Fordham University

Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 5

Code 5.7: Checking the linear space \(P_{nod,t}\)

We check that for generic \(t\), \(J_t = P_t \cap \operatorname{OG}(5,10).\) 
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition,
               ReesAlgebra, Saturation, TangentCone

i1 : K = frac(QQ[t_0,t_1,Degrees=>{0,0}]);

i2 : R=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];  

i3 : 
     JtGB = {x_2345, x_1245, x_1234, t_0*x_25-t_1*x_35-t_0*x_45, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_12-x_13, x_0+t_1*x_25-t_1*x_35-t_1*x_45, x_24*x_1345, x_13*x_1345, x_45*x_1235+x_25*x_1345, x_34*x_1235-x_25*x_1345, x_24*x_1235, x_13*x_45+x_25*x_45+x_34*x_45+t_1*x_25*x_1345-t_1*x_45*x_1345, (t_0-t_1)*x_34*x_35-t_0*x_34*x_45-t_1*x_35*x_45-t_0*x_45^2-t_0*t_1*x_35*x_1345, x_24*x_35-x_34*x_35+t_1*x_35*x_1345, x_13*x_35+x_25*x_35+x_34*x_35-t_1*x_35*x_1235-t_1*x_35*x_1345, x_25*x_34-x_34*x_35+x_25*x_45+t_1*x_35*x_1345, x_24*x_25-x_34*x_35-x_24*x_45+x_25*x_45+x_34*x_45+t_1*x_25*x_1345+t_1*x_35*x_1345-t_1*x_45*x_1345, x_13*x_25+x_25^2+x_34*x_35-x_25*x_45-t_1*x_25*x_1235-t_1*x_25*x_1345-t_1*x_35*x_1345, x_13*x_24-x_13*x_34, t_0*x_34*x_45*x_1345+t_1*x_35*x_45*x_1345+t_0*x_45^2*x_1345+t_1^2*x_35*x_1345^2, x_25*x_45*x_1345+x_34*x_45*x_1345+t_1*x_25*x_1345^2-t_1*x_45*x_1345^2, t_1*x_35^2*x_1345+t_0*x_35*x_45*x_1345-t_0*t_1*x_35*x_1235*x_1345, x_34*x_35*x_1345-t_1*x_35*x_1345^2, x_25*x_35*x_1345-t_1*x_35*x_1235*x_1345, x_25^2*x_1345+x_34*x_45*x_1345-t_1*x_25*x_1235*x_1345-t_1*x_45*x_1345^2, x_24*x_34*x_45-x_34^2*x_45+x_24*x_45^2-x_34*x_45^2+t_1*x_34*x_45*x_1345+t_1*x_45^2*x_1345, x_35^2*x_1235*x_1345-t_0*x_35*x_1235^2*x_1345-t_0*x_35*x_1235*x_1345^2, x_34^2*x_45*x_1345+x_34*x_45^2*x_1345-t_1*x_34*x_45*x_1345^2-t_1*x_45^2*x_1345^2};

i4 : Jt = ideal(JtGB);

o4 : Ideal of R

i5 : Pt = ideal {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};

o5 : Ideal of R

i6 : 
     OG510 = ideal({x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
     x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
     x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
     x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
     x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
     x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
     x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
     x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
     x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
     x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235});

o6 : Ideal of R

i7 : Jt == Pt+OG510

o7 = true

i8 : hilbertPolynomial(Jt,Projective=>false)

o8 = 12i - 6