Hi. I have a question about automorphisms of smooth ratonal surfaces. (I am not an algebraic geometer, so my question could be stupid. I am sorry about that.)

Let $X_k$ be a blow-up of $\mathbb{P}^2$ at $k$ generic points. (Especially for $k \leq 3$.) Fix a homology class $A \in H_2(X_k, \mathbb{Z})$, and let $C_1, C_2$ be two algebraic curves on $X_k$ such that $[C_1] = [C_2] = A$. Then my question is,

Q1 : Can we always find an automorphism of $X_k$ which sends $C_1$ to $C_2$ ?

Q2 : If Q1 is not true, the same question Q1 when $C_1$ and $C_2$ are rational.

For a given symplectic manifold $(M,\omega)$ and two symplectic surfaces $C_1$ and $C_2$ with $[C_1]=[C_2] \in H_2(M,\mathbb{Z})$, I wanted to know whether there exists a symplectomorphism of $(M,\omega)$ which sends $C_1$ to $C_2$ or not.

On $\mathbb{P}^2$, B. Seibert and G. Tian proved that any symplectic curve of degree less than 18 is symplectically isotopic to an algebraic curve (along the Hamiltonian isotopy on $\mathbb{P}^2$). And they also prove that for the Hirzebruch surface or $\mathbb{P}^1 \times \mathbb{P}^1$ when a degree of curve is less than 8. (ON THE HOLOMORPHICITY OF GENUS TWO LEFSCHETZ FIBRATIONS)

And V. Shevcishin proved that any two curves with the same degree less than 7 on $\mathbb{P}^2$ is symplectically isotopic each other.

My question is much weaker problem than the symplectic isotopy problem, so I hope that there would be an answer for the Question 1 and 2.

Thank you for reading and I appriciate for any comment.