-- -*- M2-comint -*- {* hash: -2018080312 *}
i1 : R = ZZ/31991[a,b,c] -- the coordinate ring of P^2
o1 = R
o1 : PolynomialRing
i2 : ipoint1 = ideal matrix({{a,b}})
o2 = ideal (a, b)
o2 : Ideal of R
i3 : ipoint1 = ideal(a,b)
o3 = ideal (a, b)
o3 : Ideal of R
i4 : ipoint2 = ideal(a,c)
o4 = ideal (a, c)
o4 : Ideal of R
i5 : ipoint3 = ideal(b,c)
o5 = ideal (b, c)
o5 : Ideal of R
i6 : icurves1 = intersect(
ipoint1^2,
ipoint2^2,
ipoint3^2
)
2 2 2 2 2 2
o6 = ideal (a*b*c, b c , a c , a b )
o6 : Ideal of R
i7 : Icurves1 = gens icurves1
o7 = | abc b2c2 a2c2 a2b2 |
1 4
o7 : Matrix R <--- R
i8 : F1 = Icurves1 * random(source Icurves1, R^{-6})
o8 = | -14408a4b2-4368a3b3+11739a2b4+7753a4bc-3903a3b2c-10843a2b3c-13639ab4c+
------------------------------------------------------------------------
12102a4c2+2678a3bc2-3566a2b2c2+3963ab3c2-691b4c2-3006a3c3+12633a2bc3-
------------------------------------------------------------------------
7645ab2c3-12006b3c3-6647a2c4-15210abc4-1315b2c4 |
1 1
o8 : Matrix R <--- R
i9 : betti F1
0 1
o9 = total: 1 1
0: 1 .
1: . .
2: . .
3: . .
4: . .
5: . 1
o9 : BettiTally
i10 : Icurves2 = gens (ipoint1^3)
o10 = | a3 a2b ab2 b3 |
1 4
o10 : Matrix R <--- R
i11 : F2 = Icurves2 * random(source Icurves2, R^{-6})
o11 = | 1645a6+2187a5b-2910a4b2+8938a3b3-9733a2b4+2213ab5-708b6-6215a5c-
-----------------------------------------------------------------------
14576a4bc-6183a3b2c-13531a2b3c-2542ab4c-8608b5c+3910a4c2-5312a3bc2-
-----------------------------------------------------------------------
11162a2b2c2-9878ab3c2-12636b4c2+10256a3c3+5857a2bc3-14066ab2c3-781b3c3
-----------------------------------------------------------------------
|
1 1
o11 : Matrix R <--- R
i12 : betti F2
0 1
o12 = total: 1 1
0: 1 .
1: . .
2: . .
3: . .
4: . .
5: . 1
o12 : BettiTally
i13 : i = ideal((a-b)^2) + (ipoint1^4)
2 2 4 3 2 2 3 4
o13 = ideal (a - 2a*b + b , a , a b, a b , a*b , b )
o13 : Ideal of R
i14 : icurves3 = intersect(i, ipoint3^2)
2 2 2 2 2 2 2 3 4 3 2 2
o14 = ideal (a c - 2a*b*c + b c , a b*c - 2a*b c + b c, b , a*b , a b )
o14 : Ideal of R
i15 : Icurves3 = gens icurves3
o15 = | a2c2-2abc2+b2c2 a2bc-2ab2c+b3c b4 ab3 a2b2 |
1 5
o15 : Matrix R <--- R
i16 : F3 = Icurves3 * random(source Icurves3, R^{-6})
o16 = | -11643a4b2+11517a3b3-13891a2b4+1400ab5+7429b6+11673a4bc-15413a3b2c-
-----------------------------------------------------------------------
15778a2b3c-8109ab4c-5828b5c-4238a4c2-4791a3bc2-146a2b2c2+1111ab3c2-
-----------------------------------------------------------------------
2606b4c2+3312a3c3+15340a2bc3-8625ab2c3-10027b3c3-2671a2c4+5342abc4-
-----------------------------------------------------------------------
2671b2c4 |
1 1
o16 : Matrix R <--- R
i17 : betti F3
0 1
o17 = total: 1 1
0: 1 .
1: . .
2: . .
3: . .
4: . .
