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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the canonical rings of covers of surfaces of minimal degree


Authors: Francisco Javier Gallego and Bangere P. Purnaprajna
Journal: Trans. Amer. Math. Soc. 355 (2003), 2715-2732
MSC (2000): Primary 14J29
Published electronically: March 19, 2003
MathSciNet review: 1975396
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Abstract: In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi $ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.


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Additional Information

Francisco Javier Gallego
Affiliation: Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: FJavier_Gallego@mat.ucm.es

Bangere P. Purnaprajna
Affiliation: Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, Kansas 66045-2142
Email: purna@math.ukans.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03200-8
PII: S 0002-9947(03)03200-8
Received by editor(s): July 5, 2002
Published electronically: March 19, 2003
Additional Notes: The first author was partially supported by MCT project number BFM2000-0621 and by UCM project number PR52/00-8862. The second author was partially supported by the General Research Fund of the University of Kansas at Lawrence. The first author is grateful for the hospitality of the Department of Mathematics of the University of Kansas at Lawrence.
Article copyright: © Copyright 2003 American Mathematical Society