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On the degree of the canonical maps of 3-folds. (English) Zbl 1068.14046

Summary: We prove the following result that answers a question of M. Chen: Let \(X\) be a Gorenstein minimal complex projective 3-fold of general type with locally factorial terminal singularities. If \(|K_X|\) defines a generically finite map \(\varphi:X \to\mathbb{P}^{p_g-1}\), then \(\deg (\varphi)\leq 576\). For any positive integer \(m>0\), we give infinitely many examples of (non-Gorenstein) 3-folds of general type with canonical map of degree \(m\).

MSC:

14J30 \(3\)-folds
14E20 Coverings in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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[1] Chen, M.: Weak boundedness theorems for canonically fibered Gorenstein minimal threefolds. (Preprint math. AG/0206097).
[2] Ein, L., and Lazarsfeld, R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Amer. Math. Soc., 10 (1), 243-258 (1997). · Zbl 0901.14028 · doi:10.1090/S0894-0347-97-00223-3
[3] Hacon, C.: A derived category approach to generic vanishing. Preprint math. AG/0308198. (To appear in J. Reine Angew. Math., 575 ). (2004). · Zbl 1137.14012 · doi:10.1515/crll.2004.078
[4] Kollár, J.: Higher direct images of dualizing sheaves I. Ann. of Math. (2), 123 (1), 11-42 (1986). · Zbl 0598.14015 · doi:10.2307/1971351
[5] Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, pp. 449-476 (1987). · Zbl 0648.14006
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