Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire

We determine explicitly the irreducible components of the singular locus of any Schubert variety for being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along each of them.The case of covexillary Schubert varieties was solved in an earlier work of the author [Ann. Inst. Fourier 51 (2) (2001) 375]. Here, we first exhibit some irreducible components of the singular locus of X_w, by describing the generic singularity along each of them. Let Σ_w be the union of these components. As mentioned above, the equality is known for covexillary varieties, and we base our proof of the general case on this result. More precisely, we study the exceptional locus of certain quasi-resolutions of a non-covexillary Schubert variety X_w, and we relate the intersection of these loci to Σ_w. Then, by induction on the dimension, we can establish the equality.     

La topologie à l'infini des variétés à géométrie bornée et croissance linéaire

We study the topology at infinity of a non compact riemannian manifold with bounded geometry and linear growth-type.     

Improvement on Hwang et al.’s generalization of proxy signature schemes based on elliptic curves

Hwang et al. proposed their generalization of proxy signature schemes based on elliptic curves. However, two attacks are proposed to show that their schemes have serious security flaws. By the first attack, an adversary can forge an illegal proxy signature that verifiers cannot actually find out the original signers of proxy signatures. The second attack is used to change proxy signatures into multi-signatures belonging to the group that actually generates the proxy signatures. To overcome these flaws, our improvement on Hwang et al.’s scheme is also proposed.