Interpolation of curves on Fano hypersurfaces

A BSTRACT . On a general hypersurface of degree d ≤ n in P n or P n itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number t of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through t fixed points has general moduli as family of t -pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the set of numerical characters ( n , d , curve degree, genus) to which our results apply.