Rational curves in Fano hypersurfaces and tropical curves

TL;DR

This paper proves that Fano hypersurfaces and complete intersections are covered by rational curves of degree at most two, using tropical geometry and degeneration techniques.

Contribution

It introduces a novel application of tropical geometry to demonstrate the existence of low-degree rational curves on Fano varieties.

Findings

Fano hypersurfaces are swept by quadratic rational curves

Extension of results to Fano complete intersections

Application of tropical and degeneration methods

Abstract

Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.