# E-Polytopes in Picard Groups of Smooth Rational Surfaces

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## Abstract

**:**

## 1. Introduction

n | 3 | 4 | 5 | 6 | 7 | 8 |

E_{n} | A_{1} × A_{2} | A_{4} | D_{5} | E_{6} | E_{7} | E_{8} |

## 2. Preliminary

#### 2.1. Smooth Rational Surfaces with ${K}^{2}>0$

**Remark 1.**

#### 2.2. $ADE$-Polytopes

#### 2.2.1. Regular and Semiregular Polytopes

#### 2.2.2. Coxeter–Dynkin Diagrams

#### 2.2.3. $ADE$-Type Coxeter Groups and Isotropy Groups

#### 2.2.4. $ADE$-Polytopes

**regular simplex**${\alpha}_{n}$ is an n-dimensional simplex with equilateral edges. Inductively, ${\alpha}_{n}$ is constructed as a pyramid based on an $(n-1)$-dimensional simplex ${\alpha}_{n-1}$. The facets of a regular simplex ${\alpha}_{n}$ are regular simplexes ${\alpha}_{n-1}$, and the vertex figure of ${\alpha}_{n}$ is also ${\alpha}_{n-1}$. For a regular simplex ${\alpha}_{n}$, only regular simplexes ${\alpha}_{k}$, $0\le k\le n-1$ appear as subpolytopes.

**crosspolytope**${\beta}_{n}$ is an n-dimensional polytope whose $2n$-vertices are given as the intersections between an n-dimensional Cartesian coordinate frame and a sphere centered at the origin. ${\beta}_{n}$ is also inductively constructed as a bipyramid based on an ($n-1$)-dimensional crosspolytope ${\beta}_{n-1}$, and the n-vertices in ${\beta}_{n}$ form a simplex ${\alpha}_{n-1}$ if any two vertices are not chosen from the same Cartesian coordinate line. The vertex figure of a crosspolytope ${\beta}_{n}$ is also a crosspolytope ${\beta}_{n-1}$, and the facets of ${\beta}_{n}$ are simplexes ${\alpha}_{n-1}$. For a crosspolytope ${\beta}_{n}$, only regular simplexes ${\alpha}_{k}$, $0\le k\le n-1$ appear as subpolytopes.

**Gosset polytopes**${k}_{21}\phantom{\rule{4pt}{0ex}}(k=-1,0,1,2,3,4)\phantom{\rule{4pt}{0ex}}$are $(k+4)$-dimensional semiregular polytopes of the Coxeter groups ${E}_{k+4}\phantom{\rule{4pt}{0ex}}$discovered by Gosset. The vertex figure of ${k}_{21}$ is ${\left(k-1\right)}_{21}$. For $k\ne -1$ the facets of ${k}_{21}$-polytopes are the regular simplexes ${\alpha}_{k+3}$ and the crosspolytopes ${\beta}_{k+3}$, but all the lower dimensional subpolytopes are regular simplexes. In fact, Coxeter called ${4}_{21}$, ${3}_{21}\phantom{\rule{4pt}{0ex}}$and ${2}_{21}\phantom{\rule{4pt}{0ex}}$ Gosset polytopes. We extend the list of Gosset polytopes along the extended list of ${E}_{n}$. Note that a Gosset polytope ${(-1)}_{21}$, a triangular prism, especially has an isosceles triangle as the vertex figure different from an equilateral triangle.

#### 2.2.5. Subpolytopes in Gosset Polytopes ${k}_{21}$

#### 2.3. Del Pezzo Surfaces and Gosset Polytopes

**Theorem 2**

**Remark 3.**

**Theorem 4**

**Remark 5.**

**Theorem 6**

**Theorem 7**

## 3. Special Divisors of Blown-up Hirzebruch Surfaces

#### 3.1. Special Divisors of ${\mathbf{F}}_{p,r}$

#### 3.2. Blown-Up Hirzebruch Surfaces and Gosset Polytopes

**Theorem 8.**

**Proof.**

**Theorem 9.**

**Remark 10.**

**Theorem 11.**

**Proof.**

**Note**: For the proof of Theorem 11, we apply a fact that for two distinct lines ${l}_{1}\phantom{\rule{4pt}{0ex}}$and ${l}_{2}$ in ${\mathbf{F}}_{\phantom{\rule{-0.166667em}{0ex}}r}$, ${l}_{1}\xb7{l}_{2}=0\phantom{\rule{4pt}{0ex}}$if and only if the corresponding vertices in ${(r-3)}_{21}\phantom{\rule{4pt}{0ex}}$are joined by an edge. This fact can be obtained by simple calculations as in the case of del Pezzo surfaces ([10]).

