Table Of ContentConfiguration of lines in del Pezzo surfaces
0 with Gosset Polytopes
1
0
2 Jae-Hyouk Lee
n
a January 23, 2010
J
3
2
Abstract
]
G Inthisarticle,westudythedivisorclassesofdelPezzosurfaces,which
are written as the sum of distinct lines with fixed intersection according
A
to theinscribed simplexes and crosspolytopes in Gosset polytopes.
. We introducethe k-Steinersystem and cornered simplexes, and char-
h
t acterize theconfigurations of inscribed m(<4)-simplexes with them.
a Higher dimensional inscribed m(4≤m≤7)-simplexes exist in 421 in
m the Picard group of del Pezzo surface S8 of degree 1.The configurations
[ of 4-and7-simplexesarerelated torulingsin S8. Andtheconfigurations
of 5- and 6-simplexes correspond the skew 3-lines and skew 7-lines in S8.
1
In particular, theseven lines in a 6-simplex produce a Fano plane.
v
WealsostudytheinscribedcrosspolytopesandhypercubesintheGos-
4
7 set polytopes.
1
4
.
1
0
0
1
:
v
i
X
r
a
1
Contents
1 Introduction 3
2 Preliminaries 6
2.1 Regular Polytopes and Gosset Polytopes . . . . . . . . . . . . . . 6
2.2 Gosset Polytopes in the Picard Groups of del Pezzo Surfaces . . 8
2.3 Monoidal transform of del Pezzo surfaces . . . . . . . . . . . . . 11
3 Steiner Systems of del Pezzo Surfaces 13
4 Inscribed regular polytopes in Gosset polytopes 15
4.1 Inscribed 1-degree simplexes in Gosset polytopes . . . . . . . . . 15
4.1.1 Inscribed 1-degree 1-simplexes and 2-simplexes . . . . . . 16
4.1.2 Inscribed 1-degree 3-simplexes . . . . . . . . . . . . . . . 18
4.2 Inscribed 1-degree 4,5,6 and 7-simplexes in 4 . . . . . . . . . . 25
21
4.2.1 Rulings and inscribed 1-degree 4-simplexes . . . . . . . . 25
4.2.2 Skew 3-lines and inscribed 1-degree 5-simplexes . . . . . . 28
4.2.3 Fano planes and inscribed 1-degree 6-simplexes . . . . . . 29
4.2.4 Rulings and inscribed 1-degree 7-simplexes . . . . . . . . 32
4.3 Inscribed 2-degree simplexes and 3-degree simplexes . . . . . . . 33
4.4 Inscribed crosspolytopes in Gosset polytopes . . . . . . . . . . . 35
5 Hypercubes in Gosset polytopes 36
5.1 3-cubes in 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . 36
21 21
5.2 4-cubes in 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
21
2
1 Introduction
In this article, we study the divisor classes of del Pezzo surfaces which are
written as the sum of distinct lines with fixed intersection, and we explore the
configurationoflines producingthedivisorsalongthe geometryofGossetpoly-
topes. As the vertices of the Gosset polytopes are corresponded to lines, the
divisorclassesweconsiderarerelatedtopolytopesintheGossetpolytopes. Be-
causeGossetpolytopesarecompactconvexpolytopeswithsymmetries,divisor
classes in this article are naturally finite, and the corresponding configurations
have symmetries leading us interesting results hereinafter.
A del Pezzo surface is a smooth irreducible surface S whose anticanonical
r
class K is ample. Each del Pezzo surfaces can be constructed by blowing
− Sr
up r 8 points from P2 unless it is P1 P1. A line in Pic S is a divisor class
r
≤ ×
l with l2 =l K = 1. By the adjunctionformula,a line l contains arational
· Sr −
smooth curve in S . This rational curve is embedded to a line in P9−r along
r
the embedding given by K , when S is very ample. The set of lines L in
|− Sr| r r
Pic S is finite, and its symmetry group is the Weyl group E .
r r
TheGosset polytopes (r 4) ,3 r 8,arether-dimensionalsemiregular
21
− ≤ ≤
polytopes discovered by Gosset whose symmetry groups are the Coxeter group
E . Thevertexfigureof(r 4) is(r 5) andthefacetsof(r 4) areregular
r − 21 − 21 − 21
(r 1)-dimensionalsimplexesα and(r 1)-dimensionalcrosspolytopesβ .
