Abstract:
Schubert varieties are subvarieties of the flag variety F‘n(C), a smooth complex
projective variety consisting of sequences of sublinear subspaces of an n-dimensional
complex vector space, ordered by inclusion. They are indexed by permutation matrices and studied in various types with important roles in algebraic geometry
due to their combinatorial structures. The smoothness and singularity of Schubert variety have been characterised by various methods using the elements of the
n-dimensional symmetric group. However, characterising smoothness using the exponents of the monomial of the Schubert variety and Plücker coordinate which
uniquely and clearly identifies the symmetry of the Poincaré polynomial have not
been established. Hence this research aims at establishing smoothness and singularity of type A Schubert varieties using the exponents of the monomials of the
Schubert variety and the Jacobian criterion on the equations of the ideals of the
Schubert variety obtained via the Plücker embedding.
For the Schubert varieties Xσ, the cohomology of the flag varieties
f : Hn−k(F‘n(C); Z) ! Hk(F‘n(C); Z) defined by f[Xσ] = [Xσ] 2 Hk(F‘n(C))
was considered, to obtain its monomials. The Poincaré polynomial was determined in order to compute the symmetry of the Schubert varieties. The flag varieties are embedded into the product of Grassmanians which is also embedded into
the product of projective spaces given by the embedding map F‘n(C) = Xσ ,!
Qn−1
k=1 Gr(k; n) ,! Qn−1
k=1 P
0B@
n k
1CA
−1
: defined by A 7! [P12; P13; · · · ; P(n−1)n] , with
Pij; 1 ≤ i < j ≤ n being the
n k
! minors for Ak;n in Gr(k; n). The equations
of the ideal of the Schubert varieties were obtained by taking all the minors of the
matrix Schubert varieties. The rank of the Jacobian matrix and the co-dimension
of the Schubert varieties were determined.
The Schubert classes forms additive Z basis that generates the cohomology ring
Hk(F‘4(C); Z). The basis for the cohomology ring are the geometric and algebraic
basis. The algebraic basic classes xi 11xi 22 · · · ; xi mm with exponents ij = m − j forms
Z basis for the cohomology ring and these basic classes are the monomials. The
vPoincaré polynomial Pσ(t) = Pv≤σ tl(v) , defined with respect to the length function
and via the Bruhat order, v ≤ σ =) l(v) ≤ l(σ) shows that the symmetry
Pσ
(t) = trPσ(t−1) Of the Poincaré polynomial is palindromic or not palindromic.
The rank of the Jacobian matrix obtained using the equations of the ideal I(Xσ)
derived through the embedding map is found to be equal to the co-dimension of
the varieties which indicates smoothness.
The exponent of the monomials xi 11xi 22 · · · xi mm of the Schubert variety Xσ have
uniquely satisfied the symmetry of its Poincaré polynomial for smooth Schubert varieties and have successfully extended the underlying group from Sn to Z+n. Smoothness has successfully been generalised in terms of the differential equations using
the equations defining the ideals, of the Schubert varieties through the Plücker
coordinates.
