Log-uniruled affine varieties without cylinder-like open subsets

TL;DR

This paper demonstrates that in dimensions three and higher, there exist smooth affine varieties that are affine-uniruled but not affine-ruled, contrasting the known equivalence in dimension two.

Contribution

It provides the first examples of higher-dimensional smooth affine varieties that are affine-uniruled but not affine-ruled, challenging previous assumptions.

Findings

Existence of higher-dimensional affine-uniruled but not affine-ruled varieties

Breakdown of the equivalence between uniruledness and ruledness in dimension three

Contrasts with classical results for surfaces

Abstract

A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira dimension. Here we show in contrast that starting from dimension three, there exists smooth affine varieties which are affine-uniruled but not affine-ruled.