Rational Curves on Del Pezzo Manifolds

TL;DR

This paper investigates rational curves on certain Fano manifolds, providing a counting formula for rational curves in three dimensions and exploring their deformation and normal bundle properties.

Contribution

It introduces an elementary specialization technique to analyze rational curves on index n-1 Fano n-folds and derives a counting formula for rational curves in three dimensions.

Findings

Derived a simple counting formula for rational curves in three-dimensional Fano manifolds.

Established deformation theoretic consequences related to the normal bundles of these curves.

Provided insights into the properties of rational curves on index n-1 Fano n-folds.

Abstract

We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case $n=3$. The arguments have immediate deformation theoretic consequences which translate to properties of the normal bundles of such curves.