Let $f: X \to Y$ be a regular map of smooth complex irreducible algebraic varieties such that every fiber of $f$ is isomorphic to $\mathbb{C}^{*}$ (the affine line with one puncture). Assume that $(X,f,Y)$ is a locally trivial (wrt Zariski topology)  fiber bundle.

Is it true that there exists a rank two vector bundle $E$ on $Y$ such that $X$ is biregular to the complement of two sections of $E$ in  the projectivization of $E$ (may be, under certain additional assumptions like the fiberwise $\mathbb{C}^{*}$-action on $X$)? Any relevant references would be greatly appreciated.

  • 1
    $\begingroup$ It is not true unless you add in the fiberwise $\mathbb{C}^*$-action. For instance, let $Y\subset \mathbb{P}^2$ be the open subset of all point $[y_0,y_1,y_2]$ such that $y_0y_1y_2\neq 0$. Let $X\subset \mathbb{P}^2\times \mathbb{P}^2$ be the set of all pairs $([x_0,x_1,x_2],[y_0,y_1,y_2])$ such that $y_0y_1y_2\neq 0$ and $y_0x_0^2 + y_1x_1^2 + y_2x_2^2 = 0$, yet $x_2\neq 0$. Edit Oops, that is not Zariski locally trivial. Let me keep thinking about this. $\endgroup$ Nov 8, 2016 at 18:13


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