Configurations of lines in del Pezzo surfaces with Gosset polytopes

Lee, Jae-Hyouk

doi:10.1090/S0002-9947-2014-06098-4

Configurations of lines in del Pezzo surfaces with Gosset polytopes
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Trans. Amer. Math. Soc. 366 (2014), 4939-4967 Request permission

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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