Abstract
In this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives.
First we study in Section 2 and Section 3 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of X also shows that, if the dimension of X is greater than or equal to 3, there does not exist a converse to a theorem of M. Green that bounds the degree of the generators of the canonical ring of X. This is in sharp contrast with the situation in dimension 2 where such converse exists, as proved by the authors in a previous work.
Second, we study in Section 4, the structure of the canonical morphism of X. We use this to show among other things the non-existence of some a priori plausible examples of triple canonical covers of varieties of minimal degree. We also characterize the targets of flat canonical covers of varieties of minimal degree. Some of the results of Section 4 are more general and apply to varieties X which are not necessarily regular, and to targets Y that are scrolls which are not of minimal degree.
Dedicated to C.S. Seshadri on his 70th birthday.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Catanese, Equations of pluriregular varieties of general type, Geometry today (Rome, 1984), 47–67, Progr. Math., 60, Birkhauser Boston, 1985.
F. Catanese, Commutative algebra methods and equations of regular surfaces, Algebraic geometry, (Bucharest, 1982), 68–111, Lecture Notes in Math., 1056, Springer Berlin, 1984.
C. Ciliberto, Sul grado dei generatori dell’anello di una superficie di tipo generale, Rend. Sem. Mat. Univ. Politec. Torino 41 (1983), 83–111.
T. Fujita, Triple covers by smooth manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 35 (1988), 169–175.
F.J. Gallego and B.P. Purnaprajna, On the canonical ring of covers of surfaces of minimal degree, to appear in Transactions of the American Mathematical Society.
M.L. Green, The canonical ring of a variety of general type, Duke Math. J. 49 (1982), 1087–1113.
E. Horikawa, Algebraic surfaces of general type with small \(c_1^2\). I, Ann. of Math. (2) 104 (1976), 357–387.
E. Horikawa, Algebraic surfaces of general type with small \(c_1^2\), II, Invent. Math. 37 (1976), 121–155.
E. Horikawa, Algebraic surfaces of general type with small \(c_1^2\), III, Invent. Math. 47 (1978), 209–248.
E. Horikawa, Algebraic surfaces of general type with small \(c_1^2\), IV, Invent. Math. 50 (1978/79), 103–128.
D. Hahn and R. Miranda, Quadruple covers of algebraic varieties, J. Algebraic Geom. 8 (1999), 1–30.
K. Konno, Algebraic surfaces of general type with \(c_1^2\) = 3pg − 6, Math. Ann. 290 (1991), 77–107.
M. Mendes Lopes and R. Pardini, Triple canonical surfaces of minimal degree, International J. Math. 11 (2000), 553–578.
R. Miranda, Triple covers in Algebraic Geometry, Amer. J. Math. 107 (1985), 1123–1158.
P. Supino, Triple covers of 3-folds as canonical maps, Communications in Algebra 26, (1998), 1475–1487.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2003 Hindustan Book Agency
About this chapter
Cite this chapter
Gallego, F.J., Purnaprajna, B.P. (2003). Triple Canonical Covers of Varieties of Minimal Degree. In: Lakshmibai, V., et al. A Tribute to C. S. Seshadri. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-11-8_17
Download citation
DOI: https://doi.org/10.1007/978-93-86279-11-8_17
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-39-5
Online ISBN: 978-93-86279-11-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)