+ M2 --no-readline --print-width 99
Macaulay2, version 1.9.2
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition,
               ReesAlgebra, TangentCone

i1 : loadPackage("MatFacCurvesP4")

o1 = MatFacCurvesP4

o1 : Package

i2 : ------------------------------------------------------------------
     -- Theorem 3.1, Remark 3.4, Corollary 3.5                       --
     -- For each pair (g,d), we construct a matrix factorization of  --
     -- the given shape giving rise to monads and smooth curves as   --
     -- claimed in Theorem 3.1	    	    	       	     	--
     -- For (g,d)=(12,14) or (13,15) we show that a general matrix   --
     -- factorization has a component with kernel of the expected    --
     -- dimension, giving thus rise to monads parameterized by a     --
     -- rational variety. We show that the general monad produces a  --
     -- smooth curve of the right genus and degree, providing hence  --
     -- a proof of Remark 3.4 and Corollary 3.5    	       	     	--
     ------------------------------------------------------------------
     --
     p=32009; -- a prime number

i3 : Fp=ZZ/p; -- a prime field

i4 : S=Fp[x_0..x_4]; -- homogeneous coordinate ring of P4

i5 : --
     ---------- (g,d) = (12,14) ---------------------------------------
     -- We construct a matrix factorization starting from a nodal
     -- curve, as explained in Corollary 3.5
     time singC=singularCurveInP4(S,12,14); -- takes 4 sec
     -- used 4.29917 seconds

o5 : Ideal of S

i6 : -- We check that the curve is singular in P4:
     isSmoothCurve(singC) == false
bad 
	    projection center

o6 = true

i7 : -- time codim(singularLocus(singC)) == 4 -- takes about 90 sec
     omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C

i8 : fomegaSing=res omegaSingC;

i9 : sM=S^{ -5}**coker transpose fomegaSing.dd_3;

i10 : (psi,phi)=matrixFactorizationFromModule(sM);

i11 : X=ideal ring phi;

o11 : Ideal of S

i12 : codim X, degree X

o12 = (1, 3)

o12 : Sequence

i13 : -- We cheack that (phi,psi) is a matrix factorization on the cubic
      -- threefold X of the correct shape: ker phi = image psi 
      prune ((ker psi) / (image phi)) == 0

o13 = true

i14 : -- and phi is the presentation of a MCM module over S
      betti res (coker psi ** S)

              0  1
o14 = total: 17 17
          0: 15  2
          1:  2 15

o14 : BettiTally

i15 : betti psi

              0  1
o15 = total: 17 17
          0: 15  2
          1:  2 15

o15 : BettiTally

i16 : -- We check that the dimension of the kernel of the last row
      -- of the Betti table of psi is 5, i.e., the expected one
      betti syz((psi_{2..16})^{15,16},DegreeLimit=>2)

              0 1
o16 = total: 15 5
          1:  . 5
          2: 15 .

o16 : BettiTally

i17 : -- We construct the curve C
      monadShape=betti map(S^{2:-1},S^{2:-1,2:-2},0);

i18 : IC=idealFromMatFac(psi, monadShape);

o18 : Ideal of S

i19 : -- We check that C is a smooth curve of genus 12 and degree 14
      -- of maximal rank. We check also that the Betti table of its
      -- section module is as expected over S and over the ring of
      -- a general supporting cubic threefold
      (codim IC, genus IC, degree IC) == (3, 12, 14)

o19 = true

i20 : isSmoothCurve(IC)

o20 = true

i21 : betti res IC -- expected Betti table

             0 1  2  3 4
o21 = total: 1 9 18 12 2
          0: 1 .  .  . .
          1: . .  .  . .
          2: . 4  .  . .
          3: . 5 18 12 2

o21 : BettiTally

i22 : omegaC=Ext^2(IC,S^{ -5}); -- canonical module of C

i23 : fomega=res omegaC;

i24 : sM=S^{ -5}**coker transpose fomega.dd_3;

i25 : betti res sM

             0  1  2 3
o25 = total: 3 14 15 4
          0: 1  .  . .
          1: .  .  . .
          2: 2 14 15 2
          3: .  .  . 2

o25 : BettiTally

i26 : gIE=gens IC;     

