Let X denote an equivariant embedding of a connected reductive group G over an algebraically closed field k. Let B denote a Borel subgroup of G and let Z denote a B B-orbit closure in X. When the characteristic of k is positive and X is projective we prove that Z is globally F-regular. As a consequence, Z is normal and CohenMacaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that Z has rational singularities. This extends earlier results by the second author and M. Brion.

In the CT-track of the 2006 RSA conference, a new multivariate public key cryptosystem, which is called the Medium Field Equation (MFE) multivariate public key cryptosystem, is proposed by Wang, Yang, Hu and Lai. We use the second order linearization equation attack method by Patarin to break MFE. Given a ciphertext, we can derive the plaintext within 2^{23} F_{2^16}-multiplications, after performing once for any given public key a computation of complexity less than 2^{52}. We also propose a high order linearization equation (HOLE) attack on multivariate public key cryptosystems, which is a further generalization of the (first and second order) linearization equation (LE). This method can be used to attack extensions of the current MFE.

The Cox rings of del Pezzo surfaces are closely related to the Lie groups En. In this paper, we generalize the definition of Cox rings to G-surfaces defined by us earlier, where the Lie groups G = An,Dn or En. We show that the Cox ring of a G-surface S is closely related to an irreducible representation V of G, and is generated by degree one elements. The Proj of the Cox ring of S is a sub-variety of the orbit of the highest weight vector in V , and both are closed sub-varieties of P(V ) defined by quadratic equations. The GIT quotient of the Spec of such a Cox ring by a natural torus action is considered.

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