r−1 r−1
− −
But all the lower dimensional subpolytopes are regular simplexes. The set of
verticesin (r 4) is bijective to L the setoflines in Pic S ([6][14]). In fact,
21 r r
−
author[11] showedthatthe convexhullof L in Pic S is the Gossetpolytopes
r r
(r 4) . These are the Gosset polytopes considered in this article. As the
21
−
subsets of vertices in (r 4) , equivalently lines in L , representthe polytopes
21 r
−
in(r 4) , the divisorclassesunder ourconsiderationandthe configurationof
21
−
lines in them are closely related to the the polytopes in n (r 4) .
21
−
The sum of m-lines with 0-intersection in Pic S is called a skew m-line
r
D in S ([11] or subsection 2.2). According to the correspondence between
m r
the lines in L and vertices in (r 4) , Lm the set of skew m-lines in S is
r − 21 r r
bijectively related to the set of the regular m-simplexes in (r 4) . Thus the
21
−
divisor classes D in Pic S exist for 1 m r and satisfy D2 = m and
m r ≤ ≤ m −
K D = m. In fact, skew m-lines are equivalent to the divisor classes with
Sr m −
D2 = m and K D = m, when 1 m 3. As a line in Pic S gives a
− Sr − ≤ ≤ r
rational map from S to S by blowing down the exceptional curve in the
r r−1
line,the skewr-lines inPicS iscorrespondedtoarationalmapfromS toP2.
r r
Therefrom,werelatetheskewr-linesinPicS todivisorclassesD withD2 =1
r t t
and K D = 3. This divisor class in Pic S is called an exceptional system,
Sr t − r
and its linear system gives a regular map to P2. By a transformation given by
K +3D , the set of exceptional system in Pic S and Lr are bijectively
Sr t Er r r
related for 3 r 7. When r = 8, the set of exceptional systems has two
≤ ≤
orbits of E action. One orbit corresponds to the set of skew 8-lines in S , and
8 8
the other orbit corresponds to the set of E -roots. Thus an exceptional system
8
01991Mathematics SubjectClassification: 51M20,14J26
3
D in satisfies either 3K +2d=D for a root d or 3D +K =D for
t E8 − S8 t t S8 1
D is a skew 8-lines. These two orbits in play key roles later on.
1 8
E
As the regular (r 1)-simplexes, a type of facets in (r 4) , are related to
21
− −
exceptional systems, the (r 1)-crosspolytopes in (r 4) , the other type of
21
− −
facets, are also connected to special divisor classes in Pic S so called rulings.
r
ArulingisadivisorclassD inPicS withD2 =0andK D = 2whichgives
r Sr −
a fibration of S over P1. The F set of rulings in Pic S is bijective to the set
r r r
of (r 1)-crosspolytopes in the (r 4) ([11]). Rulings are studied in many
21
− −
different directions. As the lines play the generators in Cox rings, the rulings
determines the relations for the rings by Batyrev and Popov ([1]). And rulings
are also applied to the geometry of the line bundles over del Pezzo surfaces via
therepresentationtheorybyLeungandZhang([12][13]).EachrulinginPicS is
r
bijectively related to the sum of two lines L with 1-intersection. Here, each of
r
the sum canbe writtenby (r 1)different pairof lines whichare corresponded
−
to the (r 1)-bipolar pairs of vertices in an (r 1)-crosspolytope in (r 4) .
21
− − −
Now, the remain cases ofthe sums oflines in L have positive intersections.
