In this paper it is shown (Corollary 1.9) that if for a 3-dimensional variety $X\subset \mathbb{P}^r$ there is an open set of hyperplanes $Y\in(\mathbb{P}^r)^{\vee}$ such that $\forall H, H'\in Y$ the hyperplane sections $H\cap X,\ H'\cap X$ are projectively isomorphic, then either the generic section of $X$ is a rational surface, or $X$ is a plane scroll over a curve.

However, the term plane scroll is not defined in this paper, and I am not sure how standard of a term this is. The definition I found is that $X$ is a plane scroll over a curve if it is birational to some $\mathbb{P}^2\times Z,$ where $Z$ is a curve.

Assuming this is the correct definition, what are the possible ways to identify whether a given variety is a plane scroll over a curve? I know that the Kodaira dimension of a plane scroll must be equal to $-\infty$, but what other ways are there to show that a variety is not a plane scroll?

Update: There is a nice answer for the smooth case. One of the varieties for which I would like to check whether it's a plane scroll is singular (determinantal, actually), so any ideas for the non-smooth case are very welcome!