cubicsurface
system:macaulay2

<h1 style="text-align: center;">We investigate the 27 lines on a cubic surface</h1>

{{{id=1|
restart
printWidth=68
///


68
}}}

We start by constructing a cubic surface in P^3.

We choose 6 points, and blow them p

{{{id=4|
P2 = QQ[r,s,t]
p1 = trim minors(2, matrix"r,s,t;0,0,1")
p2 = trim minors(2, matrix"r,s,t;0,1,0")
p3 = trim minors(2, matrix"r,s,t;1,0,0")
p4 = trim minors(2, matrix"r,s,t;1,1,1")
p5 = trim minors(2, matrix"r,s,t;1,2,6")
p6 = trim minors(2, matrix"r,s,t;3,7,-2")
pts = {p1,p2,p3,p4,p5,p6}
I = intersect pts
netList I_*
P3 = QQ[a..d]
phi = map(P2,P3,gens I)
IX = kernel phi
RX = P3/IX
///

P2

PolynomialRing


ideal (s, r)

Ideal of P2


ideal (t, r)

Ideal of P2


ideal (t, s)

Ideal of P2


ideal (s - t, r - t)

Ideal of P2


ideal (3s - t, 6r - t)

Ideal of P2


ideal (2s + 7t, 2r + 3t)

Ideal of P2


{ideal (s, r), ideal (t, r), ideal (t, s), ideal (s - t, r -
---------------------------------------------------------------
t), ideal (3s - t, 6r - t), ideal (2s + 7t, 2r + 3t)}

List


                      2          2        2      2       2   
 ideal (157r*s*t - 65s t - 175r*t  + 83s*t , 314r t - 50s t -
 --------------------------------------------------------------
       2         2        2        2           2          2 
 509r*t  + 245s*t , 942r*s  + 1343s t - 3570r*t  + 1285s*t ,
 --------------------------------------------------------------
     2        2           2          2
 942r s + 595s t - 2625r*t  + 1088s*t )

 Ideal of P2


 +---------------------------------------+
 |              2          2        2    |
 |157r*s*t - 65s t - 175r*t  + 83s*t     |
 +---------------------------------------+
 |    2       2          2         2     |
 |314r t - 50s t - 509r*t  + 245s*t      |
 +---------------------------------------+
 |      2        2           2          2|
 |942r*s  + 1343s t - 3570r*t  + 1285s*t |
 +---------------------------------------+
 |    2        2           2          2  |
 |942r s + 595s t - 2625r*t  + 1088s*t   |
 +---------------------------------------+


 P3

 PolynomialRing


                          2          2        2      2       2          2         2        2        2           2          2      2        2           2          2
 map(P2,P3,{157r*s*t - 65s t - 175r*t  + 83s*t , 314r t - 50s t - 509r*t  + 245s*t , 942r*s  + 1343s t - 3570r*t  + 1285s*t , 942r s + 595s t - 2625r*t  + 1088s*t })

 RingMap P2 <--- P3


             3         2            2       2              
 ideal(46002a  - 41310a b + 10710a*b  + 650a c + 996a*b*c -
 --------------------------------------------------------------
     2         2        2       2                     2
 525b c + 50a*c  - 65b*c  - 628a d + 157b*c*d - 314a*d )

 Ideal of P3


 RX

 QuotientRing
}}}

{{{id=5|
X = Proj RX
dim X
degree X
codim singularLocus X
HH^0(OO_X)
for i from 0 to 10 list rank HH^0(OO_X(i))
///

X

ProjectiveVariety


2


3


infinity

InfiniteNumber


  1
QQ

QQ-module, free


{1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166}

List
}}}

Consider the strict transform of the lines through 
two of the 6 points:

{{{id=7|
I1 = preimage(phi,ideal(t))  -- these are lines in P^3 on X
I2 = preimage(phi,ideal(s))
I3 = preimage(phi,ideal(r))
///

ideal (b, a)

Ideal of P3


ideal (25c - 34d, 15a - d)

Ideal of P3


ideal (1479b + 425c - 835d, 493a + 85c - 138d)