r
Thedistinctlineswithfixedpositiveintersectionsarecorrespondedtosimplexes
whose vertices are in the vertices of (r 4) but their edge are not. Since
21
−
(r 4) isconvex,thesesimplexesareinscribedin(r 4) ,andthebarycenters
21 21
− −
of the simplexes are corresponded to the sums of the distinct lines with fixed
positiveintersection.Therefromwecallthe setofm-lines l ,...,l inL with
1 m r
{ }
fixed intersection b > 0 as an inscribed b-degree m-simplex in (r 4) or an
21
−
Ar (b)-polytope, and the sum of line l +...+l is called as the center of the
m 1 m
Ar (b)-polytope. Herethek-degreeisthecommonintersectionbetweenvertices,
m
i.e. lines, in the Ar (b)-polytope and also equivalently implies that the length
m
of edges is 2(1+b). As the maximal possible intersection between lines in
Lr is 3, thepre is no inscribed simplexes with degree m > 3. As the inscribed
1-degree 1-simplexes in (r 4) are rulings, we also assume m > 1 from now
21
−
on. When r =6, A6(1)-polytopes are the biggest possible one in 2 and there
2 21
is noAr (b)-polytope with m>1.Whenr =7, A7(1)-polytopes arethe biggest
m 3
possibleonein3 with1-degree,andA7(2)-polytopesaretheonlypossibleone
21 1
with k > 2. When r = 8,A8 (1)-polytopes exist up to m = 7 because the root
m
space of S is 8-dimensional, and A8(2), A8(2) and A8(3)-polytopes also exist.
8 1 2 1
ForAr (1)-simplexesin2 ,3 and4 form 3,theconfigurationsoflines
m 21 21 21 ≤
inthe simplexescanbe obtainedby k-Steinersystemandmonoidaltransforms.
A Steiner system S(x,y,z) is a type of block design system givenby a fam-
ily of y-element subsets, called as blocks, in z-element total set where each
x-element in the total set is contained exactly one subset in the family. For ex-
ample,S(2,3,7)representsthefamousFanoprojectiveplane. Thedeterministic
nature of blocks of Steiner systems can be found in the Steiner triplet in cubic
surfaces which may not be honest Steiner system. But we focus the determin-
istic nature of the systems and define (k,S ), k-Steiner system in S which is
r r
S
a family of subsets of L where each (k 1)-lines in L with constant intersec-
r r
−
tions to each other determines exactly one subset in the family. A typical type
ofk-Steinersystems isfoundwhenthe sumsofthe linesinallthe blocksis con-
stant. Inotherwords,allblocksinthek-Steinersystemarecorrespondedtothe
4
Ar (m)-simplexes for some m with common center. In section 3, we define k-
k−1
Steiner systems (2,S ), (2,S ), (3,S ), (3,S ), and (4,S ) which
A 7 A 8 B 6 B 8 C 7
S S S S S
are related to A7(2)-, A8(3)-, A6(1)-, A8(2)- and A7(1)-polytopes respectively.
1 1 2 2 3
To apply the monoidal transform for the inscribed simplexes, we introduce
a notion of cornered simplex. A simplex in (r 4) is called cornered if there
21
−
is a line l in L whose vertex figure contains the simplex, and the otherwise
r
is called uncornered. Here the vertex figure of l in (r 4) is the subset of
21
−
lines in L with 0-intersection with l. Thus a simplex cornered by a line l is
r
preserved by the blow down map πr : S S given by the line l. As there
l r → r−1
is no Ar (1)-polytopes in 2 , 3 for m > 3, all the A8 (1)-polytopes in 4
m 21 21 m>3 21
are uncornered. For A8(1)-polytopes in 4 , both cases appear. The centers
3 21
of A8(1)-polytopes in 4 are bijectively related to the exceptional system in
3 21
, and the centers of cornered (resp. uncornered) A8(1)-polytopes correspond
E8 3
roots (skew 8-lines) in S .
8
TheAr(1)-polytopesexistforr =6,7,8. AlltheA6(1)-polytopesin2 share
2 2 21
acentersothattheconfigurationofA6(1)-polytopesin2 isgivenby (3,S ),3-
2 21 SB 6
Steiner system in S . The set of A7(1) (resp. A8(1))-polytopes is equivalently
6 2 2
the set of lines in Pic S (resp. skew 2-lines in Pic S ). The A7(1) (resp.
7 8 2
A8(1))-polytopes are corneredby a line (resp. a skew 2-linein PicS ), andthe
2 8
configurationof A7(1) (resp. A8(1))-polytopes of a fixed center is also givenby
2 2
(3,S ) along the monomial transform.
