Here are some calculations in Macaulay2 to compute the Chow ring of $\mathrm{N}$ in Proposition 4.4.

Let $R = \mathbb{Q}[\beta_1, \beta_2, \beta_3, \delta_1, \delta_2]$ be the polynomial ring. In the following code, be_i is $\beta_i$, de_i is $\delta_i$, and D is $\Delta$. $S$ is the quotient ring by the first relation $b_1-d_1$. We generate six elements $r_1, \cdots, r_6$ in $I^a$, and compute $s_i := r_i/\Delta$.

i1 : R = QQ[be_1,be_2,be_3,de_1,de_2,Degrees=>{1,2,3,1,2}];

i2 : I = ideal(be_1+be_2+be_3-de_1-de_2);

o2 : Ideal of R

i3 : S = R/I;

i4 : D = (be_1-be_2)*(be_2-be_3)*(be_3-be_1)*(de_1-de_2);

i5 : b_1 = be_1+be_2+be_3;

i6 : b_2 = be_1*be_2+be_2*be_3+be_3*be_1;

i7 : b_3 = be_1*be_2*be_3;

i8 : d_1 = de_1+de_2;

i9 : d_2 = de_1*de_2;

i10 : r1 = (be_1-be_2)*(be_1-de_1)^3*(be_2-de_1)^3+(be_2-be_3)*(be_2-de_1)^3*(be_3-de_1)^3+(be_3-be_1)*(be_3-de_1)^3*(be_1-de_1)^3-(be_1-be_2)*(be_1-de_2)^3*(be_2-de_2)^3-(be_2-be_3)*(be_2-de_2)^3*(be_3-de_2)^3-(be_3-be_1)*(be_3-de_2)^3*(be_1-de_2)^3;

i11 : s1 = -b_1^3+4*b_1*d_2-3*b_3;

i12 : r2 = de_1*(be_1-be_2)*(be_1-de_1)^3*(be_2-de_1)^3+de_1*(be_2-be_3)*(be_2-de_1)^3*(be_3-de_1)^3+de_1*(be_3-be_1)*(be_3-de_1)^3*(be_1-de_1)^3-de_2*(be_1-be_2)*(be_1-de_2)^3*(be_2-de_2)^3-de_2*(be_2-be_3)*(be_2-de_2)^3*(be_3-de_2)^3-de_2*(be_3-be_1)*(be_3-de_2)^3*(be_1-de_2)^3;

i13 : s2 = -b_1^4+5*b_1^2*d_2-2*b_1*b_3-b_2^2+3*b_2*d_2-6*d_2^2;

i14 : r3 = (be_1^2-be_2^2)*(be_1-de_1)^3*(be_2-de_1)^3+(be_2^2-be_3^2)*(be_2-de_1)^3*(be_3-de_1)^3+(be_3^2-be_1^2)*(be_3-de_1)^3*(be_1-de_1)^3-(be_1^2-be_2^2)*(be_1-de_2)^3*(be_2-de_2)^3-(be_2^2-be_3^2)*(be_2-de_2)^3*(be_3-de_2)^3-(be_3^2-be_1^2)*(be_3-de_2)^3*(be_1-de_2)^3;

i15 : s3 = -b_1^2*b_2+b_2*d_2+3*d_2^2;

i16 : r4 = de_1*(be_1^2-be_2^2)*(be_1-de_1)^3*(be_2-de_1)^3+de_1*(be_2^2-be_3^2)*(be_2-de_1)^3*(be_3-de_1)^3+de_1*(be_3^2-be_1^2)*(be_3-de_1)^3*(be_1-de_1)^3-de_2*(be_1^2-be_2^2)*(be_1-de_2)^3*(be_2-de_2)^3-de_2*(be_2^2-be_3^2)*(be_2-de_2)^3*(be_3-de_2)^3-de_2*(be_3^2-be_1^2)*(be_3-de_2)^3*(be_1-de_2)^3;

i17 : s4 = -b_1^3*b_2+b_1^2*b_3+5*b_1*b_2*d_2-b_1*b_2^2+b_2*b_3-6*b_3*d_2;

i18 : r5 = (be_1-be_2)*(be_3-de_1)^3*(be_3-de_2)^3*(de_1-de_2)+(be_2-be_3)*(be_1-de_1)^3*(be_1-de_2)^3*(de_1-de_2)+(be_3-be_1)*(be_2-de_1)^3*(be_2-de_2)^3*(de_1-de_2);

i19 : s5 = b_1*b_3-b_2^2+3*b_2*d_2-3*d_2^2;

i20 : r6 = (be_1-be_2)*(be_2-be_3)*(be_3-be_1)*((be_1-de_1)^3*(be_2-de_1)^3+(be_2-de_1)^3*(be_3-de_1)^3+(be_3-de_1)^3*(be_1-de_1)^3-(be_1-de_2)^3*(be_2-de_2)^3-(be_2-de_2)^3*(be_3-de_2)^3-(be_3-de_2)^3*(be_1-de_2)^3);

i21 : s6 = b_1^5-3*b_1^3*b_2-4*b_1^3*d_2+6*b_1^2*b_3+9*b_1*b_2*d_2+3*b_1*d_2^2+3*b_2*b_3-21*b_3*d_2;

We check that $s_i = r_i/\Delta$.

i22 : r1//D == s1

o22 = true

i23 : r2//D == s2

o23 = true

i24 : r3//D == s3

o24 = true

i25 : r4//D == s4

o25 = true

i26 : r5//D == s5

o26 = true

i27 : r6//D == s6

o27 = true

Then we obtain a ring presentation with four generators $b_1, b_2, b_3, d_2$. We eliminate $b_3$ and find a new presentation.

i1 : T = QQ[b_3,b_1,b_2,d_2,Degrees=>{3,1,2,2},MonomialOrder=>Eliminate 1];

i2 : J = ideal(-b_1^3+4*b_1*d_2-3*b_3,-b_1^4+5*b_1^2*d_2-2*b_1*b_3-b_2^2+3*b_2*d_2-6*d_2^2,-b_1^2*b_2+b_2*d_2+3*d_2^2,-b_1^3*b_2+b_1^2*b_3+5*b_1*b_2*d_2-b_1*b_2^2+b_2*b_3-6*b_3*d_2,b_1*b_3-b_2^2+3*b_2*d_2-3*d_2^2,b_1^5-3*b_1^3*b_2-4*b_1^3*d_2+6*b_1^2*b_3+9*b_1*b_2*d_2+3*b_1*d_2^2+3*b_2*b_3-21*b_3*d_2);

o2 : Ideal of T

i3 : print toString ideal selectInSubring(1,gens gb J)

ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,2*b_2*d_2^2-3*d_2^3,2*b_2^2*d_2-5*d_2^3,2*b_2^3-9*d_2^3,b_1*d_2^3,d_2^4)

The obtained relation ideal has several redundant relations. We remove them and check that the new relation ideal is same to the above one.

i1 : T = QQ[b_1,b_2,d_2,Degrees=>{1,2,2}];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,2*b_2*d_2^2-3*d_2^3,2*b_2^2*d_2-5*d_2^3,2*b_2^3-9*d_2^3,b_1*d_2^3,d_2^4);

o2 : Ideal of T

i3 : J = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2);

o3 : Ideal of T

i4 : I == J

o4 = true

Finally, we check that the presentation $T/J$ has the expected Hilbert series.

i5 : apply(10, i -> hilbertFunction(i,T/J))

o5 = {1, 1, 3, 3, 3, 1, 1, 0, 0, 0}

o5 : List

Here are some calculations in Macaulay2 to compute the Chow ring of $\mathrm{M}^+$ in Section 5.

