Vanishing locus in a higher dimension
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Let $I vartriangleleft k[x_0,...,x_n]$ be a homogeneous ideal, such that $V_{mathbb P^n }(I)$ is a variety (that is, an irreducible closed set in the projective space $mathbb P^n$).
Must $V_{ mathbb P^{n+1} }(I) subseteq mathbb P^{n+1}$ be a projective variety?
algebraic-geometry
asked Nov 17 at 13:13
zinR
407
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show 1 more comment
up vote
0
down vote
favorite
Let $I vartriangleleft k[x_0,...,x_n]$ be a homogeneous ideal, such that $V_{mathbb P^n }(I)$ is a variety (that is, an irreducible closed set in the projective space $mathbb P^n$).
Must $V_{ mathbb P^{n+1} }(I) subseteq mathbb P^{n+1}$ be a projective variety?
algebraic-geometry
asked Nov 17 at 13:13
zinR
407
What is your definition of projective variety?
– random123
Nov 18 at 13:10
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $I vartriangleleft k[x_0,...,x_n]$ be a homogeneous ideal, such that $V_{mathbb P^n }(I)$ is a variety (that is, an irreducible closed set in the projective space $mathbb P^n$).
Must $V_{ mathbb P^{n+1} }(I) subseteq mathbb P^{n+1}$ be a projective variety?
algebraic-geometry
asked Nov 17 at 13:13
zinR
407
Let $I vartriangleleft k[x_0,...,x_n]$ be a homogeneous ideal, such that $V_{mathbb P^n }(I)$ is a variety (that is, an irreducible closed set in the projective space $mathbb P^n$).
Must $V_{ mathbb P^{n+1} }(I) subseteq mathbb P^{n+1}$ be a projective variety?
algebraic-geometry
algebraic-geometry
asked Nov 17 at 13:13
zinR
407
asked Nov 17 at 13:13
zinR
407
asked Nov 17 at 13:13
zinR
407
asked Nov 17 at 13:13
zinR
407
asked Nov 17 at 13:13
zinR
407
407
What is your definition of projective variety?
– random123
Nov 18 at 13:10
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38
|
show 1 more comment
What is your definition of projective variety?
– random123
Nov 18 at 13:10
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38
What is your definition of projective variety?
– random123
Nov 18 at 13:10
What is your definition of projective variety?
– random123
Nov 18 at 13:10
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38
|
show 1 more comment
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What is your definition of projective variety?
– random123
Nov 18 at 13:10
@random123 A projective variety $X$ is a projective algebraic set $X=V_{mathbb P^n} (I)$, where $I subseteq k[x_0, ...,x_n]$ is a homogeneous ideal, such that $X$ is an irreducible topological space (via the subspace topology that is induced by the Zariski-topology on $mathbb P^n$.
– zinR
Nov 18 at 17:15
What do you mean by the $I$, when you write $V_{P^{n+1}}(I)$. Does it mean the ideal generated by the set $I$ in $k[x_0, dots x_{n+1}]$? I am assuming $k[x_0, dots x_n] subset k[x_0,dots , x_{n}, x_{n+1}]$.
– random123
Nov 18 at 17:26
yes. In particular, the ideal generated by $I$ will be homogeneous again, since $I$ was generated by homogeneous polynomials
– zinR
Nov 18 at 17:35
But isn't that the definition of a projective variety? That it is of the form $V(I)$ for a homogeneous ideal I.
– random123
Nov 18 at 17:38