Interpolation of curves on Fano hypersurfaces

TL;DR

This paper proves the existence of various curves on Fano hypersurfaces with specified genus and degree, passing through general points or subvarieties, and explores implications for intersection theory on moduli spaces.

Contribution

It establishes the existence of curves of any genus and high degree on Fano hypersurfaces, and analyzes the moduli of such curves, extending previous results in algebraic geometry.

Findings

Existence of curves of any genus and high degree passing through general points.

Family of curves through fixed points has general moduli.

Positivity of intersection numbers on Kontsevich spaces.

Abstract

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 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.