              1       9
o26 : Matrix S  <--- S

i27 : Y=ideal(gIE * random(source gIE,S^{-3}));

o27 : Ideal of S

i28 : betti res (sM**(S/Y)) -- non-minimal resolution

             0  1  2  3  4  5  6
o28 = total: 3 14 18 17 17 17 17
          0: 1  .  .  .  .  .  .
          1: .  .  1  .  .  .  .
          2: 2 14 15  2  .  .  .
          3: .  .  2 15 15  2  .
          4: .  .  .  .  2 15 15
          5: .  .  .  .  .  .  2

o28 : BettiTally

i29 : betti res prune (sM**(S/Y)) -- expected resolution

             0  1  2  3  4  5  6
o29 = total: 3 13 17 17 17 17 17
          0: 1  .  .  .  .  .  .
          1: .  .  .  .  .  .  .
          2: 2 13 15  2  .  .  .
          3: .  .  2 15 15  2  .
          4: .  .  .  .  2 15 15
          5: .  .  .  .  .  .  2

o29 : BettiTally

i30 : --
      ---------- (g,d) = (13,15) ---------------------------------------
      -- We construct a matrix factorization starting from a nodal
      -- curve, as explained in Corollary 3.5
      time singC=singularCurveInP4(S,13,15); -- takes 3 sec
     -- used 2.93518 seconds

o30 : Ideal of S

i31 : -- We check that the curve is singular in P4:
      isSmoothCurve(singC) == false
bad 
	    projection center

o31 = true

i32 : -- time codim(singularLocus(singC)) == 4 -- takes about 379 sec
      omegaSingC=Ext^2(singC,S^{ -5}); -- canonical module of C

i33 : fomegaSing=res omegaSingC;

i34 : sM=S^{ -5}**coker transpose fomegaSing.dd_3;

i35 : (psi,phi)=matrixFactorizationFromModule(sM);

i36 : X=ideal ring phi;

o36 : Ideal of S

i37 : codim X, degree X

o37 = (1, 3)

o37 : Sequence

i38 : -- We cheack that (phi,psi) is a matrix factorization on the cubic
      -- threefold X of the correct shape: ker phi = image psi 
      prune ((ker psi) / (image phi)) == 0

o38 = true

i39 : -- and phi is the presentation of a MCM module over S
      betti res (coker psi ** S)

              0  1
o39 = total: 21 21
          0: 18  3
          1:  3 18

o39 : BettiTally

i40 : betti psi

              0  1
o40 = total: 21 21
          0: 18  3
          1:  3 18

o40 : BettiTally

i41 : -- We check that the dimension of the kernel of the last row
      -- of the Betti table of psi is 3, i.e., the expected one
      betti syz((psi_{3..20})^{18,19,20},DegreeLimit=>2)

              0 1
o41 = total: 18 3
          1:  . 3
          2: 18 .

o41 : BettiTally

i42 : -- We construct the curve C
      monadShape=betti map(S^{3:-1},S^{3:-1,2:-2},0);

i43 : IC=idealFromMatFac(psi, monadShape);

o43 : Ideal of S

i44 : -- We check that C is a smooth curve of genus 13 and degree 15
      -- of maximal rank. We check also that the Betti table of its
      -- section module is as expected over S and over the ring of
      -- a general supporting cubic threefold
      (codim IC, genus IC, degree IC) == (3, 13, 15)

o44 = true

i45 : isSmoothCurve(IC)

o45 = true

i46 : betti res IC -- expected Betti table

             0  1  2  3 4
o46 = total: 1 14 27 17 3
          0: 1  .  .  . .
          1: .  .  .  . .
          2: .  2  .  . .
          3: . 12 27 17 3

o46 : BettiTally

i47 : omegaC=Ext^2(IC,S^{1:-5}); -- canonical module of C

i48 : fomega=res omegaC;

i49 : sM=S^{ -5}**coker transpose fomega.dd_3;

i50 : betti res sM

             0  1  2 3
o50 = total: 4 17 18 5
          0: 1  .  . .
          1: .  .  . .
          2: 3 17 18 3
          3: .  .  . 2

o50 : BettiTally

i51 : gIE=gens IC;     

