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On the canonical rings of covers of surfaces of minimal degree.
- Source :
-
Transactions of the American Mathematical Society . Jul2003, Vol. 355 Issue 7, p2715-2732. 18p. - Publication Year :
- 2003
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 355
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 10043567
- Full Text :
- https://doi.org/10.1090/S0002-9947-03-03200-8