3, pp. 365-394.

Résumé
Abstract

Soit $V$ une variété non singulière complexe de type général et de dimension 3. Nous montrons $P_{12} (V) : = dim H^{0} (V, 12 K_{V}) > 0$ et $P_{m_{0}} (V) > 1$ pour un certain entier $m_{0} \leq 24$ . Une conséquence directe est la birationalité de l’application pluricanonique $ϕ_{m}$ pour tout $m \geq 126$ . De plus, le volume canonique $Vol (V)$ a un minorant universel $ν (3) \geq \frac{1}{63 \cdot 126^{2}}$ .

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12} (V) : = dim H^{0} (V, 12 K_{V}) > 0$ and $P_{m_{0}} (V) > 1$ for some positive integer $m_{0} \leq 24$ . A direct consequence is the birationality of the pluricanonical map $ϕ_{m}$ for all $m \geq 126$ . Besides, the canonical volume $Vol (V)$ has a universal lower bound $ν (3) \geq \frac{1}{63 \cdot 126^{2}}$ .

MR Zbl | 3 citations dans Numdam

DOI : 10.24033/asens.2124

Classification : 14J30, 14B05
Keywords: 3-folds, plurigenus
Mot clés : variétés de dimension 3, plurigenre

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     author = {Chen, Jungkai A. and Chen, Meng},
     title = {Explicit birational geometry of threefolds of general type, {I}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {365--394},
     publisher = {Soci\'et\'e math\'ematique de France},
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Chen, Jungkai A.; Chen, Meng. Explicit birational geometry of threefolds of general type, I. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 3, pp. 365-394. doi : 10.24033/asens.2124. https://www.numdam.org/articles/10.24033/asens.2124/

Bibliographie
Cité par

[1] A. Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series 68, Cambridge Univ. Press, 1983. | MR | Zbl

[2] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. I.H.É.S. 42 (1973), 171-219. | Numdam | MR | Zbl

[3] J. A. Chen & M. Chen, The canonical volume of 3-folds of general type with $χ \leq 0$ , J. Lond. Math. Soc. 78 (2008), 693-706. | MR | Zbl

[4] J. A. Chen, M. Chen & D.-Q. Zhang, The 5-canonical system on 3-folds of general type, J. reine angew. Math. 603 (2007), 165-181. | MR | Zbl

[5] J. A. Chen & C. D. Hacon, Pluricanonical systems on irregular 3-folds of general type, Math. Z. 255 (2007), 343-355. | MR | Zbl

[6] M. Chen, Canonical stability of 3-folds of general type with $p_{g} \geq 3$ , Internat. J. Math. 14 (2003), 515-528. | MR | Zbl

[7] M. Chen, On the $ℚ$ -divisor method and its application, J. Pure Appl. Algebra 191 (2004), 143-156. | MR | Zbl

[8] M. Chen, A sharp lower bound for the canonical volume of 3-folds of general type, Math. Ann. 337 (2007), 887-908. | MR | Zbl

[9] M. Chen & K. Zuo, Complex projective 3-fold with non-negative canonical Euler-Poincaré characteristic, Comm. Anal. Geom. 16 (2008), 159-182. | MR | Zbl

[10] L. Ein & R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc. 6 (1993), 875-903. | MR | Zbl

[11] A. R. Fletcher, Contributions to Riemann-Roch on projective $3$ -folds with only canonical singularities and applications, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 221-231. | MR | Zbl

[12] A. R. Fletcher, Inverting Reid's exact plurigenera formula, Math. Ann. 284 (1989), 617-629. | MR | Zbl

[13] C. D. Hacon & J. Mckernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), 1-25. | MR | Zbl

[14] A. R. Iano-Fletcher, Working with weighted complete intersections, in Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, 2000, 101-173. | MR | Zbl

[15] Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261 (1982), 43-46. | MR | Zbl

[16] Y. Kawamata, On the plurigenera of minimal algebraic $3$ -folds with $K \equiv 0$ , Math. Ann. 275 (1986), 539-546. | MR | Zbl

[17] Y. Kawamata, K. Matsuda & K. Matsuki, Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, 1987, 283-360. | MR | Zbl

[18] J. Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. 123 (1986), 11-42. | MR | Zbl

[19] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998. | MR | Zbl

[20] T. Luo, Global $2$ -forms on regular $3$ -folds of general type, Duke Math. J. 71 (1993), 859-869. | MR | Zbl

[21] M. Reid, Canonical $3$ -folds, in Journées de Géométrie Algébrique d'Angers, juillet 1979, Sijthoff & Noordhoff, 1980, 273-310. | MR | Zbl

[22] M. Reid, Minimal models of canonical $3$ -folds, in Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, 1983, 131-180. | MR | Zbl

[23] M. Reid, Young person's guide to canonical singularities, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., 1987, 345-414. | MR | Zbl

[24] S. Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551-587. | MR | Zbl

[25] H. Tsuji, Pluricanonical systems of projective varieties of general type. I, Osaka J. Math. 43 (2006), 967-995. | MR | Zbl

[26] E. Viehweg, Vanishing theorems, J. reine angew. Math. 335 (1982), 1-8. | MR | Zbl

