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# Gosset Polytopes in Picard Groups of del Pezzo Surfaces

Published:2011-09-15
Printed: Feb 2012
• Jae-Hyouk Lee,
Korea Institute for Advanced Study, KIAS Hoegiro 87(207-43 Cheongnyangni-dong), Dongdaemun-gu, Seoul 130-722, Korea
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## Abstract

In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively.
 MSC Classifications: 51M20 - Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 14J26 - Rational and ruled surfaces 22E99 - None of the above, but in this section