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},
     volume = {Ser. 4, 43},
     number = {3},
     year = {2010},
     doi = {10.24033/asens.2124},
     mrnumber = {2667020},
     zbl = {1194.14060},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2124/}
}

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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. http://www.numdam.org/articles/10.24033/asens.2124/

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