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On the canonical rings of covers of surfaces of minimal degree

Gallego Rodrigo, Francisco Javier and Purnaprajna, Bangere P (2003) On the canonical rings of covers of surfaces of minimal degree. Transactions of the American Mathematical Society, 355 (7). pp. 2715-2732. ISSN 1088-6850

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Abstract

Let S be a regular surface of general type with at worst canonical singularities and with basepoint-free canonical system. Let X be its canonical image. It is well known that X must be a canonical surface or a minimal degree surface. The main result of the authors completely describes the number and degree of the generators of the canonical ring of S in the second case. More concretely, if r = deg(X) and n is the degree of the canonical map, then (1) if n = 2 and r = 1, the canonical ring is generated in degree 1, plus one generator in degree 4; (2) in the other cases, the canonical ring is generated in degree 1, plus r(n−2) generators in degree 2 and r −1 generators in degree 3.
This result, together with previous results of Ciliberto and Green, describes when the canonical ring of S is generated in degree less than or equal to 2: X is not a surface of minimal degree other than the plane and, in this last case, n 6= 2.
The authors also construct a series of non-trivial examples of the theorem and prove that some expected ones do not exist.
Finally, the authors apply their results to Calabi-Yau threefolds, obtaining analogous results. The key point here is that, for a Calabi-Yau threefold, the general member of a big and base-point-free linear system is a surface of general type.


Item Type:Article
Additional Information:

First published in Transactions of the American Mathematical Society in Volume 355, Number 7, 2003, published by the American Mathematical Society

Uncontrolled Keywords:Surfaces of general type, Calabi-Yau threefolds, Covering, Varieties of minimal degree, Canonical ring
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:12605
Deposited On:25 Apr 2011 20:57
Last Modified:06 Feb 2014 09:28

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