              1       14
o51 : Matrix S  <--- S

i52 : Y=ideal(gIE * random(source gIE,S^{1:-3}));

o52 : Ideal of S

i53 : betti res (sM**(S/Y)) -- non-minimal resolution

             0  1  2  3  4  5  6
o53 = total: 4 17 22 21 21 21 21
          0: 1  .  .  .  .  .  .
          1: .  .  1  .  .  .  .
          2: 3 17 18  3  .  .  .
          3: .  .  3 18 18  3  .
          4: .  .  .  .  3 18 18
          5: .  .  .  .  .  .  3

o53 : BettiTally

i54 : betti res prune (sM**(S/Y)) -- expected resolution

             0  1  2  3  4  5  6
o54 = total: 4 16 21 21 21 21 21
          0: 1  .  .  .  .  .  .
          1: .  .  .  .  .  .  .
          2: 3 16 18  3  .  .  .
          3: .  .  3 18 18  3  .
          4: .  .  .  .  3 18 18
          5: .  .  .  .  .  .  3

o54 : BettiTally

i55 : --
      ---------- (g,d) = (16,17) ---------------------------------------
      -- We construct a matrix factorization starting from a curve on
      -- the Alexander surface Y, as explained in Section 5.2
      time alexC=first curveOnAlexanderSurface(S,16,17); -- takes 23 sec
     -- used 23.0043 seconds

o55 : Ideal of S

i56 : omegaAlexC=Ext^2(alexC,S^{1:-5}); -- canonical module of C

i57 : fomegaAlex=res omegaAlexC;

i58 : sM=S^{1:-5}**coker transpose fomegaAlex.dd_3;

i59 : (psi,phi)=matrixFactorizationFromModule(sM);

i60 : X=ideal ring phi;

o60 : Ideal of S

i61 : codim X, degree X

o61 = (1, 4)

o61 : Sequence

i62 : -- We cheack that (phi,psi) is a matrix factorization on the 
      -- quartic threefold X of the correct shape: ker phi = image psi 
      time prune ((ker psi) / (image phi)) == 0 -- takes 31 sec
     -- used 31.0433 seconds

o62 = true

i63 : -- and phi is the presentation of a MCM module  
      betti res (coker psi ** S)

              0  1
o63 = total: 23 23
          0: 19  1
          1:  .  3
          2:  4 19

o63 : BettiTally

i64 : -- We construct the curve C
      monadShape=betti map(S^{4:-2},S^{3:-2,1:-1},0);

i65 : time IC=idealFromMatFac(psi, monadShape); -- takes 434 sec
     -- used 434.463 seconds

o65 : Ideal of S

i66 : -- We check that C is a smooth curve of genus 13 and degree 15
      -- of maximal rank. We check also that the Betti table of its
      -- section module is as expected over S and over the ring of
      -- a general supporting cubic threefold
      (codim IC, genus IC, degree IC) == (3, 16, 17)

o66 = true

i67 : isSmoothCurve(IC)

o67 = true

i68 : betti res IC -- expected Betti table

             0  1  2  3 4
o68 = total: 1 17 29 14 1
          0: 1  .  .  . .
          1: .  .  .  . .
          2: .  .  .  . .
          3: . 17 29 13 .
          4: .  .  .  1 1

o68 : BettiTally

i69 : omegaC=Ext^2(IC,S^{1:-5}); -- canonical module of C

i70 : fomega=res omegaC;

i71 : sM=S^{1:-5}**coker transpose fomega.dd_3;

i72 : betti res sM

             0  1  2 3
o72 = total: 5 19 18 4
          0: 1  .  . .
          1: .  .  . .
          2: 4 19 18 1
          3: .  .  . 3

o72 : BettiTally

i73 : gIE=gens IC;     

              1       17
o73 : Matrix S  <--- S

i74 : Y=ideal(gIE * random(source gIE,S^{-4}));

o74 : Ideal of S

i75 : betti res (sM**(S/Y)) -- expected resolution

             0  1  2  3  4  5  6
o75 = total: 5 19 23 23 23 23 23
          0: 1  .  .  .  .  .  .
          1: .  .  .  .  .  .  .
          2: 4 19 19  1  .  .  .
          3: .  .  .  3  .  .  .
          4: .  .  4 19 19  1  .
          5: .  .  .  .  .  3  .
          6: .  .  .  .  4 19 19
          7: .  .  .  .  .  .  .
          8: .  .  .  .  .  .  4

o75 : BettiTally

i76 : --
      ---------- (g,d) = (17,18) ---------------------------------------
      -- We construct a matrix factorization starting from a curve on
      -- the Alexander surface Y, as explained in Section 5.2
      time alexC=first curveOnAlexanderSurface(S,17,18); -- takes 25 sec
     -- used 25.4965 seconds

