I would like to understand the irreducible components of a projective algebraic set.
Given an irreducible and homogeneous polynomial $H(w,x,y)\in \mathbb{C}[w,x,y]$ we define 
$H_i(w,x_0,x_i):=H(w,x_0,x_i)\in \mathbb{C}[w,x_0,x_1,...,x_n]$ and the projective algebraic set $Z(H_1,...,H_n)\subseteq \mathbb{P}^{n+1}$.

Home many irreducible components of dimension one does this set have, are all of them isomorphic?
Does $H$ give some informations about the function field of those curves?


What I suppose is that there should be some symmetries between the these curves, but I don't know how to attack this problem. 
You may know some references dealing with the same kind of questions (maybe some intersection theory)?

This problem arises from the following:
I'm interested in finding explicitly non trivial embeddings of curves in a higher dimensional projective space. 
(By trivial embedding I mean $Z(𝐻(𝑤,𝑥_0,𝑥_1),𝑥_2−𝑥_1,…,𝑥_𝑛−𝑥_1)$, I would like the curve to "spread" among all coordinates).

Thanks in advance