First of all, we compute the total Chern class of $\mathcal{U}$ in Proposition 5.1. We used Schubert2 package to compute Chern classes.

i1 : loadPackage "Schubert2";

i2 : R = QQ[b_1,b_2,d_2,Degrees=>{1,2,2}];

i3 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2);

o3 : Ideal of R

i4 : S = R/I;

i5 : b_3 = (-b_1^3+4*b_1*d_2)/3;

i6 : X = abstractVariety(6,S);

i7 : E = abstractSheaf(X,ChernClass=>1+b_1+b_2+b_3,Rank=>3);

i8 : F = abstractSheaf(X,ChernClass=>1+b_1+d_2,Rank=>2);

i9 : U = E^6-F^3;

i10 : print toString chern(U)

1+3*b_1+(3*b_1^2+6*b_2-3*d_2)+(-b_1^3+12*b_1*b_2+2*b_1*d_2)+(27*b_2^2-48*b_2*d_2+51*d_2^2)+28*b_1*d_2^2+56*d_2^3

Now we show Proposition 5.9. We show that the ideal $J$ is equal to $\ker i^*$ in the Chow ring $\mathrm{A}^*(\mathrm{Q}) = S$, by computing the Hilbert polynomial of $S/J$.

i1 : R = QQ[p,b_1,b_2,d_2,Degrees=>{1,1,2,2}];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3);

o2 : Ideal of R

i3 : S = R/I;

i4 : J = ideal(b_2-d_2,b_1^2-3*d_2,d_2-b_1*p+3*p^2,b_1*b_2,d_2^2);

o4 : Ideal of S

i5 : apply(5, i -> hilbertFunction(i,S/J))

o5 = {1, 2, 2, 1, 0}

o5 : List

The Hilbert polynomial is equal to that of $\mathrm{Fl}(V)$.

We check that the ring presentation in Proposition 5.9 has the expected Hilbert polynomial.

i1 : R = QQ[t,p,b_1,b_2,d_2,Degrees=>{1,1,1,2,2}];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2));

o2 : Ideal of R

i3 : apply(20, i -> hilbertFunction(i,R/I))

o3 = {1, 3, 8, 13, 17, 18, 19, 19, 19, 19, 19, 19, 18, 17, 13, 8, 3, 1, 0, 0}

o3 : List

Here are some calculations in Macaulay2 to compute the Chow ring of $\mathrm{M}$ in Section 6.

By elimination, we compute the relation ideal of the invariant subring. In the code below, t is $\tau$, p is $\rho$, aa is $\alpha$, and bb is $\beta$.

i1 : R = QQ[t,p,b_1,b_2,d_2,aa,bb,x,y,z,Degrees=>{1,1,1,2,2,1,1,2,2,2},MonomialOrder=>Eliminate 5];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2),aa-(p+t),bb-(b_1+3*t),x-(b_2-(3*p-b_1)*t),y-(d_2-(3*p-b_1)*t),z-t*(3*t+3*p+b_1)/3);

o2 : Ideal of R

i3 : print toString ideal selectInSubring(1,gens gb I);

ideal(x*z-y*z,bb^2*z-3*y*z-9*z^2,3*aa^2*z-aa*bb*z+y*z,bb^2*y-3*y^2-9*y*z,bb^2*x-x*y-3*y^2-3*aa*bb*z-9*y*z+9*z^2,bb^4+3*x^2-9*x*y-3*y^2-54*y*z-81*z^2,bb*y*z+9*aa*z^2-3*bb*z^2,2*bb*x*y-3*bb*y^2-9*aa*y*z-27*aa*z^2+9*bb*z^2,3*bb*x^2-7*bb*y^2-36*aa*y*z-108*aa*z^2+36*bb*z^2,y^2*z+3*aa*bb*z^2-9*z^3,2*x*y^2-3*y^3,2*x^2*y-5*y^3,2*x^3-9*y^3,bb*y^3,y^4,aa^12+3*aa^11*bb+3*aa^10*bb^2-aa^9*bb^3+6*aa^10*x+12*aa^9*bb*x+27*aa^8*x^2-3*aa^10*y+2*aa^9*bb*y-48*aa^8*x*y+51*aa^8*y^2+28*aa^7*bb*y^2+56*aa^6*y^3+201*aa*bb*z^5-19*y*z^5-613*z^6,6*aa^10*x*y-12*aa^10*y^2-10*aa^9*bb*y^2-45*aa^8*y^3-104*aa*bb*z^6+2*y*z^6+310*z^7,3*aa^10*bb*y^2+30*aa^9*y^3+15*aa*y*z^6+63*aa*z^7-19*bb*z^7,2*aa*bb*z^7+y*z^7-7*z^8,3*aa^10*y^3-y*z^7+3*z^8,aa*z^8,3*aa*y*z^7-2*bb*z^8,z^9,y*z^8)

There are many redundant relations, in this relation ideal. To simplify the presentation, we find a smaller set of relations.

i1 : S = QQ[aa,bb,x,y,z,Degrees=>{1,1,2,2,2}];

i2 : J = ideal(x*z-y*z,bb^2*z-3*y*z-9*z^2,3*aa^2*z-aa*bb*z+y*z,bb^2*y-3*y^2-9*y*z,bb^2*x-x*y-3*y^2-3*aa*bb*z-9*y*z+9*z^2,bb^4+3*x^2-9*x*y-3*y^2-54*y*z-81*z^2,bb*y*z+9*aa*z^2-3*bb*z^2,2*bb*x*y-3*bb*y^2-9*aa*y*z-27*aa*z^2+9*bb*z^2,3*bb*x^2-7*bb*y^2-36*aa*y*z-108*aa*z^2+36*bb*z^2,y^2*z+3*aa*bb*z^2-9*z^3,2*x*y^2-3*y^3,2*x^2*y-5*y^3,2*x^3-9*y^3,bb*y^3,y^4,aa^12+3*aa^11*bb+3*aa^10*bb^2-aa^9*bb^3+6*aa^10*x+12*aa^9*bb*x+27*aa^8*x^2-3*aa^10*y+2*aa^9*bb*y-48*aa^8*x*y+51*aa^8*y^2+28*aa^7*bb*y^2+56*aa^6*y^3+201*aa*bb*z^5-19*y*z^5-613*z^6,6*aa^10*x*y-12*aa^10*y^2-10*aa^9*bb*y^2-45*aa^8*y^3-104*aa*bb*z^6+2*y*z^6+310*z^7,3*aa^10*bb*y^2+30*aa^9*y^3+15*aa*y*z^6+63*aa*z^7-19*bb*z^7,2*aa*bb*z^7+y*z^7-7*z^8,3*aa^10*y^3-y*z^7+3*z^8,aa*z^8,3*aa*y*z^7-2*bb*z^8,z^9,y*z^8);

o2 : Ideal of S

i3 : K = ideal(x*z-y*z,bb^2*z-3*y*z-9*z^2,3*aa^2*z-aa*bb*z+y*z,bb^2*y-3*y^2-9*y*z,bb^2*x-x*y-3*y^2-3*aa*bb*z-9*y*z+9*z^2,bb^4+3*x^2-9*x*y-3*y^2-54*y*z-81*z^2,bb*y*z+9*aa*z^2-3*bb*z^2,2*bb*x*y-3*bb*y^2-9*aa*y*z-27*aa*z^2+9*bb*z^2,3*bb*x^2-7*bb*y^2-36*aa*y*z-108*aa*z^2+36*bb*z^2,aa^12+3*aa^11*bb+3*aa^10*bb^2-aa^9*bb^3+6*aa^10*x+12*aa^9*bb*x+27*aa^8*x^2-3*aa^10*y+2*aa^9*bb*y-48*aa^8*x*y+51*aa^8*y^2+28*aa^7*bb*y^2+56*aa^6*y^3+201*aa*bb*z^5-19*y*z^5-613*z^6,6*aa^10*x*y-12*aa^10*y^2-10*aa^9*bb*y^2-45*aa^8*y^3-104*aa*bb*z^6+2*y*z^6+310*z^7);

o3 : Ideal of S

i4 : J==K

o4 = true

We check that the obtained Chow ring presentation $S/K$ has the expected Hilbert polynomial.

i5 : apply(21,i->hilbertFunction(i,S/K))

o5 = {1, 2, 6, 10, 14, 15, 16, 16, 16, 16, 16, 16, 15, 14, 10, 6, 2, 1, 0, 0, 0}

o5 : List

Here are some calculations in Macaulay2 to compute the total Chern class of $\mathcal{T}_{\mathrm{M}}$ in Section 7.