B 6
S
TheAr(1)-polytopesexistforr =7,8. AlltheA7(1)-polytopesin2 sharea
3 3 21
centersothattheconfigurationofA7(1)-polytopesin3 isgivenby (4,S ),4-
3 21 SC 7
Steiner system in S . The set of cornered A8(1)-polytopes is equivalently the
7 3
set of roots in Pic S , and the configuration of cornered A8(1)-polytopes of a
8 3
fixed center is also given by (4,S ) along the monomial transform. On the
C 7
S
other hand, the set of uncornered A8(1)-polytopes is equivalent to the set of
3
7-simplexes in 4 , and the configuration of uncornered A8(1)-polytopes of a
21 3
fixed center is obtained as the set of 4-skew edges in a 7-simplex in 4 .
21
The Ar (1)-polytopes for m > 3 exist only in 4 . As A8 (1)-polytopes
m 21 m>3
are uncornered, the configuration for these are far from being uniform. But
as the uncornered A8(1)-polytopes are related to the 7-simplexes in 4 ,the
3 21
A8(1) (resp.A8(1))-polytopes are related to skew 3-lines (resp. skew 7-lines),
5 6
and the A8(1) and A8(1)-polytopes are related to rulings in Pic S .
4 7 8
An A8(1)-polytope can be obtained by adding proper line to an uncornered
4
A8(1)-polytope. AndthisA8(1)-polytopecontainsuniquecornedA8(1)-polytope
3 4 3
in it, which gives a line l in L according to the correspondence between the
8
roots (in fact, lines when r = 8) in Pic S and the centers of cornered A8(1)-
8 3
polytope in 4 . This line l intersects by one with the line l′ remained line in
21
A8(1)-polytope after taking off the corned A8(1)-polytope. Thus l+l′ gives a
4 3
ruling in Pic S . It turns out this is true for all the A8(1)-polytopes. Thus for
8 4
each center D of A8(1)-polytope, there is unique line lS8 in L and the center
5 4 D5 8
A of an cornered A8(1)-polytope such that D = A +lS8. Furthermore,
D5 3 5 D5 D5
the set of the centers of A8(1)-polytope in 4 is bijective to the following set
4 21
5
of order pairs of lines with 1-intersection defined as
F˜ := (l ,l ) l ,l L with l l =1 .
8 1 2 1 2 8 1 2
{ | ∈ · }
Moreover, all the A8(1)-polytopes with center D in 4 share a common line
4 5 21
l ,andD l isthe commoncenterfortheunique corneredA8(1)-polytope
D5 5− D5 3
in each A8(1)-polytope.
4
The set of centers of A8(1)-polytopes is equivalent to the set of skew 3-lines
5
inPicS . As askew3-lineisgivenbyuniquetriple oflines with0-intersection,
8
anA8(1)-polytopecontainsuniquetripleofcorneredA8(1)-polytopesinit. This
5 3
characterizationgives the configuration of A8(1)-polytopes.
5
JustlikeA8(1)-polytopes,eachcenterofA8(1)-polytopegivesaskew7-lines
5 6
in 4 which is given by the unique choice of seven lines with 0-intersection,
21
and the uniqueness is the key to study the configuration of A8(1)-polytopes in
6
4 . Thus for each A8(1)-polytope, there are seven cornered A8(1)-polytopes
21 6 3
in it. Furthermore, for each cornered A8(1)-polytope in a A8(1)-polytope,
3 6
the remained three lines form an A8(1)-polytope which is not contained in
2
any corneredA8(1)-polytope in the 6-simplex. We call the triplet of lines Fano
3
block andshowthateachA8(1)-polytopecontainssevenFanoblocksinit. More-
6
over,the seven lines in a A8(1)-polytope and its Fano blocks produce a Steiner
6
system S(2,3,7) which is known as Fano plane.
TheconfigurationofA8(1)-polytopesin4 isclosetothatofA8(1)-polytopes.
7 21 4
Here each center of A8(1)-polytopes gives a ruling in 4 . Furthermore, each
7 21
center of A8(1)-polytopes can be written as the sum of two centers of cornered
7
A8(1)-polytopes where they are in the vertex figures of the bipolar pair of lines
3
in the corresponding ruling.
The configurations of inscribed Ar (b)-polytopes in (r 4) in degree b >
m − 21
1 are obtained by the k-Steiner system and the monoidal transforms.
We also consider the inscribed crosspolytopes in (r 4) which exists only
21
−
for r = 8. And as the applications of the configurations of lines for Ar (b)-
m
polytopes in (r 4) and crosspolytopes in 4 , we study the hypercubes in
21 21
−
3 and 4 .
