Milena Hering

Equations defining Schubert varieties and Frobenius spliiting of diagonals

I will follow Ramanathan's paper with the same title. Let X be a projective variety defined over an algebraically closed field of positive characteristic. We will show how Frobenius splittings of certain products of X compatible with certain (partial) diagonals imply that the ideal defining any embedding of X in some projective spaceis generated by quadratic equations. We will use this to see that the homogeneous ideal of a homogeneous space G/P in any embedding given by a very ample line bundle is generated by quadrics and that the homogenous ideal of any Schubert subvariety of G/P is generated by linear homogenuous polynomials.