o76 : Ideal of S

i77 : omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C

i78 : fomegaAlex=res omegaAlexC;

i79 : sM=S^{1:-5}**coker transpose fomegaAlex.dd_3;

i80 : (psi,phi)=matrixFactorizationFromModule(sM);

i81 : X=ideal ring phi;

o81 : Ideal of S

i82 : codim X, degree X

o82 = (1, 4)

o82 : Sequence

i83 : -- We cheack that (phi,psi) is a matrix factorization on the 
      -- quartic threefold X of the correct shape: ker phi = image psi 
      time prune ((ker psi) / (image phi)) == 0 -- takes 148 sec
     -- used 148.293 seconds

o83 = true

i84 : -- and phi is the presentation of a MCM module  
      betti res (coker psi ** S)

              0  1
o84 = total: 27 27
          0: 22  2
          1:  .  3
          2:  5 22

o84 : BettiTally

i85 : -- We construct the curve C
      monadShape=betti map(S^{5:-2},S^{3:-2,2:-1},0);

i86 : time IC=idealFromMatFac(psi, monadShape); -- takes 1134 sec
     -- used 1134.03 seconds

o86 : Ideal of S

i87 : -- We check that C is a smooth curve of genus 13 and degree 15
      -- of maximal rank. We check also that the Betti table of its
      -- section module is as expected over S and over the ring of
      -- a general supporting cubic threefold
      (codim IC, genus IC, degree IC) == (3, 17, 18)

o87 = true

i88 : isSmoothCurve(IC)

o88 = true

i89 : betti res IC -- expected Betti table

             0  1  2  3 4
o89 = total: 1 14 20 10 3
          0: 1  .  .  . .
          1: .  .  .  . .
          2: .  .  .  . .
          3: . 14 18  . .
          4: .  .  2 10 3

o89 : BettiTally

i90 : omegaC=Ext^2(IC,S^{-5}); -- canonical module of C

i91 : fomega=res omegaC;

i92 : sM=S^{1:-5}**coker transpose fomega.dd_3;

i93 : betti res sM

             0  1  2 3
o93 = total: 6 22 21 5
          0: 1  .  . .
          1: .  .  . .
          2: 5 22 21 2
          3: .  .  . 3

o93 : BettiTally

i94 : gIE=gens IC;     

              1       14
o94 : Matrix S  <--- S

i95 : Y=ideal(gIE * random(source gIE,S^{-4}));

o95 : Ideal of S

i96 : betti res (sM**(S/Y)) -- expected resolution

             0  1  2  3  4  5  6
o96 = total: 6 22 27 27 27 27 27
          0: 1  .  .  .  .  .  .
          1: .  .  .  .  .  .  .
          2: 5 22 22  2  .  .  .
          3: .  .  .  3  .  .  .
          4: .  .  5 22 22  2  .
          5: .  .  .  .  .  3  .
          6: .  .  .  .  5 22 22
          7: .  .  .  .  .  .  .
          8: .  .  .  .  .  .  5

o96 : BettiTally

i97 : --
      ---------- (g,d) = (18,19) ---------------------------------------
      -- We construct a matrix factorization starting from a curve on
      -- the Alexander surface Y, as explained in Section 5.2
      time alexC=first curveOnAlexanderSurface(S,18,19); -- takes 23 sec
     -- used 23.3045 seconds

o97 : Ideal of S

i98 : omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C

i99 : fomegaAlex=res omegaAlexC;

i100 : sM=S^{1:-5}**coker transpose fomegaAlex.dd_3;

i101 : (psi,phi)=matrixFactorizationFromModule(sM);

i102 : X=ideal ring phi;

o102 : Ideal of S

i103 : codim X, degree X

o103 = (1, 4)

o103 : Sequence

i104 : -- We cheack that (phi,psi) is a matrix factorization on the 
       -- quartic threefold X of the correct shape: ker phi = image psi 
       time prune ((ker psi) / (image phi)) == 0 -- takes 437 sec
     -- used 436.645 seconds

o104 = true

i105 : -- and phi is the presentation of a MCM module  
       betti res (coker psi ** S)