By using the exact sequence in [Tjo98, Equation (3.4)], we can compute the Chern class of $\mathcal{T}_{\mathrm{N}}$.

i1 : loadPackage "Schubert2";

i2 : R = QQ[b_1,b_2,d_2,Degrees=>{1,2,2}];

i3 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2);

o3 : Ideal of R

i4 : S = R/I;

i5 : b_3 = (-b_1^3+4*b_1*d_2)/3;

i6 : X = abstractVariety(6,S);

i7 : E = abstractSheaf(X,ChernClass=>1+b_1+b_2+b_3,Rank=>3);

i8 : F = abstractSheaf(X,ChernClass=>1+b_1+d_2,Rank=>2);

i9 : T = (E*dual(F))^3-dual(E)*E-dual(F)*F+1;

i10 : print toString chern(T)

1-3*b_1+(3*b_1^2+5*d_2)+(-3*b_1^3-4*b_1*d_2)+(-9*b_2^2+27*b_2*d_2+4*d_2^2)-(17/2)*b_1*d_2^2+(13/2)*d_2^3

By [Ful98, Examples 3.2.2 and 3.2.11], we obtain the total Chern class of \(\mathcal{T}_{\mathrm{Q}}\).

i1 : loadPackage "Schubert2";

i2 : R = QQ[p,b_1,b_2,d_2,Degrees=>{1,1,2,2}];

i3 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3);

o3 : Ideal of R

i4 : S = R/I;

i5 : X = abstractVariety(17,S);

i6 : TN = abstractSheaf(X,ChernClass=>1-3*b_1+(3*b_1^2+5*d_2)+(-3*b_1^3-4*b_1*d_2)+(-9*b_2^2+27*b_2*d_2+4*d_2^2)-(17/2)*b_1*d_2^2+(13/2)*d_2^3,Rank=>6);

i7 : U = abstractSheaf(X,ChernClass=>1+3*b_1+(3*b_1^2+6*b_2-3*d_2)+(-b_1^3+12*b_1*b_2+2*b_1*d_2)+(27*b_2^2-48*b_2*d_2+51*d_2^2)+28*b_1*d_2^2+56*d_2^3,Rank=>12);

i8 : L = abstractSheaf(X,ChernClass=>1+p,Rank=>1);

i9 : TQ = U*L+TN-1;

i10 : print toString chern(TQ)

1+12*p+(66*p^2-3*p*b_1-3*b_1^2+6*b_2+2*d_2)+(220*p^3-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2+30*p*d_2+22*b_1*d_2)+(495*p^4-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2)+(792*p^5-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2)+(924*p^6-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3)+(792*p^7-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3)+(495*p^8-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3)+(220*p^9-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3)+(66*p^10-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3)+(12*p^11-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3)+(-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3)+(36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3)+(-36*p^11*b_1^3-297*p^10*b_2^2-48*p^11*b_1*d_2+891*p^10*b_2*d_2-429*p^10*d_2^2+(25/2)*p^9*b_1*d_2^2+126*p^8*d_2^3)+(-108*p^11*b_2^2+324*p^11*b_2*d_2+48*p^11*d_2^2+(429/2)*p^10*b_1*d_2^2+(1315/2)*p^9*d_2^3)+(-102*p^11*b_1*d_2^2-(825/2)*p^10*d_2^3)+78*p^11*d_2^3

To apply the blow-up formula ([Ful98, Theorem 15.4]), we need to know the Chern classes of the blow-up center $\mathrm{Fl}(V)$.

i1 : loadPackage "Schubert2";

i2 : R = QQ[x,h,Degrees=>{1,1}];

i3 : I = ideal(h^3,x^2+h*x+h^2);

o3 : Ideal of R

i4 : S = R/I;

i5 : X = abstractVariety(3,S);

i6 : TP = abstractSheaf(X,ChernClass=>1+3*h+3*h^2,Rank=>2);

i7 : H = abstractSheaf(X,ChernClass=>1+h,Rank=>1);

i8 : L = abstractSheaf(X,ChernClass=>1+x,Rank=>1);

i9 : TFV = TP*dual(H)*L+TP-1;

i10 : print toString chern(TFV);

1+(2*x+4*h)+(6*x*h+6*h^2)+6*x*h^2

Then by blow-up formula ([Ful98, Theorem 15.4]), we obtain $c(\mathcal{T}_{\mathrm{M}^+})$. We compute the difference $c(\mathcal{T}_{\mathrm{M}^+})-q^*c(\mathcal{T}_{\mathrm{Q}})$.

i1 : R = QQ[t,x,h,Degrees=>{1,1,1}];

i2 : k = h+x;

i3 : I = ideal(h^3,x^2-k*x+k^2,t^14+(10*k-14*h)*t^13+(-75*k*h+36*h^2)*t^12-120*k*h^2*t^11);

o3 : Ideal of R

i4 : S = R/I;

i5 : cN = {1,10*x-4*h,-75*x*h-39*h^2,120*x*h^2,0,0,0,0,0,0,0,0,0,0,0,0};

i6 : cFl = 1+(2*x+4*h)+(6*x*h+6*h^2)+6*x*h^2;

i7 : alphaQ = 0;

i8 : for j to 14 do for i to 14-j do alphaQ = alphaQ + (binomial(14-j,i)-binomial(14-j,i+1))*t^i*cN_j;

i9 : dif = cFl*alphaQ;

i10 : print toString dif

78*t^13*x*h^2+78*t^13*x*h+78*t^13*h^2+174*t^12*x*h^2+26*t^13*x+52*t^13*h+606*t^12*x*h+30*t^12*h^2-2190*t^11*x*h^2+13*t^13+274*t^12*x+260*t^12*h+1593*t^11*x*h-1131*t^11*h^2-12282*t^10*x*h^2+77*t^12+1196*t^11*x+832*t^11*h+2268*t^10*x*h-5772*t^10*h^2-34122*t^9*x*h^2+273*t^11+3354*t^10*x+1716*t^10*h-396*t^9*x*h-15444*t^9*h^2-57222*t^8*x*h^2+637*t^10+6292*t^9*x+2288*t^9*h-7755*t^8*x*h-25311*t^8*h^2-55638*t^7*x*h^2+1001*t^9+7722*t^8*x+1716*t^8*h-14553*t^7*x*h-24453*t^7*h^2-16830*t^6*x*h^2+1001*t^8+5148*t^7*x-12474*t^6*x*h-7722*t^6*h^2+34386*t^5*x*h^2+429*t^7-858*t^6*x-1716*t^6*h-1254*t^5*x*h+14586*t^5*h^2+59598*t^4*x*h^2-429*t^6-6292*t^5*x-2288*t^5*h+9405*t^4*x*h+26169*t^4*h^2+49374*t^3*x*h^2-1001*t^5-7722*t^4*x-1716*t^4*h+11655*t^3*x*h+22347*t^3*h^2+25266*t^2*x*h^2-1001*t^4-5564*t^3*x-832*t^3*h+7416*t^2*x*h+11856*t^2*h^2+8130*t*x*h^2-637*t^3-2626*t^2*x-260*t^2*h+2808*t*x*h+3984*t*h^2+1518*x*h^2-273*t^2-804*t*x-48*t*h+603*x*h+783*h^2-77*t-146*x-4*h-13

By adding it to $q^*(c(\mathcal{T}_{\mathrm{Q}}))$, we obtain $c(\mathcal{T}_{\mathrm{M}^+})$. Note that the below computation is valid because the restriction map $\mathrm{A}^*(\mathrm{Q}) \to \mathrm{A}^*(\mathrm{Fl}(V))$ is surjective.

