B10-P03.ipynb

Kernel: SageMath 9.8

%run '/home/user/Equivariant_Vector_Bundles_On_Homogeneous_Varieties__0-0-1/src/Initialize.ipynb'

from IPython.display import clear_output , display , HTML

k = 3
n = 10
X = Orthogonal_Grassmannian( k , 2*n+1 )
fw = X.Basis('fw')
print( 'Base space:' , X.__str__( Output_Style='Short' ) )
print( 'rk K0(X):' , X.K0().rank() )
print( 'Fano index:' , X.Fano_Index() )
print()

h = X.Fano_Index()

# Kuznetsov's spinor bundle filtration
calS = X.calS()
F3 = ( X.calU().Exterior_Power(3) * calS )
F2 = ( X.calU().Exterior_Power(2) * calS ).Extend_Equivariantly_By( F3 )
F1 = ( X.calU().Exterior_Power(1) * calS ).Extend_Equivariantly_By( F2 )
F0 = ( X.calU().Exterior_Power(0) * calS ).Extend_Equivariantly_By( F1 )

LC  = X.Lefschetz_Collection( Starting_Block=[] , Support_Pattern='Trivial' )

# Tautological collection
LC += X.Tautological_Collection()

print( 'Objects of starting block:' )
Body = [ [ str(Counter)+':' , str(Object) , str(sum([ 1 for Row in LC.Support_Pattern().values() if Row[Counter-1] == True ])) ]
         for Counter , Object in enumerate( LC.Starting_Block() , start=1 )
       ]
show( table(Body) )
print()

print( 'Number of objects:' , len(LC) )
print( 'LC has maximal expected length?' , LC.Has_Maximal_Expected_Length() )
print()

#print( 'Grid of LC:' )
#show( LC.Grid() )
#print()

LC_Is_Numercially_Exceptional = LC.Is_Numerically_Exceptional()
print( 'Is LC numerially exceptional?' , LC_Is_Numercially_Exceptional )
print()

Base space: OGr(3;21)
rk K0(X): 960
Fano index: 17

Objects of starting block:

Number of objects: 612
LC has maximal expected length? False

Is LC numerially exceptional? True