We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a -nodal curve X sitting on a primitively polarized K3 surface S of degree 2p − 2, for 2 ≤ g = p − < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs S X to the moduli stack of curves g that associates to X the isomorphism class C of its normalization.

Abstract. In this article, we study the geometry of k-dimensional subvarieties with geometric genus zero of a general projective hyper-where k is an integer such that 1 ≤ k ≤ n − 5. As a corollary of our main result we obtain that the only rational curves lying on the general hypersuface X 2n−3 ⊂ P n , for n ≥ 6, are the lines.

We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.Date: October 24, 2018. Key-words : big and pseff adjoints divisors; stable, non-ample and non-nef base locus; rational curves. A.M.S. classification : 14J40.