CHARACTERISING SMOOTHNESS OF TYPE A SCHUBERT VARIETY USING PALINDROMIC POINCARÉ POLYNOMIAL AND PLÜCKER COORDINATE METHODS.

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.