**Theorem 12.**

**Remark 13.**

## 4. E-Polytopes in Picard Groups of Smooth Rational Surfaces

#### 4.1. ${2}_{k1}$ and Rulings

**Theorem 14.**

**Proof.**

#### 4.2. ${1}_{k2}$ and Exceptional Systems

**Theorem 15.**

**Proof.**

**Corollary 16.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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r | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

${\mathbf{\alpha}}_{\mathbf{1}}$ | 9 | 30 | 80 | 216 | 756 | 6720 |

${\mathbf{\alpha}}_{\mathbf{2}}$ | 2 | 30 | 160 | 720 | 4032 | $60,480$ |

${\mathbf{\alpha}}_{\mathbf{3}}$ | 0 | 5 | 120 | 1080 | $10,080$ | $241,920$ |

${\mathbf{\alpha}}_{\mathbf{4}}$ | 0 | 0 | 16 | 648 | $12,096$ | $483,840$ |

${\mathbf{\alpha}}_{\mathbf{5}}$ | 0 | 0 | 0 | 72 | 6048 | $483,840$ |

${\mathbf{\alpha}}_{\mathbf{6}}$ | 0 | 0 | 0 | 0 | 576 | $207,360$ |

${\mathbf{\alpha}}_{\mathbf{7}}$ | 0 | 0 | 0 | 0 | 0 | $17,280$ |

r | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

${\mathbf{\beta}}_{\mathit{r}-\mathbf{1}}$ | 3 | 5 | 10 | 27 | 126 | 2160 |

${(\mathit{r}-\mathbf{4})}_{\mathbf{21}}$ | $-{\mathbf{1}}_{\mathbf{21}}$ | ${\mathbf{0}}_{\mathbf{21}}$ | ${\mathbf{1}}_{\mathbf{21}}$ | ${\mathbf{2}}_{\mathbf{21}}$ | ${\mathbf{3}}_{\mathbf{21}}$ | ${\mathbf{4}}_{\mathbf{21}}$ |
---|---|---|---|---|---|---|

total # | 9 | 30 | 120 | 648 | 6048 | $207,360$ |

A, B | $3,6$ | $10,20$ | $40,80$ | $216,432$ | $2016,4032$ | $69,120,138,240$ |

r | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

total # | 3 | 10 | 40 | 216 | 2072 | $82,560$ |

I, II | $3,0$ | $10,0$ | $40,0$ | $216,0$ | $2016,56$ | $69,120,13,440$ |

r | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

$\mathbf{R}\left({\mathbf{F}}_{\mathit{p},\mathit{r}}\right)$ | 2 | 8 | 20 | 40 | 72 | 126 | 240 |

$\mathbf{L}\left({\mathbf{F}}_{\mathit{p},\mathit{r}}\right)$ | 3 | 6 | 10 | 16 | 27 | 56 | 240 |

$\mathbf{M}\left({\mathbf{F}}_{\mathit{p},\mathit{r}}\right)$ | 2 | 3 | 5 | 10 | 27 | 126 | 2160 |

$\mathcal{E}\left({\mathbf{F}}_{\mathit{p},\mathit{r}}\right)$ | 1 | 2 | 5 | 16 | 72 | 576 | $17,520$ |

$\mathcal{Q}\left({\mathbf{F}}_{\mathit{p},\mathit{r}}\right)$ | 1 | 3 | 10 | 40 | 216 | 2072 | $82,560$ |

**Table 6.**Vertices of ${2}_{(r-4)1}$, crosspolytopes of ${(r-4)}_{21}$, and rulings of ${S}_{r}$$\left({\mathbf{F}}_{r-1}\right).$

