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Configurations of lines in del Pezzo surfaces with Gosset polytopes


Author: Jae-Hyouk Lee
Journal: Trans. Amer. Math. Soc. 366 (2014), 4939-4967
MSC (2010): Primary 51M20, 14J26, 14N99, 52B20
DOI: https://doi.org/10.1090/S0002-9947-2014-06098-4
Published electronically: January 30, 2014
MathSciNet review: 3217705
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Abstract: In this article, we classify and describe the configuration of the divisor classes of del Pezzo surfaces, which are written as the sum of distinct lines with fixed intersection according to combinatorial data in Gosset polytopes.

We introduce the $ k$-Steiner system and cornered simplexes, and characterize the configurations of positive degree $ m(\leq 3)$-simplexes with them via monoidal transforms.

Higher dimensional $ m\ (4\leq m\leq 7)$-simplexes of $ 1$-degree exist in $ 4_{21}$ in the Picard group of del Pezzo surface of degree $ 1$, and their configurations are nontrivial. The configurations of $ 4$- and $ 7$-simplexes are related to rulings in $ S_{8}$, and the configurations of $ 5$- and $ 6$-simplexes correspond to the skew $ 3$-lines and skew $ 7$-lines in $ S_{8}$. In particular, the seven lines in a $ 6$-simplex produce a Fano plane.


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Jae-Hyouk Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seodaemun-Gu Daehyun- dong, Seoul, Korea
Email: jaehyoukl@ewha.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-2014-06098-4
Received by editor(s): April 17, 2012
Received by editor(s) in revised form: December 12, 2012, and January 26, 2013
Published electronically: January 30, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.