i1 : R = QQ[t,p,b_1,b_2,d_2,Degrees=>{1,1,1,2,2}];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2));

o2 : Ideal of R

i3 : S = R/I;

i4 : t = t;

i5 : x = p;

i6 : h = -1/3*b_1;

i7 : dif = -t*(78*t^13*x*h^2+78*t^13*x*h+78*t^13*h^2+174*t^12*x*h^2+26*t^13*x+52*t^13*h+606*t^12*x*h+30*t^12*h^2-2190*t^11*x*h^2+13*t^13+274*t^12*x+260*t^12*h+1593*t^11*x*h-1131*t^11*h^2-12282*t^10*x*h^2+77*t^12+1196*t^11*x+832*t^11*h+2268*t^10*x*h-5772*t^10*h^2-34122*t^9*x*h^2+273*t^11+3354*t^10*x+1716*t^10*h-396*t^9*x*h-15444*t^9*h^2-57222*t^8*x*h^2+637*t^10+6292*t^9*x+2288*t^9*h-7755*t^8*x*h-25311*t^8*h^2-55638*t^7*x*h^2+1001*t^9+7722*t^8*x+1716*t^8*h-14553*t^7*x*h-24453*t^7*h^2-16830*t^6*x*h^2+1001*t^8+5148*t^7*x-12474*t^6*x*h-7722*t^6*h^2+34386*t^5*x*h^2+429*t^7-858*t^6*x-1716*t^6*h-1254*t^5*x*h+14586*t^5*h^2+59598*t^4*x*h^2-429*t^6-6292*t^5*x-2288*t^5*h+9405*t^4*x*h+26169*t^4*h^2+49374*t^3*x*h^2-1001*t^5-7722*t^4*x-1716*t^4*h+11655*t^3*x*h+22347*t^3*h^2+25266*t^2*x*h^2-1001*t^4-5564*t^3*x-832*t^3*h+7416*t^2*x*h+11856*t^2*h^2+8130*t*x*h^2-637*t^3-2626*t^2*x-260*t^2*h+2808*t*x*h+3984*t*h^2+1518*x*h^2-273*t^2-804*t*x-48*t*h+603*x*h+783*h^2-77*t-146*x-4*h-13);

i8 : cTQ = 1+12*p+(66*p^2-3*p*b_1-3*b_1^2+6*b_2+2*d_2)+(220*p^3-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2+30*p*d_2+22*b_1*d_2)+(495*p^4-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2)+(792*p^5-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2)+(924*p^6-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3)+(792*p^7-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3)+(495*p^8-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3)+(220*p^9-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3)+(66*p^10-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3)+(12*p^11-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3)+(-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3)+(36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3)+(-36*p^11*b_1^3-297*p^10*b_2^2-48*p^11*b_1*d_2+891*p^10*b_2*d_2-429*p^10*d_2^2+(25/2)*p^9*b_1*d_2^2+126*p^8*d_2^3)+(-108*p^11*b_2^2+324*p^11*b_2*d_2+48*p^11*d_2^2+(429/2)*p^10*b_1*d_2^2+(1315/2)*p^9*d_2^3)+(-102*p^11*b_1*d_2^2-(825/2)*p^10*d_2^3)+78*p^11*d_2^3;

i9 : cTMp = cTQ + dif;

i10 : print toString cTMp

117*p^11*d_2^3-162*t^13*p*d_2-141*p^11*b_1*d_2^2-(1527/2)*p^10*d_2^3+150*t^13*p*b_1-108*p^11*b_2^2-166*t^13*d_2-258*t^12*p*d_2+246*p^11*b_2*d_2+204*p^11*d_2^2+(637/2)*p^10*b_1*d_2^2+(1965/2)*p^9*d_2^3-144*t^13*p+104*t^13*b_1+856*t^12*p*b_1-36*p^11*b_1^3-297*p^10*b_2^2+208*t^12*d_2+4614*t^11*p*d_2-48*p^11*b_1*d_2+852*p^10*b_2*d_2-351*p^10*d_2^2+(155/2)*p^9*b_1*d_2^2+(837/2)*p^8*d_2^3-77*t^13-1196*t^12*p+(832/3)*t^12*b_1+756*t^11*p*b_1+36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2+1924*t^11*d_2+11374*t^10*p*d_2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3-273*t^12-3354*t^11*p+572*t^11*b_1-132*t^10*p*b_1-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2+5148*t^10*d_2+19074*t^9*p*d_2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3-637*t^11-6292*t^10*p+12*p^11+(2288/3)*t^10*b_1-2585*t^9*p*b_1-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2+8437*t^9*d_2+18546*t^8*p*d_2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3-1001*t^10-7722*t^9*p+66*p^10+572*t^9*b_1-4851*t^8*p*b_1-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2+8151*t^8*d_2+5610*t^7*p*d_2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3-1001*t^9-5148*t^8*p+220*p^9-4158*t^7*p*b_1-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+2574*t^7*d_2-11462*t^6*p*d_2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3-429*t^8+858*t^7*p+495*p^8-572*t^7*b_1-418*t^6*p*b_1-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2-4862*t^6*d_2-19866*t^5*p*d_2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3+429*t^7+6292*t^6*p+792*p^7-(2288/3)*t^6*b_1+3135*t^5*p*b_1-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2-8723*t^5*d_2-16458*t^4*p*d_2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3+1001*t^6+7722*t^5*p+924*p^6-572*t^5*b_1+3885*t^4*p*b_1-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2-7449*t^4*d_2-8422*t^3*p*d_2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3+1001*t^5+5564*t^4*p+792*p^5-(832/3)*t^4*b_1+2472*t^3*p*b_1-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2-3952*t^3*d_2-2710*t^2*p*d_2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2+637*t^4+2626*t^3*p+495*p^4-(260/3)*t^3*b_1+936*t^2*p*b_1-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2-1328*t^2*d_2-506*t*p*d_2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2+273*t^3+804*t^2*p+220*p^3-16*t^2*b_1+201*t*p*b_1-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2-261*t*d_2+30*p*d_2+22*b_1*d_2+77*t^2+146*t*p+66*p^2-(4/3)*t*b_1-3*p*b_1-3*b_1^2+6*b_2+2*d_2+13*t+12*p+1

By the same idea, we can compute the difference $c(\mathcal{T}_{\mathrm{M}^+})-r^*(c(\mathcal{T}_{\mathrm{M}}))$.

i1 : R = QQ[t,x,h,Degrees=>{1,1,1}];

i2 : k = h+x;

i3 : n=t+k-h;

i4 : I = ideal(h^3,x^2+h*x+h^2,t^14+(10*k-14*h)*t^13+(-75*k*h+36*h^2)*t^12-120*k*h^2*t^11);

o4 : Ideal of R

i5 : S = R/I;

i6 : cN = {1,k-2*n,k^2*k*n+n^2,0,0,0,0,0,0,0,0,0,0,0,0,0};

i7 : cC4 = 1+(-k+14*n)+(7*k^2-10*k*n+91*n^2)+(78*k^2*n-39*k*n^2+364*n^3)+(393*k^2*n^2-52*k*n^3+1001*n^4)+(1180*k^2*n^3+143*k*n^4+2002*n^5)+(2343*k^2*n^4+858*k*n^5+3003*n^6)+(3234*k^2*n^5+2145*k*n^6+3432*n^7)+(3201*k^2*n^6+3432*k*n^7+3003*n^8)+(2376*k^2*n^7+3861*k*n^8+2002*n^9)+(1485*k^2*n^8+3146*k*n^9+1001*n^10)+(946*k^2*n^9+1859*k*n^10+364*n^11)+(627*k^2*n^10+780*k*n^11+91*n^12)+(348*k^2*n^11+221*k*n^12+14*n^13)+((285/4)*k*n^13-(117/16)*n^14)+(21/2)*k*n^14;

i8 : alphaM = 0;

i9 : for j to 2 do for i to 2-j do alphaM = alphaM + (binomial(2-j,i)-binomial(2-j,i+1))*t^i*cN_j;

i10 : dif = cC4*alphaM;

i11 : print toString dif

42*t^13*x*h^2+506*t^13*x*h+42*t^13*h^2-7670*t^12*x*h^2+312*t^13*x+326*t^13*h+878*t^12*x*h-3835*t^12*h^2-25665*t^11*x*h^2+77*t^13+1833*t^12*x+923*t^12*h-3140*t^11*x*h-15210*t^11*h^2-71225*t^10*x*h^2+273*t^12+4992*t^11*x+2080*t^11*h-11550*t^10*x*h-35035*t^10*h^2-119504*t^9*x*h^2+637*t^11+9295*t^10*x+3289*t^10*h-22627*t^9*x*h-54769*t^9*h^2-139788*t^8*x*h^2+1001*t^10+11726*t^9*x+3718*t^9*h-25707*t^8*x*h-56628*t^8*h^2-111144*t^7*x*h^2+1001*t^9+9009*t^8*x+3003*t^8*h-12474*t^7*x*h-32604*t^7*h^2-53625*t^6*x*h^2+429*t^8+1716*t^7*x+1716*t^7*h+9900*t^6*x*h+2145*t^6*h^2-7260*t^5*x*h^2-429*t^7-5291*t^6*x+715*t^6*h+24277*t^5*x*h+23595*t^5*h^2+9658*t^4*x*h^2-1001*t^6-7722*t^5*x+286*t^5*h+22809*t^4*x*h+23738*t^4*h^2+7756*t^3*x*h^2-1001*t^5-5837*t^4*x+169*t^4*h+12824*t^3*x*h+13286*t^3*h^2+2793*t^2*x*h^2-637*t^4-2808*t^3*x+104*t^3*h+4498*t^2*x*h+4563*t^2*h^2+520*t*x*h^2-273*t^3-867*t^2*x+43*t^2*h+915*t*x*h+905*t*h^2+40*x*h^2-77*t^2-158*t*x+10*t*h+83*x*h+80*h^2-13*t-13*x+h-1