Chen, Meng; Esser, Louis; Wang, Chengxi On explicit birational geometry for minimal n $n$ ‐folds of canonical dimension n−1 $n - 1$ , Bulletin of the London Mathematical Society, Volume 56 (2024) no. 1, p. 319 | DOI:10.1112/blms.12934
Yan, Jianshi On the pluricanonical map and the canonical volume of projective 4-folds of general type, Communications in Algebra, Volume 52 (2024) no. 7, p. 2706 | DOI:10.1080/00927872.2024.2304597
Yasuda, Takehiko The isomorphism problem of projective schemes and related algorithmic problems, International Journal of Algebra and Computation, Volume 33 (2023) no. 05, p. 893 | DOI:10.1142/s021819672350039x
Zhang, Lei Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic, Algebra Number Theory, Volume 16 (2022) no. 10, p. 2339 | DOI:10.2140/ant.2022.16.2339
Chen, Meng; Jiang, Chen On the anti-canonical geometry of weak ℚ-Fano threefolds II, Annales de l'Institut Fourier, Volume 70 (2021) no. 6, p. 2473 | DOI:10.5802/aif.3367
Chen, Meng; Hu, Yong; Penegini, Matteo On projective threefolds of general type with small positive geometric genus, Electronic Research Archive, Volume 29 (2021) no. 3, p. 2293 | DOI:10.3934/era.2020117
Yan, Jianshi On minimal 4-folds of general type with $p_{g} \geq 2$ , Electronic Research Archive, Volume 29 (2021) no. 5, p. 3309 | DOI:10.3934/era.2021040
Chen, Jheng‐Jie; Chen, Jungkai Alfred; Chen, Meng; Jiang, Zhi On quint‐canonical birationality of irregular threefolds, Proceedings of the London Mathematical Society, Volume 122 (2021) no. 2, p. 234 | DOI:10.1112/plms.12348
Coughlan, Stephen K3 transitions and canonical 3-folds, Bulletin of the London Mathematical Society, Volume 50 (2018) no. 4, p. 583 | DOI:10.1112/blms.12157
Zhu, Huanping The canonical volume of minimal 3-folds of general type, International Journal of Mathematics, Volume 29 (2018) no. 03, p. 1850023 | DOI:10.1142/s0129167x18500234
Lehmann, Brian A snapshot of the Minimal Model Program, Surveys on Recent Developments in Algebraic Geometry, Volume 95 (2017), p. 1 | DOI:10.1090/pspum/095/01636
Jiang, Chen On birational geometry of minimal threefolds with numerically trivial canonical divisors, Mathematische Annalen, Volume 365 (2016) no. 1-2, p. 49 | DOI:10.1007/s00208-015-1268-y
Chen, Meng; Zhang, Qi Characterization of the 4-canonical birationality of algebraic threefolds, II, Mathematische Zeitschrift, Volume 283 (2016) no. 3-4, p. 659 | DOI:10.1007/s00209-016-1616-y
Birkar, Caucher; Zhang, De-Qi Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Publications mathématiques de l'IHÉS, Volume 123 (2016) no. 1, p. 283 | DOI:10.1007/s10240-016-0080-x
Zhang, Tong Geography of Irregular Gorenstein 3–folds, Canadian Journal of Mathematics, Volume 67 (2015) no. 3, p. 696 | DOI:10.4153/cjm-2014-033-0
Chen, Jungkai A.; Chen, Meng Explicit birational geometry of 3-folds and 4-folds of general type, III, Compositio Mathematica, Volume 151 (2015) no. 6, p. 1041 | DOI:10.1112/s0010437x14007817
Chen, Jungkai BIRATIONAL MAPS OF $3$ -FOLDS, Taiwanese Journal of Mathematics, Volume 19 (2015) no. 6 | DOI:10.11650/tjm.19.2015.5337
Chen, J.-J. Finiteness of Calabi-Yau Quasismooth Weighted Complete Intersections, International Mathematics Research Notices (2014) | DOI:10.1093/imrn/rnu049
XU, JINSONG The third and fourth pluricanonical maps of threefolds of general type, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 157 (2014) no. 2, p. 209 | DOI:10.1017/s0305004114000267
Chen, Meng Some birationality criteria on 3-folds with p g > 1, Science China Mathematics, Volume 57 (2014) no. 11, p. 2215 | DOI:10.1007/s11425-014-4890-3
Ballico, E.; Pignatelli, R.; Tasin, L. Weighted Hypersurfaces with Either Assigned Volume or Many Vanishing Plurigenera, Communications in Algebra, Volume 41 (2013) no. 10, p. 3745 | DOI:10.1080/00927872.2012.677079
Chen, Jungkai A.; Hacon, Christopher D. Factoring 3-fold flips and divisorial contractions to curves, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2011 (2011) no. 657 | DOI:10.1515/crelle.2011.056
Chen, Meng On anti-pluricanonical systems of ℚ-Fano 3-folds, Science China Mathematics, Volume 54 (2011) no. 8, p. 1547 | DOI:10.1007/s11425-010-4158-5

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