Geodesic
Gosset Polytopes in Picard Groups of del Pezzo Surfaces
Canadian journal of mathematics, Tome 64 (2012) no. 1, pp. 123-150

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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 Pic ${{S}_{r}}\,\otimes \,\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in Pic ${{s}_{r}}$ corresponding to $(a-1)$ -simplexes $(a\le r)$ , $(r-1)$ -simplexes and $(r-1)$ -crosspolytopes of the polytope ${{(r-4)}_{21}}$ . Then we explain how these classes correspond to skew $a$ -lines $(a\le r)$ , 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.
DOI : 10.4153/CJM-2011-063-6
Mots-clés : 51M20, 14J26, 22E99 
Lee, Jae-Hyouk. Gosset Polytopes in Picard Groups of del Pezzo Surfaces. Canadian journal of mathematics, Tome 64 (2012) no. 1, pp. 123-150. doi: 10.4153/CJM-2011-063-6
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     title = {Gosset {Polytopes} in {Picard} {Groups} of del {Pezzo} {Surfaces}},
     journal = {Canadian journal of mathematics},
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     year = {2012},
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     doi = {10.4153/CJM-2011-063-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-063-6/}
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