We characterize for any d the d-uple Veronese embedding of P n as the only variety that, under certain general conditions, can be projected from the Grassmannian of (d − 1)-planes in P nd+d−1 to the Grassmannian of (d − 1)-planes in P n+2d−3 in such a way that two (d − 1)-planes meet at most in one point. We also study the relation of this problem with the Steiner bundles over P n .

Abstract. This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in P n (quasi-lines). Various characterizations of a Zariski neighbourhood of a line in P n are obtained, some of them being new. Also, methods of formal geometry are applied for deducing results of birational nature.

Abstract. This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants of m copies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

Abstract. This note contains new evidence to a conjecture, related to the Nagata conjecture and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, on the cone of effective curves of blow-ups of P 2 at very general points.

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