Ideal of P3
}}}

There are actually 15 of these

{{{id=31|
netList subsets(pts,2)
///

+----------------------+------------------------+
|ideal (s, r)          |ideal (t, r)            |
+----------------------+------------------------+
|ideal (s, r)          |ideal (t, s)            |
+----------------------+------------------------+
|ideal (t, r)          |ideal (t, s)            |
+----------------------+------------------------+
|ideal (s, r)          |ideal (s - t, r - t)    |
+----------------------+------------------------+
|ideal (t, r)          |ideal (s - t, r - t)    |
+----------------------+------------------------+
|ideal (t, s)          |ideal (s - t, r - t)    |
+----------------------+------------------------+
|ideal (s, r)          |ideal (3s - t, 6r - t)  |
+----------------------+------------------------+
|ideal (t, r)          |ideal (3s - t, 6r - t)  |
+----------------------+------------------------+
|ideal (t, s)          |ideal (3s - t, 6r - t)  |
+----------------------+------------------------+
|ideal (s - t, r - t)  |ideal (3s - t, 6r - t)  |
+----------------------+------------------------+
|ideal (s, r)          |ideal (2s + 7t, 2r + 3t)|
+----------------------+------------------------+
|ideal (t, r)          |ideal (2s + 7t, 2r + 3t)|
+----------------------+------------------------+
|ideal (t, s)          |ideal (2s + 7t, 2r + 3t)|
+----------------------+------------------------+
|ideal (s - t, r - t)  |ideal (2s + 7t, 2r + 3t)|
+----------------------+------------------------+
|ideal (3s - t, 6r - t)|ideal (2s + 7t, 2r + 3t)|
+----------------------+------------------------+
}}}

{{{id=32|
apply(subsets(pts,2), pq -> (lin := (intersect(pq_0,pq_1))_0; lin))
///

{r, s, t, r - s, r - t, s - t, 2r - s, 6r - t, 3s - t, 4r - 5s
--------------------------------------------------------------
+ t, 7r - 3s, 2r + 3t, 2s + 7t, 9r - 5s - 4t, 46r - 20s - t}

List
}}}

{{{id=9|
Lines2 = apply(subsets(pts,2), pq -> ( lin := (intersect(pq_0,pq_1))_0; preimage(phi, ideal(lin))))
#Lines2
Lines2/codim
Lines2/degree
codim(Lines2_0 + Lines2_1) -- these two lines do not meet
///

{ideal (1479b + 425c - 835d, 493a + 85c - 138d), ideal (25c -
--------------------------------------------------------------
34d, 15a - d), ideal (b, a), ideal (17b - 6c + 6d, 187a - 23c
--------------------------------------------------------------
+ 23d), ideal (329b + 21c - 53d, 141a + 10c - 23d), ideal
--------------------------------------------------------------
(438b + 61c - 146d, 146a - c), ideal (102b - 19c + 38d, 34a -
--------------------------------------------------------------
3c + 6d), ideal (63b + 14c - 30d, 33a + 5c - 9d), ideal (113b
--------------------------------------------------------------
+ 69c - 113d, 113a - 4c), ideal (588b - 235c + 188d, 42a - 5c
--------------------------------------------------------------
+ 4d), ideal (833b + 141c - 329d, 119a + 6c - 14d), ideal (42b
--------------------------------------------------------------
- 13c + 2d, 42a - 5c + 4d), ideal (21b + 2c + 2d, 11a + c),
--------------------------------------------------------------
ideal (297b + 95c - 171d, 33a + 5c - 9d), ideal (1081b + 220c
--------------------------------------------------------------
- 506d, 141a + 10c - 23d)}

List


15


{2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}

List


{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

List


4
}}}

There are 6 conics through each set of 5 of the points the preimages of these are lines in P^3

{{{id=11|
intersect drop(pts,1)
J = ideal((intersect drop(pts,1))_0)
preimage(phi,J)
///

                                   2      2          2  
ideal (30r*s - 102r*t + 85s*t - 13t , 150s t - 828r*t  +
--------------------------------------------------------------
      2       3    2         2        2    3
835s*t  - 157t , 6r t - 15r*t  + 10s*t  - t )

Ideal of P2


                                  2
ideal(30r*s - 102r*t + 85s*t - 13t )

Ideal of P2


ideal (306b - 25c - 30d, 102a - 5c)

Ideal of P3
}}}

{{{id=12|
Lines3 = for i from 0 to 5 list (J = intersect drop(pts,{i,i});preimage(phi,ideal(J_0)))
netList Lines3
///

{ideal (306b - 25c - 30d, 102a - 5c), ideal (525b + 65c -
--------------------------------------------------------------
157d, a), ideal (1209b + 50c - 130d, 93a + 5c - 13d), ideal
--------------------------------------------------------------
(10220b - 63c - 146d, 146a - c), ideal (11865b - 616c - 904d,
--------------------------------------------------------------
113a - 4c), ideal (55b + 13c - 11d, 11a + c)}

List


+------------------------------------------+
|ideal (306b - 25c - 30d, 102a - 5c)       |
+------------------------------------------+
|ideal (525b + 65c - 157d, a)              |
+------------------------------------------+
|ideal (1209b + 50c - 130d, 93a + 5c - 13d)|
+------------------------------------------+
|ideal (10220b - 63c - 146d, 146a - c)     |
+------------------------------------------+
|ideal (11865b - 616c - 904d, 113a - 4c)   |
+------------------------------------------+
|ideal (55b + 13c - 11d, 11a + c)          |
+------------------------------------------+
}}}

There are 6 more lines on X: the exceptional curves of the blowup.

