B07-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 = 7
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()
# ... extension of the tautological collection ...
# Degree = 5
E = X.calU( 4*fw[1]+fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
E = X.calU( 3*fw[1]+2*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
# Degree = 6
E = X.calU( 4*fw[1]+2*fw[2] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )

# Mysterious collection
# 1
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 3*fw[1]+fw[n-1] )(-1).Extend_Equivariantly_By( X.calU( 2*fw[1]+fw[n-2] )(-1) ).Extend_Equivariantly_By( X.calU( fw[n-1] )(-1) )] , Support_Pattern=h*[1] )
# 2
E = X.calU( fw[n] )
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( fw[1]+2*fw[n] )(-1).Extend_Equivariantly_By( X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1) )] , Support_Pattern=h*[1] )
# Note: We extend by X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1).
#       However, F3 * X.calU( fw[n] ) = X.calU( 2*fw[n] )(-1) + X.calU( fw[n-1] )(-1) + ... + X.calU( fw[k+1] )(-1) + X.calO()

# Spinor collection
# Degree = 1/2
E = X.calU( fw[n] )
LC += X.Lefschetz_Collection( Starting_Block=[E] , Support_Pattern=h*[1] )
# Degree = 3/2
E = X.calO()(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 5/2
E = X.calU( fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 2*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 7/2
E = X.calU( 2*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 3*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 9/2
E = X.calU( 3*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 4*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=h*[1] )
# Degree = 11/2
E = X.calU( 4*fw[1] )(1)
LC += X.Lefschetz_Collection( Starting_Block=[X.calU( 5*fw[1]+fw[n] ).Extend_Equivariantly_By( F3 * E )] , Support_Pattern=5*[1] )


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;15)
rk K0(X): 280
Fano index: 11

Objects of starting block:

Number of objects: 280
LC has maximal expected length? True

Grid of LC: