Interpolating, moduli-interpolating and ultra-interpolating curves of any genus on Fano hypersurfaces, and positivity of some Kontsevich intersection numbers

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ 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 meeting some general collection of linear subspaces; 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