21 21
As Coxeter[7] related his study on 4 to the Cayley integral numbers, an-
21
other direct application of the configuration of lines in del Pezzo surfaces can
be found in the octonions numbers. We discuss this in another article.
2 Preliminaries
2.1 Regular Polytopes and Gosset Polytopes
In this subsection, we review the general theory on regular polytopes that we
useinthisarticleandafamilyofsemiregularpolytopesknownasGossetfigures
(k accordingtoCoxeter). Here,weonly presentgeneralfacts,andthe further
21
detail of them can be found [3][4][5]and [11].
6
Let P be a convex n-polytope in an n-dimensional Euclidean space. For
n
each vertex V, the midpoints of all the edges emanating from a vertex V in P
n
form an (n 1)-polytope if they lie in a hyperplane, and this (n 1)-polytope
− −
is called the vertex figure of P at V. In this article, the vertices on the other
n
ends of the edges emanating from the vertex V also form an (n 1)-polytope,
−
and we also call this (n 1)-polytope the vertex figure of V in P .
n
−
Aregular polytopeP (n 2)isapolytopewhosefacetsandvertexfigureat
n
≥
eachvertexare regular. In particular,a polygonP is regularif it is equilateral
2
and equiangular. Naturally, the facets of regular P are all congruent, and the
n
vertex figures are all the same.
In this article, we consider two classes of regular polytopes.
(1) A regular simplex α is an n-dimensional simplex with equilateral
n
edges. Note α is a pyramid based on α . Thus the facets of a regular
n n−1
simplex α is a regular simplex α , and the vertex figure of α is also α .
n n−1 n n−1
For example, α is a line-segment, α is an equilateral triangle, and α is a
1 2 3
tetrahedron. For a regular simplex α , only regular simplex α , 0 k n 1
n k
≤ ≤ −
appears as subpolytopes.
(2) A crosspolytope β is an n-dimensional polytope whose 2n-vertices
n
are the intersects between an n-dimensional Cartesian coordinate frame and
a sphere centered at the origin. Note β is a bipyramid based on β , and
n n−1
the n-vertices in β form α if the choice is made as one vertex from each
n n−1
Cartesian coordinate line. So the vertex figure of a crosspolytope β is also a
n
crosspolytope β , and the facets of β is α . For instance, β is a line-
n−1 n n−1 1
segment, β is a square, and β is an octahedron. For a crosspolytope β , only
2 3 n
regular simplex α , 0 k n 1 appears as subpolytopes.
k
≤ ≤ −
ApolytopeP iscalledsemiregular ifitsfacetsareregularanditsverticesare
n
equivalent, namely, the symmetry group of P acts transitively on the vertices
n
of P .
n
Here, weconsider the semiregulark polytopes discoveredby Gossetwhich
21
are(k+4)-dimensionalpolytopeswhosesymmetrygroupsaretheCoxetergroup
E . Note that the vertex figure of k is (k 1) and the facets of k are
k+4 21 − 21 21
regular simplexes α and crosspolytopes β . The list of k polytopes is
k+3 k+3 21
following.
k E k -polytopes
k+4 21
1 A A triangular prism
1 2
− ×
0 A rectified 5-cell
4
1 D demipenteract
5
2 E E -polytope
6 6
3 E E -polytope
7 7
4 E E -polytope
8 8
ListofGossetPolytopesk21
For k = 1, the facets of k -polytopes are the regular simplex α and
21 k+3
6 −
the crosspolytopeβ . But allthe lowerdimensionalsubpolytopes areregular
k+3
simplexes. When k = 1, the vertex figure in 1 is an isosceles triangle
21
− −
7
instead of an equilateral triangle, and its facets are the regular triangle α and
2
the square β .
2
Thetableoftotalnumbersofsubpolytopesink isveryusefulinthisarticle
21
and presented below.
