Identifying plane scrolls

Question

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!

Answer 1

One way to identify plane scroll could be the following. Assume that $X \subset \mathbb{P}^5$ is smooth and that it contains a $\mathbb{P}^2$, which I denote by $L$.

Consider $L^{\perp} \subset (\mathbb{P}^5)^*$, which is again a $\mathbb{P}^2$. If $X$ is not a linear space, then for any $H \in L^{\perp}$, $H \cap X$ is singular along a curve, say $C_H$, which is included in $L$. Assume that for generic $H \in L^{\perp}$, the hyperplane sections $H \cap X$ are isomorphic, then their singular locus $C_H$ are also isomorphic.

If the $C_H$ for generic $H \in L^{\perp}$ cover a dense open subset in $X$, then we are done, $X$ is birational to $L^{\perp} \times C_H$.

If the $C_H$ for $H \in L^{\perp}$ do not cover $X$. Then it means that for a fixed $C_{H_0}$, you have at last a one dimensional scheme of hyperplanes included in $L^{\perp}$ which are tangent to $X$ along $C_{H_0}$. Take $H_1$ and $H_2$ distinct in this one dimensional scheme : the intersection $H_1 \cap H_2$ is a $\mathbb{P}^3$ which is tangent to $X$ along the curve $C_{H_0}$. Since $X$ is smooth, this is impossible by Zak's Theorem on tangency. So this second case is excluded. And we have proved that $X$ is birational to $L^{\perp} \times C_H$.

I have the feeling this argument could be generalized to any smooth $X \subset \mathbb{P}^r$, as long as it contains a divisor spanning a low dimensional linear space.