Gosset Polytopes in Picard Groups of del Pezzo Surfaces

Canadian Journal of Mathematics

Jae-Hyouk Lee¹ 

DOI  :  10.4153/CJM-2011-063-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In this article, we study the correspondence between the geometry ofdel Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gossetpolytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in$operatorname{Pic} S_{r}otimesmathbb{Q}$ whose vertices are lines, and we identifydivisor classes in $operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($aleqr$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$.Then we explain how these classes correspond to skew $a$-lines($aleq r$),exceptional systems, and rulings, respectively.As an application, we work on the monoidal transform for lines to study thelocal geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformationand the Bertini transformation induce a symmetry of polytopes $3_{21}$ and$4_{21}$, respectively.

【 授权许可】

Unknown