These can be obtained in a number of ways

One way is to construct the graph of the map 
<pre>   phi:P2 ---> P3</pre>
(as a bigraded ideal).

{{{id=14|
S = QQ[r,s,t][a..d]
IS = sub(I,S)
m = matrix{{a,b,c,d},IS_*}
Gr = saturate(minors(2,m), ideal"a,b,c,d")
Gr = trim saturate(Gr,IS)
primaryDecomposition Gr
///

S

PolynomialRing


                     2          2        2      2       2   
ideal (157r*s*t - 65s t - 175r*t  + 83s*t , 314r t - 50s t -
--------------------------------------------------------------
      2         2        2        2           2          2 
509r*t  + 245s*t , 942r*s  + 1343s t - 3570r*t  + 1285s*t ,
--------------------------------------------------------------
    2        2           2          2
942r s + 595s t - 2625r*t  + 1088s*t )

Ideal of S


| a                         b                         
| 157rst-65s2t-175rt2+83st2 314r2t-50s2t-509rt2+245st2
--------------------------------------------------------------
c                              d                             |
942rs2+1343s2t-3570rt2+1285st2 942r2s+595s2t-2625rt2+1088st2 |

        2       4
Matrix S  <--- S


            2        2           2          2              2  
ideal ((942r s + 595s t - 2625r*t  + 1088s*t )c + (- 942r*s  -
--------------------------------------------------------------
     2           2          2           2        2           2
1343s t + 3570r*t  - 1285s*t )d, (942r*s  + 1343s t - 3570r*t 
--------------------------------------------------------------
         2            2       2          2         2   
+ 1285s*t )b + (- 314r t + 50s t + 509r*t  - 245s*t )c,
--------------------------------------------------------------
     2        2           2          2            2       2   
(942r s + 595s t - 2625r*t  + 1088s*t )b + (- 314r t + 50s t +
--------------------------------------------------------------
      2         2         2       2          2         2    
509r*t  - 245s*t )d, (314r t - 50s t - 509r*t  + 245s*t )a +
--------------------------------------------------------------
                 2          2        2           2        2   
(- 157r*s*t + 65s t + 175r*t  - 83s*t )b, (942r*s  + 1343s t -
--------------------------------------------------------------
       2          2                      2          2  
3570r*t  + 1285s*t )a + (- 157r*s*t + 65s t + 175r*t  -
--------------------------------------------------------------
     2         2        2           2          2       
83s*t )c, (942r s + 595s t - 2625r*t  + 1088s*t )a + (-
--------------------------------------------------------------
              2          2        2
157r*s*t + 65s t + 175r*t  - 83s*t )d)

Ideal of S


                                                            
ideal (299013t*a + (56049s - 53550t)b + (- 4710r + 6715t)c +
--------------------------------------------------------------
                                                             
(4710s - 18683t)d, 99671s*a + (- 71655s - 55335t)b + (- 4867r
--------------------------------------------------------------
                                                              
- 9673t)c + (4867s + 23885t)d, 598026r*a + (59262s + 276675t)b
--------------------------------------------------------------
                                                  3         2 
+ (- 4980r + 48365t)c + (4980s - 119425t)d, 46002a  - 41310a b
--------------------------------------------------------------
          2       2                   2         2        2  
+ 10710a*b  + 650a c + 996a*b*c - 525b c + 50a*c  - 65b*c  -
--------------------------------------------------------------
    2                     2
628a d + 157b*c*d - 314a*d )

Ideal of S


                                                             
{ideal (299013t*a + (56049s - 53550t)b + (- 4710r + 6715t)c +
--------------------------------------------------------------
                                                             
(4710s - 18683t)d, 99671s*a + (- 71655s - 55335t)b + (- 4867r
--------------------------------------------------------------
                                                              
- 9673t)c + (4867s + 23885t)d, 598026r*a + (59262s + 276675t)b
--------------------------------------------------------------
                                                  3         2 
+ (- 4980r + 48365t)c + (4980s - 119425t)d, 46002a  - 41310a b
--------------------------------------------------------------
          2       2                   2         2        2  
+ 10710a*b  + 650a c + 996a*b*c - 525b c + 50a*c  - 65b*c  -
--------------------------------------------------------------
    2                     2
628a d + 157b*c*d - 314a*d )}

List
}}}

{{{id=15|
codim Gr -- so Gr defines a surface (which is isomorphic to X)
degree Gr_0
isHomogeneous Gr
///

3


{1, 1}

List


true
}}}

{{{id=16|
trim(Gr + sub(p1,S))
primaryDecomposition oo
saturate(trim(Gr + sub(p1,S)), ideal(r_S,s,t))
select(oo_*, f -> first degree f > 0) -- equations of the line
///

