I am studying an example of symplectic Lefschetz fibrations. As far as I know, given a Weinstein manifold $F$ and a collection $V_1,\ldots,V_k$ of exact framed Lagrangian spheres of $F$, there exists a unique up to deformation Lefschetz fibration $f:W\rightarrow\mathbb{C}$ whose regular fiber is $F$ and whose vanishing cycles are $V_1,\ldots,V_k$.

Now if we apply this fact to the case where the fiber $F$ is the plumbing of a collection of $T^*S^n$ according to any tree $T$ and the collection $V_1,\ldots,V_k$ are the zero sections, then we get the corresponding Lefschetz fibration $f:W\rightarrow\mathbb{C}$.

My question is: Is it possible to write $W$ as an explicit complex affine variety (i.e. defined by some polynomials that we can write down) and $f$ as a map defined on the ambient space $\mathbb{C}^N$ (e.g. linear map)?

I have been looking at the example where our tree is $A_k$. I know the fiber is $F=A_k^{2n}=\{x_1^2+\cdots+x_n^2+t^{k+1}-1=0\}$, but I think we cannot take $f$ to be $\mathbb{C}^{n+1}\rightarrow\mathbb{C}:(\mathbf{x},t)\mapsto x_1^2+\cdots+x_n^2+t^{k+1}-1$ because it is not Morse. I don't know if we can perturb this polynomial so that it becomes Morse, but then will the fiber still be $A_k$?

Thanks in advance.