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Abstract. We determine set-theoretic defining equations for the varietyN that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni varietyshowing it is an irreducible component of the variety of hypersurfaces of degree n in C n 2 with dual of dimension at most 2n − 2. We establish additional geometric properties of the Mulmuley-Sohoni variety and prove a quadratic lower bound for the determinental border-complexity of the permanent.

We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.

Let $G$ be a complex connected reductive algebraic group and $G/B$ denote the flag variety of $G$. A $G$-homogeneous space $G/H$ is said to be {\it spherical} if $H$ acts on $G/B$ with finitely many orbits. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group $G$ (viewed as a $G\times G$-homogeneous space) has particularly nice proterties. Namely, the pair $(G,H)$ is called a {\it spherical pair of minimal rank} if there exists $x$ in $G/B$ such that the orbit $H.x$ of $x$ by $H$ is open in $G/B$ and the stabilizer $H_x$ of $x$ in $H$ contains a maximal torus of $H$. In this article, we study and classify the spherical pairs of minimal rank.Comment: Document produced in 200

Let G be a complex connected reductive group. The Parthasarathy-Ranga RaoVaradarajan (PRV) conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for 'the same reason' as the PRV ones.