                                                              
ideal (s, r, 3255t*b + 569t*c - 1405t*d, 93t*a + 5t*c - 13t*d,
--------------------------------------------------------------
      3         2            2       2                   2   
46002a  - 41310a b + 10710a*b  + 650a c + 996a*b*c - 525b c +
--------------------------------------------------------------
     2        2       2                     2
50a*c  - 65b*c  - 628a d + 157b*c*d - 314a*d )

Ideal of S


                       3         2            2       2   
{ideal (t, s, r, 46002a  - 41310a b + 10710a*b  + 650a c +
--------------------------------------------------------------
               2         2        2       2              
996a*b*c - 525b c + 50a*c  - 65b*c  - 628a d + 157b*c*d -
--------------------------------------------------------------
      2
314a*d ), ideal (s, r, 3255b + 569c - 1405d, 93a + 5c - 13d)}

List


ideal (s, r, 3255b + 569c - 1405d, 93a + 5c - 13d)

Ideal of S


{3255b + 569c - 1405d, 93a + 5c - 13d}

List
}}}

{{{id=17|
Lines1 = apply(pts, p -> (J := saturate(trim(Gr + sub(p,S)), ideal(r_S,s,t));substitute(ideal select(J_*, f -> first degree f > 0), P3)))
netList Lines1
///

{ideal (3255b + 569c - 1405d, 93a + 5c - 13d), ideal (119b +
--------------------------------------------------------------
10d, 119a + 13d), ideal (c, a), ideal (5253b + 625c - 1373d,
--------------------------------------------------------------
1751a + 10c - 78d), ideal (12002b + 1775c - 4366d, 12002a +
--------------------------------------------------------------
445c - 1196d), ideal (104601b - 14950c + 37286d, 4981a - 365c
--------------------------------------------------------------
+ 897d)}

List


+------------------------------------------------------+
|ideal (3255b + 569c - 1405d, 93a + 5c - 13d)          |
+------------------------------------------------------+
|ideal (119b + 10d, 119a + 13d)                        |
+------------------------------------------------------+
|ideal (c, a)                                          |
+------------------------------------------------------+
|ideal (5253b + 625c - 1373d, 1751a + 10c - 78d)       |
+------------------------------------------------------+
|ideal (12002b + 1775c - 4366d, 12002a + 445c - 1196d) |
+------------------------------------------------------+
|ideal (104601b - 14950c + 37286d, 4981a - 365c + 897d)|
+------------------------------------------------------+
}}}

{{{id=33|
Lines = join(Lines1,Lines2,Lines3)
netList Lines
///

{ideal (3255b + 569c - 1405d, 93a + 5c - 13d), ideal (119b +
--------------------------------------------------------------
10d, 119a + 13d), ideal (c, a), ideal (5253b + 625c - 1373d,
--------------------------------------------------------------
1751a + 10c - 78d), ideal (12002b + 1775c - 4366d, 12002a +
--------------------------------------------------------------
445c - 1196d), ideal (104601b - 14950c + 37286d, 4981a - 365c
--------------------------------------------------------------
+ 897d), ideal (1479b + 425c - 835d, 493a + 85c - 138d), ideal
--------------------------------------------------------------
(25c - 34d, 15a - d), ideal (b, a), ideal (17b - 6c + 6d, 187a
--------------------------------------------------------------
- 23c + 23d), ideal (329b + 21c - 53d, 141a + 10c - 23d),
--------------------------------------------------------------
ideal (438b + 61c - 146d, 146a - c), ideal (102b - 19c + 38d,
--------------------------------------------------------------
34a - 3c + 6d), ideal (63b + 14c - 30d, 33a + 5c - 9d), ideal
--------------------------------------------------------------
(113b + 69c - 113d, 113a - 4c), ideal (588b - 235c + 188d, 42a
--------------------------------------------------------------
- 5c + 4d), ideal (833b + 141c - 329d, 119a + 6c - 14d), ideal
--------------------------------------------------------------
(42b - 13c + 2d, 42a - 5c + 4d), ideal (21b + 2c + 2d, 11a +
--------------------------------------------------------------
c), ideal (297b + 95c - 171d, 33a + 5c - 9d), ideal (1081b +
--------------------------------------------------------------
220c - 506d, 141a + 10c - 23d), ideal (306b - 25c - 30d, 102a
--------------------------------------------------------------
- 5c), ideal (525b + 65c - 157d, a), ideal (1209b + 50c -
--------------------------------------------------------------
130d, 93a + 5c - 13d), ideal (10220b - 63c - 146d, 146a - c),
--------------------------------------------------------------
ideal (11865b - 616c - 904d, 113a - 4c), ideal (55b + 13c -
--------------------------------------------------------------
11d, 11a + c)}