And we obtain $r^*(c(\mathcal{T}_{\mathrm{M}}))$.

i1 : R = QQ[t,p,b_1,b_2,d_2,Degrees=>{1,1,1,2,2}];

i2 : I = ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2));

o2 : Ideal of R

i3 : S = R/I;

i4 : x = p;

i5 : h = -1/3*b_1;

i6 : dif = -t*(42*t^13*x*h^2+506*t^13*x*h+42*t^13*h^2-7670*t^12*x*h^2+312*t^13*x+326*t^13*h+878*t^12*x*h-3835*t^12*h^2-25665*t^11*x*h^2+77*t^13+1833*t^12*x+923*t^12*h-3140*t^11*x*h-15210*t^11*h^2-71225*t^10*x*h^2+273*t^12+4992*t^11*x+2080*t^11*h-11550*t^10*x*h-35035*t^10*h^2-119504*t^9*x*h^2+637*t^11+9295*t^10*x+3289*t^10*h-22627*t^9*x*h-54769*t^9*h^2-139788*t^8*x*h^2+1001*t^10+11726*t^9*x+3718*t^9*h-25707*t^8*x*h-56628*t^8*h^2-111144*t^7*x*h^2+1001*t^9+9009*t^8*x+3003*t^8*h-12474*t^7*x*h-32604*t^7*h^2-53625*t^6*x*h^2+429*t^8+1716*t^7*x+1716*t^7*h+9900*t^6*x*h+2145*t^6*h^2-7260*t^5*x*h^2-429*t^7-5291*t^6*x+715*t^6*h+24277*t^5*x*h+23595*t^5*h^2+9658*t^4*x*h^2-1001*t^6-7722*t^5*x+286*t^5*h+22809*t^4*x*h+23738*t^4*h^2+7756*t^3*x*h^2-1001*t^5-5837*t^4*x+169*t^4*h+12824*t^3*x*h+13286*t^3*h^2+2793*t^2*x*h^2-637*t^4-2808*t^3*x+104*t^3*h+4498*t^2*x*h+4563*t^2*h^2+520*t*x*h^2-273*t^3-867*t^2*x+43*t^2*h+915*t*x*h+905*t*h^2+40*x*h^2-77*t^2-158*t*x+10*t*h+83*x*h+80*h^2-13*t-13*x+h-1);

i7 : cTMp = 117*p^11*d_2^3-162*t^13*p*d_2-141*p^11*b_1*d_2^2-(1527/2)*p^10*d_2^3+150*t^13*p*b_1-108*p^11*b_2^2-166*t^13*d_2-258*t^12*p*d_2+246*p^11*b_2*d_2+204*p^11*d_2^2+(637/2)*p^10*b_1*d_2^2+(1965/2)*p^9*d_2^3-144*t^13*p+104*t^13*b_1+856*t^12*p*b_1-36*p^11*b_1^3-297*p^10*b_2^2+208*t^12*d_2+4614*t^11*p*d_2-48*p^11*b_1*d_2+852*p^10*b_2*d_2-351*p^10*d_2^2+(155/2)*p^9*b_1*d_2^2+(837/2)*p^8*d_2^3-77*t^13-1196*t^12*p+(832/3)*t^12*b_1+756*t^11*p*b_1+36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2+1924*t^11*d_2+11374*t^10*p*d_2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3-273*t^12-3354*t^11*p+572*t^11*b_1-132*t^10*p*b_1-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2+5148*t^10*d_2+19074*t^9*p*d_2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3-637*t^11-6292*t^10*p+12*p^11+(2288/3)*t^10*b_1-2585*t^9*p*b_1-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2+8437*t^9*d_2+18546*t^8*p*d_2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3-1001*t^10-7722*t^9*p+66*p^10+572*t^9*b_1-4851*t^8*p*b_1-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2+8151*t^8*d_2+5610*t^7*p*d_2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3-1001*t^9-5148*t^8*p+220*p^9-4158*t^7*p*b_1-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+2574*t^7*d_2-11462*t^6*p*d_2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3-429*t^8+858*t^7*p+495*p^8-572*t^7*b_1-418*t^6*p*b_1-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2-4862*t^6*d_2-19866*t^5*p*d_2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3+429*t^7+6292*t^6*p+792*p^7-(2288/3)*t^6*b_1+3135*t^5*p*b_1-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2-8723*t^5*d_2-16458*t^4*p*d_2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3+1001*t^6+7722*t^5*p+924*p^6-572*t^5*b_1+3885*t^4*p*b_1-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2-7449*t^4*d_2-8422*t^3*p*d_2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3+1001*t^5+5564*t^4*p+792*p^5-(832/3)*t^4*b_1+2472*t^3*p*b_1-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2-3952*t^3*d_2-2710*t^2*p*d_2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2+637*t^4+2626*t^3*p+495*p^4-(260/3)*t^3*b_1+936*t^2*p*b_1-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2-1328*t^2*d_2-506*t*p*d_2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2+273*t^3+804*t^2*p+220*p^3-16*t^2*b_1+201*t*p*b_1-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2-261*t*d_2+30*p*d_2+22*b_1*d_2+77*t^2+146*t*p+66*p^2-(4/3)*t*b_1-3*p*b_1-3*b_1^2+6*b_2+2*d_2+13*t+12*p+1;

i8 : cTM = cTMp - dif;

i9 : print toString cTM

96*p^11*d_2^3-(1492/3)*t^13*p*d_2+112*p^11*b_1*d_2^2+(3491/2)*p^10*d_2^3-512*t^13*p*b_1-108*p^11*b_2^2+(91/3)*t^13*d_2-4407*t^12*p*d_2+1182*p^11*b_2*d_2-1668*p^11*d_2^2-(2157/2)*p^10*b_1*d_2^2-(8815/2)*p^9*d_2^3+919*t^13*p-(919/3)*t^13*b_1-(67/3)*t^12*p*b_1-36*p^11*b_1^3-297*p^10*b_2^2-3861*t^12*d_2-(66623/3)*t^11*p*d_2-48*p^11*b_1*d_2+1083*p^10*b_2*d_2-813*p^10*d_2^2-(615/2)*p^9*b_1*d_2^2-1314*p^8*d_2^3+196*t^13+3796*t^12*p-416*t^12*b_1+4606*t^11*p*b_1+36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2-(29263/3)*t^11*d_2-(85382/3)*t^10*p*d_2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3+364*t^12+5941*t^11*p-(1573/3)*t^11*b_1+(22231/3)*t^10*p*b_1-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2-(39325/3)*t^10*d_2-27522*t^9*p*d_2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3+364*t^11+5434*t^10*p+12*p^11-(1430/3)*t^10*b_1+5984*t^9*p*b_1-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2-10439*t^9*d_2-18502*t^8*p*d_2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3+1287*t^9*p+66*p^10-429*t^9*b_1-693*t^8*p*b_1-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2-2717*t^8*d_2-12265*t^7*p*d_2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3-572*t^9-3432*t^8*p+220*p^9-572*t^8*b_1-7458*t^7*p*b_1-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+3289*t^7*d_2-13882*t^6*p*d_2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3-858*t^8-4433*t^7*p+495*p^8-(2431/3)*t^7*b_1-(25531/3)*t^6*p*b_1-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2+3003*t^6*d_2-(49940/3)*t^5*p*d_2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3-572*t^7-1430*t^6*p+792*p^7-858*t^6*b_1-4468*t^5*p*b_1-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2-(2431/3)*t^5*d_2-(41618/3)*t^4*p*d_2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3+1885*t^5*p+924*p^6-(1885/3)*t^5*b_1-(1169/3)*t^4*p*b_1-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2-(9061/3)*t^4*d_2-7491*t^3*p*d_2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3+364*t^5+2756*t^4*p+792*p^5-312*t^4*b_1+(2918/3)*t^3*p*b_1-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2-2431*t^3*d_2-(7610/3)*t^2*p*d_2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2+364*t^4+1759*t^3*p+495*p^4-101*t^3*b_1+631*t^2*p*b_1-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2-(3079/3)*t^2*d_2-(1478/3)*t*p*d_2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2+196*t^3+646*t^2*p+220*p^3-(58/3)*t^2*b_1+(520/3)*t*p*b_1-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2-(703/3)*t*d_2+30*p*d_2+22*b_1*d_2+64*t^2+133*t*p+66*p^2-(5/3)*t*b_1-3*p*b_1-3*b_1^2+6*b_2+2*d_2+12*t+12*p+1

Therefore the remaining task is to express the total Chern class as a combination of generators $\alpha, \beta, x,y,z$ of $\mathrm{A}^*(\mathrm{M})$. David Swinarski kindly made a code computing this presentation.

