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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. 27152732. ISSN 10886850

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Abstract
Let S be a regular surface of general type with at worst canonical singularities and with basepointfree 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 nontrivial examples of the theorem and prove that some expected ones do not exist.
Finally, the authors apply their results to CalabiYau threefolds, obtaining analogous results. The key point here is that, for a CalabiYau threefold, the general member of a big and basepointfree 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, CalabiYau 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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