List


+------------------------------------------------------+
|ideal (3255b + 569c - 1405d, 93a + 5c - 13d)          |
+------------------------------------------------------+
|ideal (119b + 10d, 119a + 13d)                        |
+------------------------------------------------------+
|ideal (c, a)                                          |
+------------------------------------------------------+
|ideal (5253b + 625c - 1373d, 1751a + 10c - 78d)       |
+------------------------------------------------------+
|ideal (12002b + 1775c - 4366d, 12002a + 445c - 1196d) |
+------------------------------------------------------+
|ideal (104601b - 14950c + 37286d, 4981a - 365c + 897d)|
+------------------------------------------------------+
|ideal (1479b + 425c - 835d, 493a + 85c - 138d)        |
+------------------------------------------------------+
|ideal (25c - 34d, 15a - d)                            |
+------------------------------------------------------+
|ideal (b, a)                                          |
+------------------------------------------------------+
|ideal (17b - 6c + 6d, 187a - 23c + 23d)               |
+------------------------------------------------------+
|ideal (329b + 21c - 53d, 141a + 10c - 23d)            |
+------------------------------------------------------+
|ideal (438b + 61c - 146d, 146a - c)                   |
+------------------------------------------------------+
|ideal (102b - 19c + 38d, 34a - 3c + 6d)               |
+------------------------------------------------------+
|ideal (63b + 14c - 30d, 33a + 5c - 9d)                |
+------------------------------------------------------+
|ideal (113b + 69c - 113d, 113a - 4c)                  |
+------------------------------------------------------+
|ideal (588b - 235c + 188d, 42a - 5c + 4d)             |
+------------------------------------------------------+
|ideal (833b + 141c - 329d, 119a + 6c - 14d)           |
+------------------------------------------------------+
|ideal (42b - 13c + 2d, 42a - 5c + 4d)                 |
+------------------------------------------------------+
|ideal (21b + 2c + 2d, 11a + c)                        |
+------------------------------------------------------+
|ideal (297b + 95c - 171d, 33a + 5c - 9d)              |
+------------------------------------------------------+
|ideal (1081b + 220c - 506d, 141a + 10c - 23d)         |
+------------------------------------------------------+
|ideal (306b - 25c - 30d, 102a - 5c)                   |
+------------------------------------------------------+
|ideal (525b + 65c - 157d, a)                          |
+------------------------------------------------------+
|ideal (1209b + 50c - 130d, 93a + 5c - 13d)            |
+------------------------------------------------------+
|ideal (10220b - 63c - 146d, 146a - c)                 |
+------------------------------------------------------+
|ideal (11865b - 616c - 904d, 113a - 4c)               |
+------------------------------------------------------+
|ideal (55b + 13c - 11d, 11a + c)                      |
+------------------------------------------------------+
}}}

<p>Check that they all lie on $X$:</p>

{{{id=18|
tally apply(Lines, L -> isSubset(IX, L))
///


Tally{true => 27}

Tally
}}}

<p>theory tells us that these 27 lines are all  exceptional curves, i.e. have self  intersection -1 on X.</p>
<h2><strong><strong>Let's check that. How should we find intersection numbers of curves on X?</strong></strong></h2>

{{{id=20|
LinesR = apply(Lines, L -> sub(L,RX));
L = LinesR_0
ML = sheaf Hom(L,RX)
///

ideal (3255b + 569c - 1405d, 93a + 5c - 13d)

Ideal of RX


image {-1} | 3255b+569c-1405d 44207922a2-39698910ab+10292310b2-1752120ac+3091506bc+142250c2+5576094ad-5549310bd-544710cd+477700d2 |
      {-1} | 93a+5c-13d       30225bc+1250c2-41106bd-4950cd+4420d2                                                                |

                                    2
coherent sheaf on X, subsheaf of OO  (1)
                                   X
}}}

{{{id=21|
euler RX
///

1
}}}

{{{id=22|
euler ML
///

1
}}}

{{{id=37|
euler(ML ** ML) 
///

0
}}}

{{{id=24|
ML2 = ML ** ML
HH^0 ML2
HH^1 ML2
HH^2 ML2
///


cokernel {0} | 30225bc+1250c2-41106bd-4950cd+4420d2 14367574650a2-12902145750ab+3345000750b2-569439000ac+4678750c2+1812230550ad-437080098bd-12482850cd+8322860d2 0                                    0                                                                                                            30225bc+1250c2-41106bd-4950cd+4420d2 14367574650a2-12902145750ab+3345000750b2-569439000ac+4678750c2+1812230550ad-437080098bd-12482850cd+8322860d2 0                                    0                                                                                                            |
         {1} | -93a-5c+13d                          3091506a-1057875b-18715c+24479d                                                                              0                                    0                                                                                                            0                                    0                                                                                                            30225bc+1250c2-41106bd-4950cd+4420d2 14367574650a2-12902145750ab+3345000750b2-569439000ac+4678750c2+1812230550ad-437080098bd-12482850cd+8322860d2 |
         {1} | 0                                    0                                                                                                            30225bc+1250c2-41106bd-4950cd+4420d2 14367574650a2-12902145750ab+3345000750b2-569439000ac+4678750c2+1812230550ad-437080098bd-12482850cd+8322860d2 -93a-5c+13d                          3091506a-1057875b-18715c+24479d                                                                              0                                    0                                                                                                            |
         {2} | 0                                    0                                                                                                            -93a-5c+13d                          3091506a-1057875b-18715c+24479d                                                                              0                                    0                                                                                                            -93a-5c+13d                          3091506a-1057875b-18715c+24479d                                                                              |