E -polytope(k ) 1 0 1 2 3 4
k+4 21 21 21 21 21 21 21
−
β 3 5 10 27 126 2160
k+3
vertex 6 10 16 27 56 240
α 9 30 80 216 756 6720
1
α 2 30 160 720 4032 60480
2
α 5 120 1080 10080 241920
3
α 16 648 12096 483840
4
α 72 6048 483840
5
α 576 207360
6
α 17280
7
Numberofsubpolytopesink21
2.2 Gosset Polytopes in the Picard Groups of del Pezzo
Surfaces
The del Pezzo surfaces are smooth irreducible surfaces S whose anticanonical
r
class K is ample. We can construct the del Pezzo surfaces by blowing up
− Sr
r 8 points from P2 unless it is P1 P1. In particular, it is very well known
≤ ×
thatthere are27linesona cubic surfaceS andthe configurationofthese lines
6
is acted by the Weyl groupE ([8][9][10]). The set of 27-lines in S are bijective
6 6
to the set of vertices of a Gosset 2 polytope. The similar correspondences
21
are found between the 28-bitangents in S and 3 polytopes, and between the
7 21
tritangent planes for S and 4 polytopes. The correspondence between lines
8 21
inS andverticesin2 isappliedtostudythegeometryof2 byCoxeter([6]),
6 21 21
and the correspondence is extended to each 3 r 8([14]).
≤ ≤
We denote such a del Pezzo surface by S and the corresponding blow up
r
by π : S P2. And K2 = 9 r is called the degree of the del Pezzo
r r → Sr −
surface. Each exceptional curve and the corresponding class given by blowing
up is denoted by e , and both the class of π∗(h) in S and the class of a line h
i r r
in P2 are referred as h. Then, we have
h2 =1, h e =0, e e = δ for 1 i,j r,
i i j ij
· · − ≤ ≤
and the Picard group of S is
r
Pic S Zh Ze ... Ze
r 1 r
≃ ⊕ ⊕ ⊕
r
with the signature (1, r). And K = 3h+ e .
− Sr − i=1 i
P
8
TheinnerproductgivenbytheintersectiononPicS inducesanegativedef-
r
inite metric on (ZK )⊥ in Pic S where we canalso define naturalreflections.
Sr r
To define reflections on (ZK )⊥ in Pic S , we consider a root system
Sr r
R := d Pic S d2 = 2, d K =0 ,
r { ∈ r | − · Sr }
with simple roots d = h e e e ,d = e e , 1 i r 1. Each
0 1 2 3 i i i+1
element d in R defines a r−eflect−ion o−n (ZK )⊥ in−Pic S ≤ ≤ −
r Sr r
σ (D):=D+(D d)d for D (ZK )⊥
d · ∈ Sr
andthe correspondingWeylgroupW(S ) is E where 3 r 8. Furthermore,
r r
≤ ≤
the reflection σ on (ZK )⊥ can be used to obtain a transformation both on
d Sr
Pic S and on Pic S Q Qh Qe ... Qe via the linear extension of
r r 1 r
⊗ ≃ ⊕ ⊕ ⊕
the intersections of divisors in Pic S . Here Pic S Q is a vector space with
r r
⊗
the signature (1, r).
−
In this article, we deal with divisor classes D satisfying D K =α, D2 =
· Sr
β where α and β are integers, which are preserved by the extended action of
W(S ). In particular, W(S ) acts as a reflection group on the set of divisor
r r
classes with D K =α. Therefrom, we define an affine hyperplane section in
· Sr
Pic S Q defined by
r
⊗
H˜ := D Pic S Q D K =b
b { ∈ r ⊗ |− · Sr }
where b is an arbitrary real number and an affine hyperplane section H :=
b
H˜ Pic S in Pic S . By the ample condition of K and the Hodge index
b∩ r r − Sr
theorem,the inner product onPic S induces a negative definite metric on H .
r b
In fact, the induced metric is defined on Pic S Q, and we can also consider
r
the induced norm by fixing a center b K in⊗the affine hyperplane section
9−r Sr
D K =b in Pic S Q. This norm is also negative definite. Furthermore,
−the ·reflSerction σ on Pric⊗S Q induces a reflection on H˜ . Generically, the
d r b
hyperplanes in Pic S Q in⊗duce affine hyperplanes in H˜ and they may not
r b
⊗
share a common point. But the reflection hyperplane of each reflection σ in
d
Pic S Q gives a hyperplane in H˜ containing the center because it is given
r b
by a con⊗dition K d=0. Thus, the Weyl groupW(S ) acts on H˜ and H as
Sr· r b b
an reflection group.