5: . 1
o17 : BettiTally
i18 : can1 = basis(3, intersect(ipoint1,ipoint2,ipoint3))
o18 = {2} | b c 0 0 0 0 0 |
{2} | 0 0 a c 0 0 0 |
{2} | 0 0 0 0 a b c |
o18 : Matrix
i19 : target can1
o19 = image | bc ac ab |
1
o19 : R-module, submodule of R
i20 : source can1
7
o20 = R
o20 : R-module, free, degrees {3, 3, 3, 3, 3, 3, 3}
i21 : can1 = can1 ** R
o21 = {2} | b c 0 0 0 0 0 |
{2} | 0 0 a c 0 0 0 |
{2} | 0 0 0 0 a b c |
o21 : Matrix
i22 : can1 = super can1
o22 = | b2c bc2 a2c ac2 a2b ab2 abc |
1 7
o22 : Matrix R <--- R
i23 : can2 = basis(3, ipoint1^2)
o23 = {2} | a b c 0 0 0 0 |
{2} | 0 0 0 b c 0 0 |
{2} | 0 0 0 0 0 b c |
o23 : Matrix
i24 : can2 = super (can2 ** R)
o24 = | a3 a2b a2c ab2 abc b3 b2c |
1 7
o24 : Matrix R <--- R
i25 : can3 = basis(3, intersect(ideal(a-b) + ipoint1^2, ipoint3))
o25 = {2} | a c 0 0 0 0 0 |
{2} | 0 0 b c 0 0 0 |
{2} | 0 0 0 0 a b c |
o25 : Matrix
i26 : can3 = super (can3 ** R)
o26 = | a2c-abc ac2-bc2 b3 b2c a2b ab2 abc |
1 7
o26 : Matrix R <--- R
i27 : betti can1
0 1
o27 = total: 1 7
0: 1 .
1: . .
2: . 7
o27 : BettiTally
i28 : betti can2
0 1
o28 = total: 1 7
0: 1 .
1: . .
2: . 7
o28 : BettiTally
i29 : betti can3
0 1
o29 = total: 1 7
0: 1 .
1: . .
2: . 7
o29 : BettiTally
i30 : S = (coefficientRing R)[x_0..x_6]
o30 = S
o30 : PolynomialRing
i31 : T1 = R/ideal F1
o31 = T1
o31 : QuotientRing
i32 : f1 = map(T1,S,substitute(can1, T1))
2 2 2 2 2 2
o32 = map(T1,S,{b c, b*c , a c, a*c , a b, a*b , a*b*c})
o32 : RingMap T1 <--- S
i33 : IC1 = mingens ker f1
o33 = | x_3x_5-x_6^2 x_2x_5-x_4x_6 x_1x_5-x_0x_6 x_3x_4-x_2x_6 x_1x_4-x_6^2
-----------------------------------------------------------------------
x_0x_4-x_5x_6 x_0x_3-x_1x_6 x_1x_2-x_3x_6 x_0x_2-x_6^2
-----------------------------------------------------------------------
x_0^2-14242x_0x_1+2826x_1^2+3964x_2^2-15117x_1x_3-2079x_2x_3+5843x_3^2-
-----------------------------------------------------------------------
289x_2x_4-9007x_4^2-13360x_0x_5-12216x_4x_5-15156x_5^2-14404x_0x_6-
-----------------------------------------------------------------------
3276x_1x_6-10791x_2x_6-10898x_3x_6+15978x_4x_6+15201x_5x_6-6106x_6^2 |
1 10
o33 : Matrix S <--- S
i34 : T2 = R/ideal F2
o34 = T2
o34 : QuotientRing
i35 : f2 = map(T2,S,substitute(can2, T2))
3 2 2 2 3 2
o35 = map(T2,S,{a , a b, a c, a*b , a*b*c, b , b c})
o35 : RingMap T2 <--- S
i36 : IC2 = mingens ker f2
o36 = | x_4x_5-x_3x_6 x_2x_5-x_1x_6 x_4^2-x_2x_6 x_3x_4-x_1x_6 x_1x_4-x_0x_6
-----------------------------------------------------------------------
x_3^2-x_1x_5 x_2x_3-x_0x_6 x_1x_3-x_0x_5 x_1x_2-x_0x_4 x_1^2-x_0x_3
-----------------------------------------------------------------------
x_0^2x_5+5116x_0x_1x_5+13028x_0x_3x_5-5012x_0x_5^2+8901x_1x_5^2+9375x_
-----------------------------------------------------------------------
3x_5^2+14449x_5^3-15270x_0x_1x_6-3101x_0x_3x_6-14097x_0x_4x_6+11558x_2x
-----------------------------------------------------------------------