Let the degrees of p,t,b_1,b_2,d_2 be 1,1,1,2,2. For ease of notation this will be suppressed throughout the following discussion, but we assume it throughout.

Let I be the ideal in QQ[p,t,b_1,b_2,d_2] given by b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2)

Let R = QQ[p,t,b_1,b_2,d_2]/I. R is $\mathrm{A}^*(\mathrm{M}^+)$.

Let S be the subalgebra of R generated by u,v,x,y,z where

u = (p+t),

v = (b_1+3*t),

x = (b_2-(3*p-b_1)*t),

y = (d_2-(3*p-b_1)*t),

z = t*(3*t+3*p+b_1)/3.

So S is $\mathrm{A}^*(\mathrm{M})$.

Thus the degrees of u,v,x,y,z are 1,1,2,2,2.

The ring R is zero in all sufficiently large degrees. To get a finite-dimensional $\mathbb{Q}$-vector space, we choose a degree d and use only the part of J spanned by elements of degree less than or equal to d.

Choose a basis F of this truncation of J, and write k17 ($r^*c(\mathcal{T}_{\mathrm{M}})$) as

$k17 = \sum_{f \in F} c_f f$

Now rewrite k17 and all the f in terms of t,p,b_1,b_2,d_2. This yields one equation for each monomial in t,p,b_1,b_2,d_2. If this system of linear equations in the variables $c_f$ for $f \in F$ has a solution, then we have found a way to write k17 using elements of $S$ of degree less than d. Of course, if there is no solution, we cannot conclude that k17 is not in $S$, only that k17 cannot be written in $S$ as an expression of degree <= d.

i1 : S = QQ[u,v,x,y,z]/ideal({x*z-y*z, v^2*z-3*y*z-9*z^2, 3*u^2*z-u*v*z+y*z, v^2*y-3*y^2-9*y*z, v^2*x-x*y-3*y^2-3*u*v*z-9*y*z+9*z^2, v^4+3*x^2-9*x*y-3*y^2-54*y*z-81*z^2, v*y*z+9*u*z^2-3*v*z^2, 2*v*x*y-3*v*y^2-9*u*y*z-27*u*z^2+9*v*z^2, 3*v*x^2-7*v*y^2-36*u*y*z-108*u*z^2+36*v*z^2, y^2*z+3*u*v*z^2-9*z^3, 2*x*y^2-3*y^3, 2*x^2*y-5*y^3, 2*x^3-9*y^3, v*y^3, y^4, u^12+3*u^11*v+3*u^10*v^2-u^9*v^3+6*u^10*x+12*u^9*v*x+27*u^8*x^2-3*u^10*y+2*u^9*v*y-48*u^8*x*y+51*u^8*y^2+28*u^7*v*y^2+56*u^6*y^3+201*u*v*z^5-19*y*z^5-613*z^6, 6*u^10*x*y-12*u^10*y^2-10*u^9*v*y^2-45*u^8*y^3-104*u*v*z^6+2*y*z^6+310*z^7, 3*u^10*v*y^2+30*u^9*y^3+15*u*y*z^6+63*u*z^7-19*v*z^7, 2*u*v*z^7+y*z^7-7*z^8, 3*u^10*y^3-y*z^7+3*z^8, u*z^8, 3*u*y*z^7-2*v*z^8, z^9, y*z^8})

o1 = S

o1 : QuotientRing

i2 : J = flatten entries basis(0,18,ideal(S));

i3 : J = apply(#J, i -> lift(J_i,ambient(S)));

i4 : G = unique flatten J;

i5 : G = drop(G,{0,0});

i6 : R = QQ[t,p,b_1,b_2,d_2, Degrees=>{1,1,1,2,2}]/ideal(b_1^2*d_2-3*d_2^2,b_1^2*b_2-b_2*d_2-3*d_2^2,b_1^4+3*b_2^2-9*b_2*d_2-3*d_2^2,2*b_1*b_2*d_2-3*b_1*d_2^2,3*b_1*b_2^2-7*b_1*d_2^2,p^12+3*p^11*b_1+3*p^10*(b_1^2+2*b_2-d_2)+p^9*(-b_1^3+12*b_1*b_2+2*b_1*d_2)+3*p^8*(9*b_2^2-16*b_2*d_2+17*d_2^2)+28*p^7*b_1*d_2^2+56*p^6*d_2^3,t*(b_2-d_2),t*(b_1^2-3*d_2),t*(d_2-b_1*p+3*p^2),t*b_1*b_2,t*d_2^2,t^14+(10*p+4/3*b_1)*t^13+(25*p*b_1-13*b_2)*t^12+40*p*b_2*t^11+(p+1/3*b_1)^10*(-3*b_1^2*b_2+5*b_1^2*d_2));

i7 : F=map(R,S,{(p+t),(b_1+3*t),(b_2-(3*p-b_1)*t),(d_2-(3*p-b_1)*t),t*(3*t+3*p+b_1)/3});

o7 : RingMap R  F(G_i));

