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We investigate the 27 lines on a cubic surface

restart
printWidth=68
68
We start by constructing a cubic surface in P^3. We choose 6 points, and blow them p
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
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:
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
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)| +----------------------+------------------------+
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
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
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
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 
   phi:P2 ---> P3
(as a bigraded ideal).
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
WARNING: Output truncated!
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, ... (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
codim Gr -- so Gr defines a surface (which is isomorphic to X)
degree Gr_0
isHomogeneous Gr
3 {1, 1} List true
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
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)| +------------------------------------------------------+
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) | +------------------------------------------------------+

Check that they all lie on XX:

tally apply(Lines, L -> isSubset(IX, L))
Tally{true => 27} Tally

theory tells us that these 27 lines are all exceptional curves, i.e. have self intersection -1 on X.

Let's check that. How should we find intersection numbers of curves on X?

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
euler RX
1
euler ML
1
euler(ML ** ML)
0
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
intersectionNumber = (F,G) -> euler ring F - euler F - euler G + euler(F**G)
intersectionNumber FunctionClosure
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
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!
{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
Lines13 = LinesM_{0..5,21..26}
matrix apply(Lines13, L -> apply(Lines13, M ->  intersectionNumber(L,M)))
WARNING: Output truncated!
{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