Now, we want construct Gosset polytopes (r 4) in Pic S Q as the
21 r
− ⊗
convex hull of the set of special classes in Pic S , which is known as lines. A
r
line in Pic S is equivalently a divisor class l with l2 = 1 and K l = 1,
r − Sr · −
and the set of lines is given as
L := D Pic(S ) D2 = 1,K D = 1 .
r { ∈ r | − Sr − }
As the Weyl group W(S ) acts as an affine reflection group on the affine hy-
r
perplane given by D K = 1, W(S ) acts on the set of lines in Pic S .
· Sr − r r
Therefrom, we construct a semiregular polytope in Pic S Q whose vertices
r
⊗
are exactly the lines in Pic S . Since the symmetry group of the polytope is
r
W(S ), the polytope is actually a Gosset polytope (r 4) .
r 21
−
9
Remark: Each line l in L contains a exceptional curve which produces a
r
blow down map from S to S . We denote the map as πr :S S .
r r−1 l r → r−1
For a Gosset polytope (r 4) , subpolytopes are regular simplexes excepts
21
−
the facets which consist of (r 3)-simplexes and (r 3)-crosspolytopes. Since
− −
the subpolytopes in (r 4) are basically configurations of vertices, we obtain
21
−
naturalcharacterizationofsubpolytopesin(r 4) asdivisorclassesinPicS .
21 r
−
Remark : To identify each subpolytope in (r 4) , we want to use the
21
−
barycenterofthe subpolytope. Eachvertexofthe polytope (r 4) represents
21
−
a line in S , and the honest centers of simplexes (resp. crosspolytopes ) are
r
written as (l +...+l )/k (resp.(l′ +l′)/2) in H˜ which may not be elements
1 k 1 2 1
in Pic S . Therefore, alternatively, we choose (l +...+l ) as the center of a
r 1 k
subpolytope so that (l +...+l ) is in Pic S .
1 k r
We use the algebraic geometry of del Pezzo surfaces to identify the divi-
sor classes corresponding to the subpolytopes in (r 4) . For this purpose,
21
−
we consider the following set of divisor classes which are called skew a-lines,
exceptional systems and rulings in Pic S .
r
La := D Pic(S ) D =l +...+l , l disjoint lines in S
r { ∈ r | 1 a i r}
:= D Pic(S ) D2 =1, K D = 3
Er { ∈ r | Sr · − }
F := D Pic(S ) D2 =0, K D = 2
r { ∈ r | Sr · − }
The skew a-line in La is an extension of the definition of lines in S . Each
r r
skewa-linerepresentsan(a 1)-simplexinan(r 4) polytope. Furthermore,
21
− −
each skew a-line, there is only one set of disjoint lines in L to present it. The
8
skewa-linesalsoholdsD2 = aandD K = a,andthe divisorclasseswith
− · Sr −
these conditions are equivalently skew a-lines for a 3.
≤
Theexceptional system in isadivisorclassinPicS whoselinearsystem
r r
E
givesaregularmapfromS toP2. Asthisregularmapcorrespondstoablowing
r
upfromP2 to S , naturallyexceptionalsystemsarerelatedto (r 1)-simplexes
r
−
in (r 4) polytopes, which is the one of two types of facets appearing in
21
−
(r 4) polytopes. In fact, by a transformation Φ from to Lr by
− 21 Er r
Φ(D ):=K +3D for D ,
t Sr t t ∈Er
theset isbijectivetothesetofthe(r 1)-simplexesin(r 4) polytopes,for
r 21
E − −
3 r 7. Whenr=8,thesetofexceptionalsystemshastwoorbits. Oneorbit
≤ ≤
with 17280elements corresponds to the set of skew 8-lines in S , and the other
8
orbitwith 240 elements corresponds to the set of E -roots because 3K + 2d
8 − S8
is an exceptional system for each E -root d. Thus an exceptional system D in
8 t
satisfies either 3K +2d=D for a root d or 3D +K =D for D is
Er − S8 t t S8 1 1
a skew 8-lines.
Theruling inF isadivisorclassinPicS whichgivesafibrationofS over
r r r
P1. And we show that the F is bijective to the set of (r 1)-crosspolytopes
r
−
10
