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

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 $H_i(w,x_0,x_i):=H(w,x_0,x_i)\in \mathbb{C}[w,x_0,x_1,\dotsc,x_n]$ and the projective algebraic set $Z(H_1,\dotsc,H_n)\subseteq \mathbb{P}^{n+1}$.

How 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 $Z(H(w,x_0,x_1),x_2−x_1,\dotsc,x_n−x_1)$, I would like the curve to "spread" among all coordinates.)

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 do this set have, are all of them isomorphic?

What I suppose is that there should be some symmetries between the irreducible components, 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)?

Thanks in advance

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