Suppose I have a projective variety $X$ in $\mathbb{P}^N$ ($N >2$, say) defined as the zero set of some homogeneous polynomials $f_1, \ldots, f_r$. Consider the projection map $[x_0: x_1 : \cdots :x_N] \mapsto [x_0:x_1]$ from $\mathbb{P}^N$ to $\mathbb{P}^1$. How can I find, in general, the defining equations of my variety $X$ under this projection?
To make things more explicit, how can I find the defining equations of $X$ under the above projection for the following toy examples:
- $X \subseteq \mathbb{P}^4$ defined by $\mathbb{V}(x_0^2 - x_1 x_2, x_3^2 +3 x_3 x_4 - x_4^2)$;
- $X \subseteq \mathbb{P}^3$ defined by $\mathbb{V}(x_0^2 - x_1 x_2, x_3^2 + x_0 x_2)$;
- $X \subseteq \mathbb{P}^3$ defined by $\mathbb{V}(x_0^2 - x_1 x_2, x_3^2 +3 x_3 x_2)$.
I'm trying to understand how projections of varieties work, so many details, examples, and explanations are more than welcome!