Configurations of lines in del Pezzo surfaces with Gosset polytopes
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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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Additional Information
- Jae-Hyouk Lee
- Affiliation: Department of Mathematics, Ewha Womans University, Seodaemun-Gu Daehyun- dong, Seoul, Korea
- Email: jaehyoukl@ewha.ac.kr
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3217705