i9 : k17minusone = 96*p^11*d_2^3-(39/2)*p^11*b_1*d_2^2+(861/2)*p^10*d_2^3-161*t^13*p*b_1-108*p^11*b_2^2-2093*t^12*p*d_2+909*p^11*b_2*d_2-1122*p^11*d_2^2-669*p^10*b_1*d_2^2-2815*p^9*d_2^3+555*t^13*p-185*t^13*b_1+476*t^12*p*b_1-36*p^11*b_1^3-297*p^10*b_2^2-2405*t^12*d_2-8718*t^11*p*d_2-48*p^11*b_1*d_2+1083*p^10*b_2*d_2-813*p^10*d_2^2-(615/2)*p^9*b_1*d_2^2-1314*p^8*d_2^3+196*t^13+2795*t^12*p-(247/3)*t^12*b_1+5321*t^11*p*b_1+36*p^11*b_1^2-99*p^10*b_1^3-765*p^9*b_2^2-6084*t^11*d_2-2959*t^10*p*d_2+60*p^11*d_2-99*p^10*b_1*d_2+2775*p^9*b_2*d_2-1745*p^9*d_2^2+576*p^8*b_1*d_2^2+2301*p^7*d_2^3+364*t^12+3939*t^11*p+143*t^11*b_1+8030*t^10*p*b_1-36*p^11*b_1+99*p^10*b_1^2-255*p^9*b_1^3-180*p^9*b_1*b_2-1296*p^8*b_2^2-6435*t^10*d_2+8085*t^9*p*d_2+330*p^10*d_2+35*p^9*b_1*d_2+5724*p^8*b_2*d_2-5688*p^8*d_2^2+99*p^7*b_1*d_2^2+2814*p^6*d_2^3+364*t^11+2431*t^10*p+12*p^11+(1573/3)*t^10*b_1+6127*t^9*p*b_1-165*p^10*b_1+195*p^9*b_1^2-414*p^8*b_1^3+60*p^9*b_2-702*p^8*b_1*b_2-1566*p^7*b_2^2-1430*t^9*d_2+18678*t^8*p*d_2+1070*p^9*d_2+918*p^8*b_1*d_2+9426*p^7*b_2*d_2-11346*p^7*d_2^2-644*p^6*b_1*d_2^2+2628*p^5*d_2^3-2145*t^9*p+66*p^10+715*t^9*b_1-1122*t^8*p*b_1-495*p^9*b_1+135*p^8*b_1^2-522*p^7*b_1^3+270*p^8*b_2-1728*p^7*b_1*b_2-1512*p^6*b_2^2+6435*t^8*d_2+16764*t^7*p*d_2+2340*p^8*d_2+2934*p^7*b_1*d_2+12516*p^6*b_2*d_2-16170*p^6*d_2^2-1104*p^5*b_1*d_2^2+(3783/2)*p^4*d_2^3-572*t^9-6435*t^8*p+220*p^9+429*t^8*b_1-8173*t^7*p*b_1-990*p^8*b_1-234*p^7*b_1^2-588*p^6*b_1^3+720*p^7*b_2-2772*p^6*b_1*b_2-1080*p^5*b_2^2+10296*t^7*d_2+2849*t^6*p*d_2+3600*p^7*d_2+5292*p^6*b_1*d_2+12648*p^5*b_2*d_2-16656*p^5*d_2^2-(1757/2)*p^4*b_1*d_2^2+(2403/2)*p^3*d_2^3-858*t^8-6435*t^7*p+495*p^8-143*t^7*b_1-9130*t^6*p*b_1-1386*p^7*b_1-756*p^6*b_1^2-612*p^5*b_1^3+1260*p^6*b_2-3024*p^5*b_1*b_2-459*p^4*b_2^2+7007*t^6*d_2-9735*t^5*p*d_2+3990*p^6*d_2+6282*p^5*b_1*d_2+9231*p^4*b_2*d_2-12342*p^4*d_2^2-(775/2)*p^3*b_1*d_2^2+615*p^2*d_2^3-572*t^7-2431*t^6*p+792*p^7-(1573/3)*t^6*b_1-4819*t^5*p*b_1-1386*p^6*b_1-1026*p^5*b_1^2-531*p^4*b_1^3+1512*p^5*b_2-2268*p^4*b_1*b_2-9*p^3*b_2^2+858*t^5*d_2-11940*t^4*p*d_2+3204*p^5*d_2+5112*p^4*b_1*d_2+4611*p^3*b_2*d_2-6407*p^3*d_2^2-87*p^2*b_1*d_2^2+201*p*d_2^3+1521*t^5*p+924*p^6-507*t^5*b_1-524*t^4*p*b_1-990*p^5*b_1-855*p^4*b_1^2-339*p^3*b_1^3+1260*p^4*b_2-1152*p^3*b_1*b_2+108*p^2*b_2^2-2535*t^4*d_2-7166*t^3*p*d_2+1845*p^4*d_2+2843*p^3*b_1*d_2+1464*p^2*b_2*d_2-2205*p^2*d_2^2-5*p*b_1*d_2^2+(59/2)*d_2^3+364*t^5+2665*t^4*p+792*p^5-(845/3)*t^4*b_1+939*t^3*p*b_1-495*p^4*b_1-465*p^3*b_1^2-144*p^2*b_1^3+720*p^3*b_2-378*p^2*b_1*b_2+54*p*b_2^2-2340*t^3*d_2-2513*t^2*p*d_2+740*p^3*d_2+1038*p^2*b_1*d_2+258*p*b_2*d_2-450*p*d_2^2+b_1*d_2^2+364*t^4+1745*t^3*p+495*p^4-(289/3)*t^3*b_1+626*t^2*p*b_1-165*p^3*b_1-162*p^2*b_1^2-36*p*b_1^3+270*p^2*b_2-72*p*b_1*b_2+9*b_2^2-1017*t^2*d_2-493*t*p*d_2+195*p^2*d_2+225*p*b_1*d_2+18*b_2*d_2-41*d_2^2+196*t^3+645*t^2*p+220*p^3-19*t^2*b_1+173*t*p*b_1-33*p^2*b_1-33*p*b_1^2-4*b_1^3+60*p*b_2-6*b_1*b_2-234*t*d_2+30*p*d_2+22*b_1*d_2+64*t^2+133*t*p+66*p^2-(5/3)*t*b_1-3*p*b_1-3*b_1^2+6*b_2+2*d_2+12*t+12*p;

i10 : allPolynomials = append(FG,k17minusone);

i11 : allMonomials = sort unique flatten apply(#allPolynomials, i ->  flatten entries monomials allPolynomials_i);

i12 : A = matrix apply(#allMonomials, i -> apply(#FG, j -> coefficient(allMonomials_i,FG_j)));

               231        10965

o12 : Matrix QQ     {coefficient(allMonomials_i,k17minusone)});

               231        1

o13 : Matrix QQ     if h_i !=0 then i));

i17 : k17monomials=apply(#I, i -> G_(I_i));

i18 : k17coefficients=apply(#I, i -> h_(I_i));

i19 : s=sum apply(#k17monomials, i -> (k17monomials_i)*(k17coefficients_i));

i20 : s=(1_S)*s;

i21 : F(s) == k17minusone

o21 = true

i22 : print toString(s)

-36*u^11*v^3+36*u^11*v^2-99*u^10*v^3-108*u^11*x^2-48*u^11*v*y+696*u^11*y^2-36*u^11*v+99*u^10*v^2-255*u^9*v^3-297*u^10*x^2+60*u^11*y-99*u^10*v*y+1353*u^10*y^2+(2995/2)*u^9*v*y^2-(8915/2)*u^9*y^3+12*u^11-165*u^10*v+195*u^9*v^2-414*u^8*v^3-180*u^9*v*x-765*u^9*x^2+330*u^10*y+35*u^9*v*y+2775*u^9*x*y-1745*u^9*y^2+576*u^8*v*y^2+(13617/2)*u^8*y^3+66*u^10-495*u^9*v+135*u^8*v^2-522*u^7*v^3+60*u^9*x-702*u^8*v*x-1296*u^8*x^2+1070*u^9*y+918*u^8*v*y+5724*u^8*x*y-5688*u^8*y^2+99*u^7*v*y^2+2301*u^7*y^3+220*u^9-990*u^8*v-234*u^7*v^2-588*u^6*v^3+270*u^8*x-1728*u^7*v*x-1566*u^7*x^2+2340*u^8*y+2934*u^7*v*y+9426*u^7*x*y-11346*u^7*y^2-644*u^6*v*y^2+2814*u^6*y^3+(64/3)*v*z^8+495*u^8-1386*u^7*v-756*u^6*v^2-612*u^5*v^3+720*u^7*x-2772*u^6*v*x-1512*u^6*x^2+3600*u^7*y+5292*u^6*v*y+12516*u^6*x*y-16170*u^6*y^2-1104*u^5*v*y^2+2628*u^5*y^3+4078*u*y*z^6+8370*u*z^7-3253*v*z^7+202*y*z^7-619*z^8+792*u^7-1386*u^6*v-1026*u^5*v^2-531*u^4*v^3+1260*u^6*x-3024*u^5*v*x-1080*u^5*x^2+3990*u^6*y+6282*u^5*v*y+12648*u^5*x*y-16656*u^5*y^2-(1757/2)*u^4*v*y^2+(3783/2)*u^4*y^3+6287*u*y*z^5-(10889/2)*y^2*z^5+28841*u*z^6-(26296/3)*v*z^6-1698*y*z^6+1333*z^7+924*u^6-990*u^5*v-855*u^4*v^2-339*u^3*v^3+1512*u^5*x-2268*u^4*v*x-459*u^4*x^2+3204*u^5*y+5112*u^4*v*y+9231*u^4*x*y-12342*u^4*y^2-(775/2)*u^3*v*y^2+(2403/2)*u^3*y^3-16382*u*y*z^4-(3587/3)*y^2*z^4-14284*u*z^5+(24554/3)*v*z^5-6357*y*z^5+6466*z^6+792*u^5-495*u^4*v-465*u^3*v^2-144*u^2*v^3+1260*u^4*x-1152*u^3*v*x-9*u^3*x^2+1845*u^4*y+2843*u^3*v*y+4611*u^3*x*y-6407*u^3*y^2-87*u^2*v*y^2+615*u^2*y^3-(93083/2)*u*y*z^3+9131*y^2*z^3-(118563/2)*u*z^4+(52961/2)*v*z^4-15564*y*z^4+14298*z^5+495*u^4-165*u^3*v-162*u^2*v^2-36*u*v^3+720*u^3*x-378*u^2*v*x+108*u^2*x^2+740*u^3*y+1038*u^2*v*y+1464*u^2*x*y-2205*u^2*y^2-5*u*v*y^2+201*u*y^3-17941*u*y*z^2+(39845/6)*y^2*z^2+27317*u*z^3-(952/3)*v*z^3-27565*y*z^3+26133*z^4+220*u^3-33*u^2*v-33*u*v^2-4*v^3+270*u^2*x-72*u*v*x+54*u*x^2+195*u^2*y+225*u*v*y+258*u*x*y-450*u*y^2+v*y^2+(59/2)*y^3+877*u*v*z-123*u*y*z-(4063/3)*y^2*z+16667*u*z^2-(7537/3)*v*z^2-12306*y*z^2+18894*z^3+66*u^2-3*u*v-3*v^2+60*u*x-6*v*x+9*x^2+30*u*y+22*v*y+18*x*y-41*y^2+414*u*z+22*v*z-774*y*z+163*z^2+12*u+6*x+2*y+34*z

Here are some calculations in Macaulay2 to compute the generalized K-theoretic Donaldson numbers in Section 8.

