Fordham University

Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 1

Code 1.1: Betti table of the orthogonal Grassmannian

We compute the Betti table of the orthogonal Grassmannian. These equations are given in Mukai's paper at [24, (0.1)]. 
+ M2 --no-readline --print-width 81
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup,
               PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : R=QQ[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];  

i2 : 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});

o2 : Ideal of R

i3 : betti res OG510

            0  1  2  3  4 5
o3 = total: 1 10 16 16 10 1
         0: 1  .  .  .  . .
         1: . 10 16  .  . .
         2: .  .  . 16 10 .
         3: .  .  .  .  . 1

o3 : BettiTally