Narrow Results

Let $X$ be either a general hypersurface of degree $n+1$ in ${\mathbb{P}}^n$ or a general $(2,n)$ complete intersection in ${\mathbb{P}}^{n+1},n\ge 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=4$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi–Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n+2$ in ${\mathbb{P}}^n$.

On a general hypersurface of degree $d\le n$ 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 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 A by Chang describes the set of numerical characters ($n,d$, curve degree, genus) to which our results apply.