                                    1       2           1
coherent sheaf on X, quotient of OO   ++ OO  (-1) ++ OO  (-2)
                                   X       X           X


  1
QQ

QQ-module, free


  1
QQ

QQ-module, free


0

QQ-module
}}}

{{{id=25|
intersectionNumber = (F,G) -> euler ring F - euler F - euler G + euler(F**G)
///

intersectionNumber

FunctionClosure
}}}

{{{id=26|
intersectionNumber := method()
intersectionNumber(CoherentSheaf,CoherentSheaf) := (F,G) -> euler ring F - euler F - euler G + euler(F**G)
///

{*Function*}

MethodFunction


{*Function[stdio:67:57-67:104]*}

FunctionClosure
}}}

{{{id=27|
LinesM = apply(Lines, L -> sheaf Hom(sub(L,RX),RX))
apply(LinesM, F -> intersectionNumber(F,F))
matrix for i from 0 to 26 list for j from 0 to 26 list intersectionNumber(LinesM_i, LinesM_j)
///
WARNING: Output truncated!  
<html><a target='_new' href='/home/wstein/46/cells/27/full_output.txt' class='file_link'>full_output.txt</a></html>



{image {-1} | 3255b+569c-1405d
       {-1} | 93a+5c-13d      
--------------------------------------------------------------
44207922a2-39698910ab+10292310b2-1752120ac+3091506bc+142250c2+
30225bc+1250c2-41106bd-4950cd+4420d2                          
--------------------------------------------------------------
5576094ad-5549310bd-544710cd+477700d2 |, image {-1} | 119b+10d
                                      |        {-1} | 119a+13d
--------------------------------------------------------------
38319666a2-34411230ab+8921430b2+541450ac+829668bc+41650c2-
437325bc+54145c2+974610bd-76895cd+328770d2                
--------------------------------------------------------------
4709306ad+3759210bd-59150cd+252900d2 |, image {-1} | c
                                     |        {-1} | a
--------------------------------------------------------------
46002a2-41310ab+10710b2+650ac+996bc+50c2-628ad-314d2 |, image
525b2+65bc-157bd                                     |       
--------------------------------------------------------------
{-1} | 5253b+625c-1373d
{-1} | 1751a+10c-78d   
--------------------------------------------------------------
8296598706a2-7450382430ab+1931580630b2+69847390ac+222180888bc+
35238875bc+137900c2-28681380bd-746320cd-2568540d2             
--------------------------------------------------------------
8618750c2+256318384ad-331884540bd+1647580cd-45212890d2 |,
                                                       | 
--------------------------------------------------------------
image {-1} | 12002b+1775c-4366d
      {-1} | 12002a+445c-1196d 
--------------------------------------------------------------
194896949412a2-175018324860ab+45375121260b2-4472365270ac+10708
3906651000bc+94680425c2-4521633480bd-289110190cd+93108600d2   
--------------------------------------------------------------
940526bc+377657875c2+16760841008ad-17440586280bd-1067115740cd+
                                                              
--------------------------------------------------------------
339893100d2 |, image {-1} | 104601b-14950c+37286d
            |        {-1} | 4981a-365c+897d      
--------------------------------------------------------------
67136836866a2-60289177230ab+15630527430b2+5868315340ac-2964302
-18056125bc+12280425c2+134038710bd-48623015cd+45325410d2      
--------------------------------------------------------------
682bc+502992750c2-13006815566ad+10857135510bd-2009910970cd+
                                                           
--------------------------------------------------------------
1884061580d2 |, image {-1} | 1479b+425c-835d
             |        {-1} | 493a+85c-138d  
--------------------------------------------------------------
657690594a2-590609070ab+153120870b2-104101880ac+116068962bc+
11302025bc+3732690c2-14287140bd-10593182cd+7359540d2        
--------------------------------------------------------------
18663450c2+175121488ad-165322620bd-59333440cd+44530550d2 |,
                                                         | 
--------------------------------------------------------------
image {-1} | 25c-34d
      {-1} | 15a-d  
--------------------------------------------------------------
3450150a2-3098250ab+803250b2+48750ac+74700bc+3750c2+182910ad-
23625b2+2925bc-6075bd-150cd-334d2                            

