I was aware of an algebraic geometric model of symplectic $T^* S^2$ recently, that it is $\{x_1^2+x_2^2+x_3^2=1\}$ in $\mathbb{C}^3$, which the Lagrangian $S^2$ is just the real part, and in this way we have 2 foliations of lines. First I had a hard time visualizing this fact even after I saw all the symplectomorphisms written down, and I also wonder whether there are higher genus analogues to this model so that we can see lines or conics or whatever somehow clearly?