Interpolation of rational scrolls

TL;DR

This paper investigates the existence of balanced rational scrolls with specific degrees and dimensions in projective space, expanding understanding of their geometric properties.

Contribution

It demonstrates the existence of balanced rational scrolls of certain degrees and fiber dimensions in projective spaces, providing new examples in algebraic geometry.

Findings

Existence of balanced scrolls of degree mk+1 in P^{2k+1} for all m ≥ 1

Balanced scrolls contain the expected number of general linear spaces as rulings

Results apply to a broad class of rational scrolls in algebraic geometry.

Abstract

We show in many cases that there exist rational scrolls which are balanced, i.e. they contain the expected number of general linear spaces as rulings. For example, there exist balanced scrolls of degree $mk+1$ and fibre dimension $k$ in $\P^{2k+1}$ for all $m\geq 1$.