...



| -1 0  0  0  0  0  1  1  0  1  0  0  1  0  0  0  1  0  0  0 
| 0  -1 0  0  0  0  1  0  1  0  1  0  0  1  0  0  0  1  0  0 
| 0  0  -1 0  0  0  0  1  1  0  0  1  0  0  1  0  0  0  1  0 
| 0  0  0  -1 0  0  0  0  0  1  1  1  0  0  0  1  0  0  0  1 
| 0  0  0  0  -1 0  0  0  0  0  0  0  1  1  1  1  0  0  0  0 
| 0  0  0  0  0  -1 0  0  0  0  0  0  0  0  0  0  1  1  1  1 
| 1  1  0  0  0  0  -1 0  0  0  0  1  0  0  1  1  0  0  1  1 
| 1  0  1  0  0  0  0  -1 0  0  1  0  0  1  0  1  0  1  0  1 
| 0  1  1  0  0  0  0  0  -1 1  0  0  1  0  0  1  1  0  0  1 
| 1  0  0  1  0  0  0  0  1  -1 0  0  0  1  1  0  0  1  1  0 
| 0  1  0  1  0  0  0  1  0  0  -1 0  1  0  1  0  1  0  1  0 
| 0  0  1  1  0  0  1  0  0  0  0  -1 1  1  0  0  1  1  0  0 
| 1  0  0  0  1  0  0  0  1  0  1  1  -1 0  0  0  0  1  1  1 
| 0  1  0  0  1  0  0  1  0  1  0  1  0  -1 0  0  1  0  1  1 
| 0  0  1  0  1  0  1  0  0  1  1  0  0  0  -1 0  1  1  0  1 
| 0  0  0  1  1  0  1  1  1  0  0  0  0  0  0  -1 1  1  1  0 
| 1  0  0  0  0  1  0  0  1  0  1  1  0  1  1  1  -1 0  0  0 
| 0  1  0  0  0  1  0  1  0  1  0  1  1  0  1  1  0  -1 0  0 
| 0  0  1  0  0  1  1  0  0  1  1  0  1  1  0  1  0  0  -1 0 
| 0  0  0  1  0  1  1  1  1  0  0  0  1  1  1  0  0  0  0  -1
| 0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0  0  0  0  0 
| 0  1  1  1  1  1  1  1  0  1  0  0  1  0  0  0  1  0  0  0 
| 1  0  1  1  1  1  1  0  1  0  1  0  0  1  0  0  0  1  0  0 
| 1  1  0  1  1  1  0  1  1  0  0  1  0  0  1  0  0  0  1  0 
| 1  1  1  0  1  1  0  0  0  1  1  1  0  0  0  1  0  0  0  1 
| 1  1  1  1  0  1  0  0  0  0  0  0  1  1  1  1  0  0  0  0 
| 1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1 
--------------------------------------------------------------
0  0  1  1  1  1  1  |
0  1  0  1  1  1  1  |
0  1  1  0  1  1  1  |
0  1  1  1  0  1  1  |
1  1  1  1  1  0  1  |
1  1  1  1  1  1  0  |
1  1  1  0  0  0  0  |
1  1  0  1  0  0  0  |
1  0  1  1  0  0  0  |
1  1  0  0  1  0  0  |
1  0  1  0  1  0  0  |
1  0  0  1  1  0  0  |
0  1  0  0  0  1  0  |
0  0  1  0  0  1  0  |
0  0  0  1  0  1  0  |
0  0  0  0  1  1  0  |
0  1  0  0  0  0  1  |
0  0  1  0  0  0  1  |
0  0  0  1  0  0  1  |
0  0  0  0  1  0  1  |
-1 0  0  0  0  1  1  |
0  -1 0  0  0  0  0  |
0  0  -1 0  0  0  0  |
0  0  0  -1 0  0  0  |
0  0  0  0  -1 0  0  |
1  0  0  0  0  -1 0  |
1  0  0  0  0  0  -1 |

         27        27
Matrix ZZ   <--- ZZ
}}}

{{{id=28|
Lines13 = LinesM_{0..5,21..26}
matrix apply(Lines13, L -> apply(Lines13, M ->  intersectionNumber(L,M)))
///