As a preparation, the Chow ring $\mathrm{A}^*(\mathrm{M})$, the total Chern class $c(\mathcal{T}_\mathcal{M})$, and the Todd class $td(\mathcal{T}_\mathcal{M})$ are obtained.

i1 : loadPackage "Schubert2";

i2 : R = QQ[aa,bb,x,y,z,Degrees=>{1,1,2,2,2}];

i3 : I = ideal(x*z-y*z,bb^2*z-3*y*z-9*z^2,3*aa^2*z-aa*bb*z+y*z,bb^2*y-3*y^2-9*y*z,bb^2*x-x*y-3*y^2-3*aa*bb*z-9*y*z+9*z^2,bb^4+3*x^2-9*x*y-3*y^2-54*y*z-81*z^2,bb*y*z+9*aa*z^2-3*bb*z^2,2*bb*x*y-3*bb*y^2-9*aa*y*z-27*aa*z^2+9*bb*z^2,3*bb*x^2-7*bb*y^2-36*aa*y*z-108*aa*z^2+36*bb*z^2,aa^12+3*aa^11*bb+3*aa^10*bb^2-aa^9*bb^3+6*aa^10*x+12*aa^9*bb*x+27*aa^8*x^2-3*aa^10*y+2*aa^9*bb*y-48*aa^8*x*y+51*aa^8*y^2+28*aa^7*bb*y^2+56*aa^6*y^3+201*aa*bb*z^5-19*y*z^5-613*z^6,6*aa^10*x*y-12*aa^10*y^2-10*aa^9*bb*y^2-45*aa^8*y^3-104*aa*bb*z^6+2*y*z^6+310*z^7);

o3 : Ideal of R

i4 : S = R/I;

i5 : M = abstractVariety(17, S);

i6 : TM = abstractSheaf(M, ChernClass=>-36*aa^11*bb^3+36*aa^11*bb^2-99*aa^10*bb^3-108*aa^11*x^2-48*aa^11*bb*y+696*aa^11*y^2-36*aa^11*bb+99*aa^10*bb^2-255*aa^9*bb^3-297*aa^10*x^2+60*aa^11*y-99*aa^10*bb*y+1353*aa^10*y^2+(2995/2)*aa^9*bb*y^2-(8915/2)*aa^9*y^3+12*aa^11-165*aa^10*bb+195*aa^9*bb^2-414*aa^8*bb^3-180*aa^9*bb*x-765*aa^9*x^2+330*aa^10*y+35*aa^9*bb*y+2775*aa^9*x*y-1745*aa^9*y^2+576*aa^8*bb*y^2+(13617/2)*aa^8*y^3+66*aa^10-495*aa^9*bb+135*aa^8*bb^2-522*aa^7*bb^3+60*aa^9*x-702*aa^8*bb*x-1296*aa^8*x^2+1070*aa^9*y+918*aa^8*bb*y+5724*aa^8*x*y-5688*aa^8*y^2+99*aa^7*bb*y^2+2301*aa^7*y^3+220*aa^9-990*aa^8*bb-234*aa^7*bb^2-588*aa^6*bb^3+270*aa^8*x-1728*aa^7*bb*x-1566*aa^7*x^2+2340*aa^8*y+2934*aa^7*bb*y+9426*aa^7*x*y-11346*aa^7*y^2-644*aa^6*bb*y^2+2814*aa^6*y^3+(64/3)*bb*z^8+495*aa^8-1386*aa^7*bb-756*aa^6*bb^2-612*aa^5*bb^3+720*aa^7*x-2772*aa^6*bb*x-1512*aa^6*x^2+3600*aa^7*y+5292*aa^6*bb*y+12516*aa^6*x*y-16170*aa^6*y^2-1104*aa^5*bb*y^2+2628*aa^5*y^3+4078*aa*y*z^6+8370*aa*z^7-3253*bb*z^7+202*y*z^7-619*z^8+792*aa^7-1386*aa^6*bb-1026*aa^5*bb^2-531*aa^4*bb^3+1260*aa^6*x-3024*aa^5*bb*x-1080*aa^5*x^2+3990*aa^6*y+6282*aa^5*bb*y+12648*aa^5*x*y-16656*aa^5*y^2-(1757/2)*aa^4*bb*y^2+(3783/2)*aa^4*y^3+6287*aa*y*z^5-(10889/2)*y^2*z^5+28841*aa*z^6-(26296/3)*bb*z^6-1698*y*z^6+1333*z^7+924*aa^6-990*aa^5*bb-855*aa^4*bb^2-339*aa^3*bb^3+1512*aa^5*x-2268*aa^4*bb*x-459*aa^4*x^2+3204*aa^5*y+5112*aa^4*bb*y+9231*aa^4*x*y-12342*aa^4*y^2-(775/2)*aa^3*bb*y^2+(2403/2)*aa^3*y^3-16382*aa*y*z^4-(3587/3)*y^2*z^4-14284*aa*z^5+(24554/3)*bb*z^5-6357*y*z^5+6466*z^6+792*aa^5-495*aa^4*bb-465*aa^3*bb^2-144*aa^2*bb^3+1260*aa^4*x-1152*aa^3*bb*x-9*aa^3*x^2+1845*aa^4*y+2843*aa^3*bb*y+4611*aa^3*x*y-6407*aa^3*y^2-87*aa^2*bb*y^2+615*aa^2*y^3-(93083/2)*aa*y*z^3+9131*y^2*z^3-(118563/2)*aa*z^4+(52961/2)*bb*z^4-15564*y*z^4+14298*z^5+495*aa^4-165*aa^3*bb-162*aa^2*bb^2-36*aa*bb^3+720*aa^3*x-378*aa^2*bb*x+108*aa^2*x^2+740*aa^3*y+1038*aa^2*bb*y+1464*aa^2*x*y-2205*aa^2*y^2-5*aa*bb*y^2+201*aa*y^3-17941*aa*y*z^2+(39845/6)*y^2*z^2+27317*aa*z^3-(952/3)*bb*z^3-27565*y*z^3+26133*z^4+220*aa^3-33*aa^2*bb-33*aa*bb^2-4*bb^3+270*aa^2*x-72*aa*bb*x+54*aa*x^2+195*aa^2*y+225*aa*bb*y+258*aa*x*y-450*aa*y^2+bb*y^2+(59/2)*y^3+877*aa*bb*z-123*aa*y*z-(4063/3)*y^2*z+16667*aa*z^2-(7537/3)*bb*z^2-12306*y*z^2+18894*z^3+66*aa^2-3*aa*bb-3*bb^2+60*aa*x-6*bb*x+9*x^2+30*aa*y+22*bb*y+18*x*y-41*y^2+414*aa*z+22*bb*z-774*y*z+163*z^2+12*aa+6*x+2*y+34*z+1,Rank=>17);

i7 : tTM = todd(TM);

For the class $(-3k+m)\alpha-k\beta$, by using Grothendieck-Hirzebruch-Riemann-Roch theorem, we are able to compute the Donaldson number. Note that $\mathrm{deg}(\beta z^8) = 9$.

i8 : k=1;

i9 : m=3;

i10 : print integral(ch(abstractSheaf(M,ChernClass=>1+((-3*k+m)*aa-k*bb),Rank=>1))*tTM);

         20    8

integral(--bb*z )

          9

We give a table of Donaldson_numbers for varying k and m.

A red number refers that the corresponding $\mathrm{M}(v)$ is empty.

A black number is $\mathrm{H}^0(\mathrm{M},E)$ due to Kawabata-Viehweg vanishing theorem.