ar X iv : m at h / 02 07 25 7 v 1 [ m at h . A G ] 2 7 Ju l 2 00 2 RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE , II

ar X iv : m at h / 04 09 02 9 v 1 [ m at h . A G ] 2 S ep 2 00 4 ACM BUNDLES ON GENERAL HYPERSURFACES IN P 5 OF LOW DEGREE

In this paper we show that on a general hypersurface of degree r = 3, 4, 5, 6 in P 5 a rank 2 vector bundle E splits if and only if h 1 E(n) = h 2 E(n) = 0 for all n ∈ Z. Similar results for r = 1, 2 were obtained in [15], [16] and [1].

ar X iv : m at h / 02 10 22 5 v 1 [ m at h . A G ] 1 5 O ct 2 00 2 RATIONAL CONNECTIVITY AND SECTIONS OF FAMILIES OVER CURVES TOM

ar X iv : m at h / 06 02 64 7 v 1 [ m at h . A G ] 2 8 Fe b 20 06 HIGHER FANO MANIFOLDS AND RATIONAL SURFACES

Let X be a Fano manifold of pseudo-index ≥ 3 such that c 1 (X) 2 − 2c 2 (X) is nef. Irreducibility of some spaces of rational curves on X (in fact, a weaker hypothesis) implies a general point of X is contained in a rational surface.

ar X iv : 0 71 1 . 27 58 v 1 [ m at h . A G ] 1 7 N ov 2 00 7 Rational curves of degree 11 on a general quintic threefold ∗

We prove the “strong form” of the Clemens conjecture in degree 11. Namely, on a general quintic threefold F in P, there are only finitely many smooth rational curves of degree 11, and each curve C is embedded in F with normal bundle O(−1) ⊕ O(−1). Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with ration...

ar X iv : m at h / 02 07 29 6 v 1 [ m at h . M G ] 3 1 Ju l 2 00 2 Products of hyperbolic metric spaces

Let (Xi, di), i = 1, 2, be proper geodesic hyperbolic metric spaces. We give a general construction for a " hyperbolic product " X1× h X2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.