### A Castelnuovo bound for smooth surfaces.

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Let $X$ be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of $X$ is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.

We find two basis of the spaces of rational homology of the Hilbert scheme of points in an algebraic surface, by exhibiting two candidates having as cardinalities the known Betti numbers of this scheme and showing that both intersect in a matrix of nonzero determinant.

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens...