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.