Remarks on an example of K. Ueno

Abstract

In [U75], Example 16.15, p. 207, Kenji Ueno constructs the threefold X described below, and says that it is unknown, whether or not it is unirational. After recalling Ueno’s construction, we show that X is rationally connected. The (much deeper) question of whether X is unirational (or even rational) remains open. We found it worthwhile drawing attention to this seemingly unnoticed example, on which several questions (such as unirationality vs rational connectedness), might possibly be tested more easily than on the test cases usually considered (such as quartics or double ℙ3’s ramified along sextics). We formulate some of these questions in §4.