Suppose that $$\mathcal F:\;(X^3+Y^3+Z^3)\lambda+Z^2X\mu=0$$ is a family of projective plane curves parameterized by $(\lambda:\mu)\in\mathbb P^1(\mathbb C)$. This family of curves forms a surface $S$; is there a way to obtain some equations defining the surface $S$?
Many thanks in advance.
Yes. The surface lives naturally in $\mathbb P^2 \times \mathbb P^1 \subseteq \mathbb P^5$.
As a surface in $\mathbb P^2 \times \mathbb P^1$, the equation is the same equation as you wrote.
You can embed it in $\mathbb P^5$ via the Segre embedding, and then the equations are given by the minors of a $2 \times 2$-matrix together with your original equation (in the new variables obtained by the Segre embedding).