Balanced rational curves and minimal rational connectedness of Fano hypersurfaces

TL;DR

This paper investigates the minimal degrees of rational curves passing through multiple points on general Fano hypersurfaces, revealing new bounds and density results that enhance understanding of their rational connectedness properties.

Contribution

It determines the minimal degree of rational curves through multiple points on Fano hypersurfaces for infinitely many cases, extending known results especially for index 1 hypersurfaces.

Findings

Minimal degree $e$ determined for infinitely many $k$

Results hold for all $k extgreater=1$ in index 1 case

Density of curve degrees covered is approximately given by a specific formula

Abstract

On a general Fano hypersurface in projective space, we determine for infinitely many $k$ the minimal degree $e$ of a rational curve through a general collection of $k$ points. In the case of a hypersurface of index 1, our results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an arithmetical result which implies that in the case of index $>1$, the density of the set of curve degrees $e$ covered by our method is approximately $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.