Chow ring of \(\mathrm{M}\)


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

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