Varieties Connected by Chains of Lines

TL;DR

This paper establishes a numerical criterion for the existence of chains of lines connecting two general points on an irreducible variety in projective space, based on degrees and defining polynomials, and proves its sharpness.

Contribution

It provides a precise, sharp criterion involving degrees and defining polynomials for chains of lines connecting points on varieties.

Findings

Criterion is sharp for all l > 2

Connects chain length with degrees and polynomials of the variety

Ensures existence of line chains between general points

Abstract

In this paper we give for any integer l > 2 a numerical criterion ensuring the existence of a chain of length l of lines through two general points of an irreducible variety X in P^N, involving the degrees and the number of homogeneous polynomials defining X. We show that our criterion is sharp.