               0  1
o105 = total: 31 31
           0: 25  3
           1:  .  3
           2:  6 25

o105 : BettiTally

i106 : -- We construct the curve C
       monadShape=betti map(S^{6:-2},S^{3:-2,3:-1},0);

i107 : time IC=idealFromMatFac(psi, monadShape); -- takes 2227 sec
     -- used 2227.28 seconds

o107 : Ideal of S

i108 : -- We check that C is a smooth curve of genus 13 and degree 15
       -- of maximal rank. We check also that the Betti table of its
       -- section module is as expected over S and over the ring of
       -- a general supporting cubic threefold
       (codim IC, genus IC, degree IC) == (3, 18, 19)

o108 = true

i109 : isSmoothCurve(IC)

o109 = true

i110 : betti res IC -- expected Betti table

              0  1  2  3 4
o110 = total: 1 11 24 19 5
           0: 1  .  .  . .
           1: .  .  .  . .
           2: .  .  .  . .
           3: . 11  7  . .
           4: .  . 17 19 5

o110 : BettiTally

i111 : omegaC=Ext^2(IC,S^{-5}); -- canonical module of C

i112 : fomega=res omegaC;

i113 : sM=S^{1:-5}**coker transpose fomega.dd_3;

i114 : betti res sM

              0  1  2 3
o114 = total: 7 25 24 6
           0: 1  .  . .
           1: .  .  . .
           2: 6 25 24 3
           3: .  .  . 3

o114 : BettiTally

i115 : gIE=gens IC;     

               1       11
o115 : Matrix S  <--- S

i116 : Y=ideal(gIE * random(source gIE,S^{-4}));

o116 : Ideal of S

i117 : betti res (sM**(S/Y)) -- expected resolution

              0  1  2  3  4  5  6
o117 = total: 7 25 31 31 31 31 31
           0: 1  .  .  .  .  .  .
           1: .  .  .  .  .  .  .
           2: 6 25 25  3  .  .  .
           3: .  .  .  3  .  .  .
           4: .  .  6 25 25  3  .
           5: .  .  .  .  .  3  .
           6: .  .  .  .  6 25 25
           7: .  .  .  .  .  .  .
           8: .  .  .  .  .  .  6

o117 : BettiTally

i118 : --
       ---------- (g,d) = (19,20) ---------------------------------------
       -- We construct a matrix factorization starting from a curve on
       -- the Alexander surface Y, as explained in Section 5.2
       time alexC=first curveOnAlexanderSurface(S,19,20); -- takes 22 sec
     -- used 21.8851 seconds

o118 : Ideal of S

i119 : omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C

i120 : fomegaAlex=res omegaAlexC;

i121 : sM=S^{1:-5}**coker transpose fomegaAlex.dd_3;

i122 : (psi,phi)=matrixFactorizationFromModule(sM);

i123 : X=ideal ring phi;

o123 : Ideal of S

i124 : codim X, degree X

o124 = (1, 4)

o124 : Sequence

i125 : -- We cheack that (phi,psi) is a matrix factorization on the 
       -- quartic threefold X of the correct shape: ker phi = image psi 
       time prune ((ker psi) / (image phi)) == 0 -- takes 1125 sec
     -- used 1125.38 seconds

o125 = true

i126 : -- and phi is the presentation of a MCM module  
       betti res (coker psi ** S)

               0  1
o126 = total: 35 35
           0: 28  4
           1:  .  3
           2:  7 28

o126 : BettiTally

i127 : -- We construct the curve C
       monadShape=betti map(S^{7:-2},S^{3:-2,4:-1},0);

i128 : time IC=idealFromMatFac(psi, monadShape); -- takes 4721 sec
     -- used 4721.22 seconds

o128 : Ideal of S

i129 : -- We check that C is a smooth curve of genus 13 and degree 15
       -- of maximal rank. We check also that the Betti table of its
       -- section module is as expected over S and over the ring of
       -- a general supporting cubic threefold
       (codim IC, genus IC, degree IC) == (3, 19, 20)

o129 = true

i130 : isSmoothCurve(IC)

o130 = true

i131 : betti res IC -- expected Betti table

              0  1  2  3 4
o131 = total: 1 12 32 28 7
           0: 1  .  .  . .
           1: .  .  .  . .
           2: .  .  .  . .
           3: .  8  .  . .
           4: .  4 32 28 7

o131 : BettiTally

i132 : omegaC=Ext^2(IC,S^{-5}); -- canonical module of C

i133 : fomega=res omegaC;

i134 : sM=S^{1:-5}**coker transpose fomega.dd_3;

i135 : betti res sM

              0  1  2 3
o135 = total: 8 28 27 7
           0: 1  .  . .
           1: .  .  . .
           2: 7 28 27 4
           3: .  .  . 3

o135 : BettiTally

i136 : gIE=gens IC;     

