Abstract. In this paper, we associate an invariant α x (L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ǫ x (L) measuring local positivity of L at x, and in particular that Roth's theorem on P 1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

Abstract. We prove that for n > 2 and d < n+1 2 , a general complex hypersurface X ⊂ P n of degree d has the property that for each integer e the scheme R e (X) parametrizing degree e, smooth rational curves on X is an integral, local complete intersection scheme of "expected" dimension (n + 1 − d)e + (n − 4).The techniques used in the proof include: 1. Classical results about lines on hypersurfaces including a new result about flatness of the projection map from the space of pointed lines. 2. The Kontsevich moduli space of stable maps, M 0,r (X, e). In particular we use the deformation theory of stable maps, properness of the stack M 0,r (X, e), and the decomposition of M 0,r (X, e) described in [2]. 3. A version of Mori's bend-and-break lemma.

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.Comment: 39 pages, 1 figure; final version with a few expository changes suggested by the refere

In this article we consider the spaces H d,g (X) parametrizing smooth curves of degree d and genus g on a smooth cubic threefold X ⊂ P 4 . For 1 ≤ d ≤ 5, we show that each variety H d,g (X) is irreducible of dimension 2d.

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).2000 Mathematics Subject Classification. 13D40, 13D02, 05C90.