Given n general points p 1 , p 2 , . . . , p n ∈ P r , it is natural to ask when there exists a curve C ⊂ P r , of degree d and genus g, passing through p 1 , p 2 , . . . , p n . In this paper, we give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N C of a general nonspecial curve of degree d and genus g in P r (with d ≥ g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H 0 (N C (−D)) = 0 or H 1 (N C (−D)) = 0), with exactly three exceptions.

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree d hypersurface in P n with n ≥ d + 2, the space M 0,0 (X, e) of degree e Kontsevich stable maps from a rational curve to X is an irreducible local complete intersection stack of dimension e(n − d + 1) + n − 4. This resolves all but one case of a conjecture of Coskun, Harris and Starr, and also proves that the Gromov-Witten invariants of these hypersurfaces are enumerative.

Purpose: We give a new interpretation of Koszul cohomology, which is equivalent under the Bridgeland-King-Reid equivalence to Voisin's Hilbert scheme interpretation in dimensions 1 and 2 but is different in higher dimensions. Methods: We show that an explicit resolution of a certain S n -equivariant sheaf is equivalent to a resolution appearing in the theory of Koszul cohomology. Results: Our methods easily show that the dimension

Abstract. We give a criterion for the vanishing of the weight one syzygies associated to a line bundle B in a sufficiently positive embedding of a smooth complex projective variety of arbitrary dimension.

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