$\begin{array}{c}\hfill \mathbf{Del}\phantom{\rule{4pt}{0ex}}\mathbf{Pezzo}\mathbf{Surface}{\mathit{S}}_{\mathit{r}}\hfill \\ \hfill \left(\begin{array}{c}\hfill \mathbf{Blown}-\mathbf{up}\hfill \\ \hfill \mathbf{Hirzebruch}\phantom{\rule{4pt}{0ex}}{\mathbf{F}}_{\mathit{r}-\mathbf{1}}\hfill \end{array}\right)\hfill \end{array}$ | ${\mathit{S}}_{\mathbf{3}}$ (${\mathbf{F}}_{\mathbf{2}}$) | ${\mathit{S}}_{\mathbf{4}}$ (${\mathbf{F}}_{\mathbf{3}}$) | ${\mathit{S}}_{\mathbf{5}}$ (${\mathbf{F}}_{\mathbf{4}}$) | ${\mathit{S}}_{\mathbf{6}}$ (${\mathbf{F}}_{\mathbf{5}}$) | ${\mathit{S}}_{\mathbf{7}}$ (${\mathbf{F}}_{\mathbf{6}}$) | ${\mathit{S}}_{\mathbf{8}}$ (${\mathbf{F}}_{\mathbf{7}}$) |
---|---|---|---|---|---|---|

rulings | 3 | 5 | 10 | 27 | 126 | 2160 |

${2}_{\left(r-4\right)1}$ | ${2}_{-11}$ | ${2}_{01}$ | ${2}_{11}$ | ${2}_{21}$ | ${2}_{31}$ | ${2}_{41}$ |

vertices of ${2}_{\left(r-4\right)1}$ | 3 | 5 | 10 | 27 | 126 | 2160 |

${(r-4)}_{21}$ | $-{1}_{21}$ | ${0}_{21}$ | ${1}_{21}$ | ${2}_{21}$ | ${3}_{21}$ | ${4}_{21}$ |

midrule $(r-1)$-crosspolytopes | 3 | 5 | 10 | 27 | 126 | 2160 |

**Table 7.**Vertices of ${1}_{(}r-4)2$, $(r-1)$-simplexes in ${(r-4)}_{21}$ and exceptional systems of ${S}_{r}$$\left({\mathbf{F}}_{r-1}\right)$.

$\begin{array}{c}\hfill \mathbf{Del}\hfill \\ \hfill \mathbf{Pezzo}\mathbf{Surface}{\mathit{S}}_{\mathit{r}}\hfill \\ \hfill \left(\begin{array}{c}\hfill \mathbf{Blown}-\mathbf{up}\hfill \\ \hfill \mathbf{Hirzebruch}\phantom{\rule{4pt}{0ex}}{\mathbf{F}}_{\mathit{r}-\mathbf{1}}\hfill \end{array}\right)\hfill \end{array}$ | ${\mathit{S}}_{\mathbf{3}}$ (${\mathbf{F}}_{\mathbf{2}}$) | ${\mathit{S}}_{\mathbf{4}}$ (${\mathbf{F}}_{\mathbf{3}}$) | ${\mathit{S}}_{\mathbf{5}}$ (${\mathbf{F}}_{\mathbf{4}}$) | ${\mathit{S}}_{\mathbf{6}}$ (${\mathbf{F}}_{\mathbf{5}}$) | ${\mathit{S}}_{\mathbf{7}}$ (${\mathbf{F}}_{\mathbf{6}}$) | ${\mathit{S}}_{\mathbf{8}}$ (${\mathbf{F}}_{\mathbf{7}}$) |
---|---|---|---|---|---|---|

exceptional systems | 2 | 5 | 16 | 72 | 576 | $17,520$ |

${1}_{\left(r-4\right)2}$ | ${1}_{-11}$ | ${1}_{01}$ | ${1}_{11}$ | ${1}_{21}$ | ${1}_{31}$ | ${1}_{41}$ |

vertices of ${1}_{\left(r-4\right)2}$ | 2 | 5 | 16 | 72 | 576 | $17,280$ |

${(r-4)}_{21}$ | $-{1}_{21}$ | ${0}_{21}$ | ${1}_{21}$ | ${2}_{21}$ | ${3}_{21}$ | ${4}_{21}$ |

$(r-1)$-simplexes | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2$ | 5 | 16 | 72 | 576 | $17,280$ |

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Lee, J.-H.; Shin, Y.
E-Polytopes in Picard Groups of Smooth Rational Surfaces. *Symmetry* **2016**, *8*, 27.
https://doi.org/10.3390/sym8040027

**AMA Style**

Lee J-H, Shin Y.
E-Polytopes in Picard Groups of Smooth Rational Surfaces. *Symmetry*. 2016; 8(4):27.
https://doi.org/10.3390/sym8040027

**Chicago/Turabian Style**

Lee, Jae-Hyouk, and YongJoo Shin.
2016. "E-Polytopes in Picard Groups of Smooth Rational Surfaces" *Symmetry* 8, no. 4: 27.
https://doi.org/10.3390/sym8040027