Feel like I'm asking a really simple question, but I'm self studying this so please offer me any help!
Suppose I have two conics $E_1$, $E_2$ in $\mathbb{C}\mathbb{P}^2$ and their real parts are ellipses in $\mathbb{R}^2$. I want to know the expression for the union of these two ellipses, and the dual of this union.
I think the way that works is that I take the product of $E_1E_2$ and this would be an algebraic curve of degree 4 that has the real part as the union of ellipses. Then I take the dual of $E_1E_2$ to get the dual. My question is: is $dual(E_1E_2)=dual(E_1)dual(E_2)$, or would it factor into anything? Is this identity generally true for the product of two algebraic curves?
Also, it would be nice to know whether $dual(E_1E_2)$ is degenerate or not, and I'm not sure how to check it.