_4x_6-12489x_0x_5x_6-15255x_1x_5x_6-2977x_3x_5x_6-12296x_5^2x_6-13772x_
-----------------------------------------------------------------------
0x_6^2-12317x_1x_6^2-13843x_2x_6^2-8699x_3x_6^2-767x_4x_6^2+9541x_5x_6^
-----------------------------------------------------------------------
2+11104x_6^3 x_0^2x_3+8210x_0x_1x_5+12984x_0x_3x_5-6489x_0x_5^2-4948x_
-----------------------------------------------------------------------
1x_5^2+6458x_3x_5^2+10117x_5^3-15270x_0^2x_6-3803x_0x_1x_6-14097x_0x_2x
-----------------------------------------------------------------------
_6+11558x_2^2x_6-15309x_0x_3x_6-1234x_0x_4x_6+6788x_2x_4x_6-7558x_0x_5x
-----------------------------------------------------------------------
_6+15554x_1x_5x_6-9680x_3x_5x_6+12030x_5^2x_6+1053x_0x_6^2+14794x_1x_6^
-----------------------------------------------------------------------
2-8053x_2x_6^2+14144x_3x_6^2+183x_4x_6^2+6510x_5x_6^2+7952x_6^3
-----------------------------------------------------------------------
x_0^2x_1-15270x_0^2x_4-14097x_0x_2x_4+11558x_2^2x_4+14807x_0x_1x_5+
-----------------------------------------------------------------------
11535x_0x_3x_5+3146x_0x_5^2-3308x_1x_5^2+11713x_3x_5^2-3662x_5^3-3803x_
-----------------------------------------------------------------------
0^2x_6+10653x_0x_1x_6-1234x_0x_2x_6+6788x_2^2x_6-13184x_0x_3x_6-6015x_
-----------------------------------------------------------------------
0x_4x_6-13927x_2x_4x_6-12902x_0x_5x_6-10895x_1x_5x_6+12076x_3x_5x_6-
-----------------------------------------------------------------------
13436x_5^2x_6-5271x_0x_6^2+13163x_1x_6^2-12810x_2x_6^2-10603x_3x_6^2+
-----------------------------------------------------------------------
2795x_4x_6^2+14349x_5x_6^2+10510x_6^3
-----------------------------------------------------------------------
x_0^3-15270x_0^2x_2-14097x_0x_2^2+11558x_2^3-3803x_0^2x_4-1234x_0x_2x_4
-----------------------------------------------------------------------
+6788x_2^2x_4+13611x_0x_1x_5+3280x_0x_3x_5-9744x_0x_5^2-14465x_1x_5^2-
-----------------------------------------------------------------------
10338x_3x_5^2+9465x_5^3+10653x_0^2x_6+9309x_0x_1x_6-6015x_0x_2x_6-
-----------------------------------------------------------------------
13927x_2^2x_6-3480x_0x_3x_6-12267x_0x_4x_6-266x_2x_4x_6+5748x_0x_5x_6+
-----------------------------------------------------------------------
4410x_1x_5x_6+15396x_3x_5x_6+6091x_5^2x_6-7458x_0x_6^2-13475x_1x_6^2+
-----------------------------------------------------------------------
9759x_2x_6^2-7315x_3x_6^2+10674x_4x_6^2-1331x_5x_6^2-15179x_6^3 |
1 14
o36 : Matrix S <--- S
i37 : T3 = R/ideal F3
o37 = T3
o37 : QuotientRing
i38 : f3 = map(T3,S,substitute(can3, T3))
2 2 2 3 2 2 2
o38 = map(T3,S,{a c - a*b*c, a*c - b*c , b , b c, a b, a*b , a*b*c})
o38 : RingMap T3 <--- S
i39 : IC3 = mingens ker f3
o39 = | x_3x_5-x_2x_6 x_1x_5+x_3x_6-x_6^2 x_0x_5-x_4x_6+x_5x_6 x_3x_4-x_5x_6
