×

On polarized canonical Calabi-Yau threefolds. (English) Zbl 0813.14029

We call a normal projective complex threefold \(X\) a canonical Calabi-Yau threefold if \({\mathcal O}_ X (K_ X) = {\mathcal O}_ X\), \(h^ 1({\mathcal O}_ X) = 0\) and \(X\) has only canonical singularities, and say that \(X\) is a minimal Calabi-Yau threefold if, in addition, \(X\) has only \(\mathbb{Q}\)- factorial terminal singularities. A pair of a normal projective variety and a line bundle is called a polarized variety if the line bundle is ample. A line bundle \(L\) on a normal variety \(X\) is said to be simply- generated, if the graded \(\mathbb{C}\)-algebra \(\oplus_{k \geq 0} H^ 0 ({\mathcal O}_ X (kL))\) is generated by the linear piece \(H^ 0 ({\mathcal O}_ X (L))\). It is well-known that an ample simple-generated line bundle is very ample. Our main theorem are as follows:
Theorem I. Let \((X,L)\) be a polarized minimal Calabi-Yau threefold. Then, (1) \(| mL|\) gives a birational map when \(m \geq 5\), (2) \(| mL |\) is free when \(m \geq 5\), (3) \(mL\) is simply-generated, in particular very ample, when \(m \geq 10\).
Theorem II. Let \((X,L)\) be a polarized canonical Calabi-Yau threefold. Then, (1) \(| mL |\) gives a birational map when \(m \geq 5\), (2) \(| mL |\) is free when \(m \geq 7\), (3) \(mL\) is simply-generated, in particular very ample, when \(m \geq 14\).

MSC:

14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] L. Ein, R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. Preprint (1992) · Zbl 0803.14004
[2] J.P. Demailly, A numerical criterion for very ample line bundles. J. Differ. Geom.37 (1993), 323-374 · Zbl 0783.32013
[3] T. Fujita, On hyperelliptic polarized varieties. Tohoku Math. J.35 (1983), 1-44 · Zbl 0505.14003 · doi:10.2748/tmj/1178229099
[4] T. Fujita, Classification of polarized manifolds of sectional genus two, Algebraic geometry and commutative algebra, in honor of M. Nagata (1987), Kinokuniya, Tokyo, 73-98
[5] M. Green, Koszul cohomology and the geometry of projective varieties. J. Differ. Geom.19 (1984), 125-171 · Zbl 0559.14008
[6] Y. Kawamata, The crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces. Ann. Math.127 (1988), 93-163 · Zbl 0651.14005 · doi:10.2307/1971417
[7] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem. Adv. Stud. Pure Math.10 (1987), 283-360 · Zbl 0672.14006
[8] J. Kollár, Effective base point freeness. Math. Ann.296 (1993), 595-605 · Zbl 0818.14002 · doi:10.1007/BF01445123
[9] K. Oguiso, On polarized Calabi-Yau 3-folds. J. Fac. Sci. Univ. Tokyo38 (1991), 395-429 · Zbl 0766.14034
[10] K. Oguiso, On algebraic fiber space structures on a Calabi-Yau 3-fold. Intern. Math.4 (1993), 439-465 · Zbl 0793.14030 · doi:10.1142/S0129167X93000248
[11] M. Reid, Canonical 3-folds, in Géométrie Algébriques Anger 1979. Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands (1980), 273-310
[12] M. Reid, Minimal models of canonical 3-folds. Adv. Stud. Pure Math.1 (1983), 131-180 · Zbl 0558.14028
[13] M. Reid, 1-2-3 (1992), lecture note at Utah University. (Preprint)
[14] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math.127 (1988), 309-316 · Zbl 0663.14010 · doi:10.2307/2007055
[15] P.M.H. Wilson, Calabi-Yau manifolds with large Picard number. Invent. Math.98 (1989), 139-155 · Zbl 0688.14032 · doi:10.1007/BF01388848
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.