               1       12
o136 : Matrix S  <--- S

i137 : Y=ideal(gIE * random(source gIE,S^{-4}));

o137 : Ideal of S

i138 : betti res (sM**(S/Y)) -- expected resolution

              0  1  2  3  4  5  6
o138 = total: 8 28 35 35 35 35 35
           0: 1  .  .  .  .  .  .
           1: .  .  .  .  .  .  .
           2: 7 28 28  4  .  .  .
           3: .  .  .  3  .  .  .
           4: .  .  7 28 28  4  .
           5: .  .  .  .  .  3  .
           6: .  .  .  .  7 28 28
           7: .  .  .  .  .  .  .
           8: .  .  .  .  .  .  7

o138 : BettiTally

i139 : --
       ---------- (g,d) = (20,20) ---------------------------------------
       -- We construct a matrix factorization starting from a curve on
       -- the Alexander surface Y, as explained in Section 5.2
       time alexC=first curveOnAlexanderSurface(S,20,20); -- takes 22 sec
     -- used 22.2438 seconds

o139 : Ideal of S

i140 : omegaAlexC=Ext^2(alexC,S^{-5}); -- canonical module of C

i141 : fomegaAlex=res omegaAlexC;

i142 : sM=S^{1:-5}**coker transpose fomegaAlex.dd_3;

i143 : (psi,phi)=matrixFactorizationFromModule(sM);

i144 : X=ideal ring phi;

o144 : Ideal of S

i145 : codim X, degree X

o145 = (1, 4)

o145 : Sequence

i146 : -- We cheack that (phi,psi) is a matrix factorization on the 
       -- quartic threefold X of the correct shape: ker phi = image psi 
       time prune ((ker psi) / (image phi)) == 0 -- takes 7 sec
     -- used 6.8099 seconds

o146 = true

i147 : -- and phi is the presentation of a MCM module  
       betti res (coker psi ** S)

               0  1
o147 = total: 28 28
           0: 22  .
           1:  .  4
           2:  6 24

o147 : BettiTally

i148 : -- We construct the curve C
       monadShape=betti map(S^{6:-2},S^{4:-2},0);

i149 : time IC=idealFromMatFac(psi, monadShape); -- takes 3522 sec
     -- used 3521.88 seconds

o149 : Ideal of S

i150 : -- We check that C is a smooth curve of genus 13 and degree 15
       -- of maximal rank. We check also that the Betti table of its
       -- section module is as expected over S and over the ring of
       -- a general supporting cubic threefold
       (codim IC, genus IC, degree IC) == (3, 20, 20)

o150 = true

i151 : isSmoothCurve(IC)

o151 = true

i152 : betti res IC -- expected Betti table

              0 1  2  3 4
o152 = total: 1 9 26 24 6
           0: 1 .  .  . .
           1: . .  .  . .
           2: . .  .  . .
           3: . 9  .  . .
           4: . . 26 24 6

o152 : BettiTally

i153 : omegaC=Ext^2(IC,S^{-5}); -- canonical module of C

i154 : fomega=res omegaC;

i155 : sM=S^{1:-5}**coker transpose fomega.dd_3;

i156 : betti res sM

              0  1  2 3
o156 = total: 7 24 21 4
           0: 1  .  . .
           1: .  .  . .
           2: 6 24 21 .
           3: .  .  . 4

o156 : BettiTally

i157 : gIE=gens IC;     

               1       9
o157 : Matrix S  <--- S

i158 : Y=ideal(gIE * random(source gIE,S^{-4}));

o158 : Ideal of S

i159 : betti res (sM**(S/Y)) -- expected resolution

              0  1  2  3  4  5  6
o159 = total: 7 24 28 28 28 28 28
           0: 1  .  .  .  .  .  .
           1: .  .  .  .  .  .  .
           2: 6 24 22  .  .  .  .
           3: .  .  .  4  .  .  .
           4: .  .  6 24 22  .  .
           5: .  .  .  .  .  4  .
           6: .  .  .  .  6 24 22
           7: .  .  .  .  .  .  .
           8: .  .  .  .  .  .  6

o159 : BettiTally

i160 :