{image {-1} | 3255b+569c-1405d
       {-1} | 93a+5c-13d      
--------------------------------------------------------------
44207922a2-39698910ab+10292310b2-1752120ac+3091506bc+142250c2+
30225bc+1250c2-41106bd-4950cd+4420d2                          
--------------------------------------------------------------
5576094ad-5549310bd-544710cd+477700d2 |, image {-1} | 119b+10d
                                      |        {-1} | 119a+13d
--------------------------------------------------------------
38319666a2-34411230ab+8921430b2+541450ac+829668bc+41650c2-
437325bc+54145c2+974610bd-76895cd+328770d2                
--------------------------------------------------------------
4709306ad+3759210bd-59150cd+252900d2 |, image {-1} | c
                                     |        {-1} | a
--------------------------------------------------------------
46002a2-41310ab+10710b2+650ac+996bc+50c2-628ad-314d2 |, image
525b2+65bc-157bd                                     |       
--------------------------------------------------------------
{-1} | 5253b+625c-1373d
{-1} | 1751a+10c-78d   
--------------------------------------------------------------
8296598706a2-7450382430ab+1931580630b2+69847390ac+222180888bc+
35238875bc+137900c2-28681380bd-746320cd-2568540d2             
--------------------------------------------------------------
8618750c2+256318384ad-331884540bd+1647580cd-45212890d2 |,
                                                       | 
--------------------------------------------------------------
image {-1} | 12002b+1775c-4366d
      {-1} | 12002a+445c-1196d 
--------------------------------------------------------------
194896949412a2-175018324860ab+45375121260b2-4472365270ac+10708
3906651000bc+94680425c2-4521633480bd-289110190cd+93108600d2   
--------------------------------------------------------------
940526bc+377657875c2+16760841008ad-17440586280bd-1067115740cd+
                                                              
--------------------------------------------------------------
339893100d2 |, image {-1} | 104601b-14950c+37286d
            |        {-1} | 4981a-365c+897d      
--------------------------------------------------------------
67136836866a2-60289177230ab+15630527430b2+5868315340ac-2964302
-18056125bc+12280425c2+134038710bd-48623015cd+45325410d2      
--------------------------------------------------------------
682bc+502992750c2-13006815566ad+10857135510bd-2009910970cd+
                                                           
--------------------------------------------------------------
1884061580d2 |, image {-1} | 306b-25c-30d
             |        {-1} | 102a-5c     
--------------------------------------------------------------
4692204a2-4213620ab+1092420b2+296310ac-104958bc+19625c2-
3925c2-5338cd                                           
--------------------------------------------------------------
64056ad-3140cd-32028d2 |, image {-1} | 525b+65c-157d
                       |        {-1} | a            
--------------------------------------------------------------
46002a2-41310ab+10710b2+650ac+996bc+50c2-628ad-314d2 |, image
bc                                                   |       
--------------------------------------------------------------
{-1} | 1209b+50c-130d
{-1} | 93a+5c-13d    
--------------------------------------------------------------
44207922a2-39698910ab+10292310b2-1752120ac+3091506bc+142250c2+
81375bc+14225c2-110670bd-54471cd+47770d2                      
--------------------------------------------------------------
5576094ad-5549310bd-544710cd+477700d2 |, image {-1} |
                                      |        {-1} |
--------------------------------------------------------------
10220b-63c-146d 490289316a2-440281980ab+114147180b2+10285846ac
146a-c          68766bc+9577c2-22922cd                        
--------------------------------------------------------------
+7599738bc+603351c2-6693224ad-45844cd-3346612d2 |, image {-1}
                                                |        {-1}
--------------------------------------------------------------
| 11865b-616c-904d 587399538a2-527487390ab+136755990b2+
| 113a-4c          17741bc+10833c2-17741cd             
--------------------------------------------------------------
29092754ac-5954196bc+1668282c2-8018932ad-283856cd-4009466d2 |,
                                                            | 
--------------------------------------------------------------
image {-1} | 55b+13c-11d
      {-1} | 11a+c      
--------------------------------------------------------------
506022a2-454410ab+117810b2-38852ac+52266bc+4082c2-6908ad+628cd
3297bc+314c2+314cd                                            
--------------------------------------------------------------
-3454d2 |}
        |

List


| -1 0  0  0  0  0  0  1  1  1  1  1  |
| 0  -1 0  0  0  0  1  0  1  1  1  1  |
| 0  0  -1 0  0  0  1  1  0  1  1  1  |
| 0  0  0  -1 0  0  1  1  1  0  1  1  |
| 0  0  0  0  -1 0  1  1  1  1  0  1  |
| 0  0  0  0  0  -1 1  1  1  1  1  0  |
| 0  1  1  1  1  1  -1 0  0  0  0  0  |
| 1  0  1  1  1  1  0  -1 0  0  0  0  |
| 1  1  0  1  1  1  0  0  -1 0  0  0  |
| 1  1  1  0  1  1  0  0  0  -1 0  0  |
| 1  1  1  1  0  1  0  0  0  0  -1 0  |
| 1  1  1  1  1  0  0  0  0  0  0  -1 |

         12        12
Matrix ZZ   <--- ZZ
}}}

This configuration of 12 lines is called Schlaffi's double six

{{{id=30|

///
}}}

{{{id=39|

///
}}}