-----------------------------------------------------------------------
x_2x_4-x_5^2 x_1x_4-x_0x_6 x_0x_3+x_3x_6-x_6^2 x_1x_2+x_3^2-x_3x_6
-----------------------------------------------------------------------
x_0x_2+x_2x_6-x_5x_6 x_0^2-8576x_0x_1-3902x_1^2+9200x_2^2+4791x_1x_3+
-----------------------------------------------------------------------
409x_2x_3+8757x_3^2+5938x_0x_4+9748x_4^2+6884x_2x_5-5249x_4x_5+15878x_5
-----------------------------------------------------------------------
^2-3356x_0x_6-4791x_1x_6+8479x_2x_6-11995x_3x_6-14734x_4x_6+1755x_5x_6+
-----------------------------------------------------------------------
12012x_6^2 |
1 10
o39 : Matrix S <--- S
i40 : IC1res = res(coker IC1)
1 10 25 25 10 1
o40 = S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6
o40 : ChainComplex
i41 : betti IC1res
0 1 2 3 4 5
o41 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o41 : BettiTally
i42 : IC2res = res(coker IC2)
1 14 35 35 14 1
o42 = S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6
o42 : ChainComplex
i43 : betti IC2res
0 1 2 3 4 5
o43 = total: 1 14 35 35 14 1
0: 1 . . . . .
1: . 10 20 15 4 .
2: . 4 15 20 10 .
3: . . . . . 1
o43 : BettiTally
i44 : IC3res = res(coker IC3)
1 10 25 25 10 1
o44 = S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6
o44 : ChainComplex
i45 : betti IC3res
0 1 2 3 4 5
o45 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o45 : BettiTally
i46 : IC1 = matrix entries IC1
o46 = | x_3x_5-x_6^2 x_2x_5-x_4x_6 x_1x_5-x_0x_6 x_3x_4-x_2x_6 x_1x_4-x_6^2
-----------------------------------------------------------------------
x_0x_4-x_5x_6 x_0x_3-x_1x_6 x_1x_2-x_3x_6 x_0x_2-x_6^2
-----------------------------------------------------------------------
x_0^2-14242x_0x_1+2826x_1^2+3964x_2^2-15117x_1x_3-2079x_2x_3+5843x_3^2-
-----------------------------------------------------------------------
289x_2x_4-9007x_4^2-13360x_0x_5-12216x_4x_5-15156x_5^2-14404x_0x_6-
-----------------------------------------------------------------------
3276x_1x_6-10791x_2x_6-10898x_3x_6+15978x_4x_6+15201x_5x_6-6106x_6^2 |
1 10
o46 : Matrix S <--- S
i47 : IC1res = res(coker IC1, DegreeLimit => {1})
1 10 25 25 10 1
o47 = S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6
o47 : ChainComplex
i48 : betti IC1res
0 1 2 3 4 5
o48 = total: 1 10 25 25 10 1
0: 1 . . . . .
1: . 10 16 9 . .
2: . . 9 16 10 .
3: . . . . . 1
o48 : BettiTally
i49 : ff1 = map(R,S,can1)
2 2 2 2 2 2
o49 = map(R,S,{b c, b*c , a c, a*c , a b, a*b , a*b*c})
o49 : RingMap R <--- S
i50 : G = map(coker F1,ff1)
o50 = | 1 |
o50 : Matrix
i51 : trim coimage G
o51 = cokernel | x_3x_5-x_6^2 x_2x_5-x_4x_6 x_1x_5-x_0x_6 x_3x_4-x_2x_6 x_1x_4-x_6^2 x_0x_4-x_5x_6 x_0x_3-x_1x_6 x_1x_2-x_3x_6 x_0x_2-x_6^2 x_0^2-14242x_0x_1+2826x_1^2+3964x_2^2-15117x_1x_3-2079x_2x_3+5843x_3^2-289x_2x_4-9007x_4^2-13360x_0x_5-12216x_4x_5-15156x_5^2-14404x_0x_6-3276x_1x_6-10791x_2x_6-10898x_3x_6+15978x_4x_6+15201x_5x_6-6106x_6^2 |
1
o51 : S-module, quotient of S
i52 :
