Map of Grassmannians associated with a Veronese embedding
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I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.
Let $V$ be a finite-dimensional vector space (over $mathbb{C}$, say) and let $v_n colon mathbb{P}(V) longrightarrow mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.
If $ell$ is a line in $mathbb{P}(V)$, then $v_n(ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $Pi_{ell} subset mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$psi_n colon mathbb{G}(1, , mathbb{P}(V)) longrightarrow mathbb{G}(n, , mathbb{P}(S^nV)),$$
given by $psi_n(ell) :=Pi_{ell}$.
Question. Is $psi_n$ an embedding?
ag.algebraic-geometry reference-request grassmannians
add a comment |
up vote
9
down vote
favorite
I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.
Let $V$ be a finite-dimensional vector space (over $mathbb{C}$, say) and let $v_n colon mathbb{P}(V) longrightarrow mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.
If $ell$ is a line in $mathbb{P}(V)$, then $v_n(ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $Pi_{ell} subset mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$psi_n colon mathbb{G}(1, , mathbb{P}(V)) longrightarrow mathbb{G}(n, , mathbb{P}(S^nV)),$$
given by $psi_n(ell) :=Pi_{ell}$.
Question. Is $psi_n$ an embedding?
ag.algebraic-geometry reference-request grassmannians
Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago
add a comment |
up vote
9
down vote
favorite
up vote
9
down vote
favorite
I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.
Let $V$ be a finite-dimensional vector space (over $mathbb{C}$, say) and let $v_n colon mathbb{P}(V) longrightarrow mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.
If $ell$ is a line in $mathbb{P}(V)$, then $v_n(ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $Pi_{ell} subset mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$psi_n colon mathbb{G}(1, , mathbb{P}(V)) longrightarrow mathbb{G}(n, , mathbb{P}(S^nV)),$$
given by $psi_n(ell) :=Pi_{ell}$.
Question. Is $psi_n$ an embedding?
ag.algebraic-geometry reference-request grassmannians
I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to relevant books or papers will be highly appreciated.
Let $V$ be a finite-dimensional vector space (over $mathbb{C}$, say) and let $v_n colon mathbb{P}(V) longrightarrow mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.
If $ell$ is a line in $mathbb{P}(V)$, then $v_n(ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $Pi_{ell} subset mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$psi_n colon mathbb{G}(1, , mathbb{P}(V)) longrightarrow mathbb{G}(n, , mathbb{P}(S^nV)),$$
given by $psi_n(ell) :=Pi_{ell}$.
Question. Is $psi_n$ an embedding?
ag.algebraic-geometry reference-request grassmannians
ag.algebraic-geometry reference-request grassmannians
edited 21 hours ago
asked yesterday
Francesco Polizzi
47.2k3125202
47.2k3125202
Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago
add a comment |
Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago
Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago
Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago
add a comment |
1 Answer
1
active
oldest
votes
up vote
12
down vote
I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).
The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
up vote
12
down vote
I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).
The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
add a comment |
up vote
12
down vote
I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).
The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
add a comment |
up vote
12
down vote
up vote
12
down vote
I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).
The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.
I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).
The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.
answered yesterday
Sasha
20.1k22654
20.1k22654
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
add a comment |
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
1
1
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
And both are special cases of the Borel-Weil-Tits embedding of $G/P_2$ into projective space of its geometric quantization, when endowed with its symplectic structure as coadjoint orbit of $omega_2$, resp. ${n+1choose 2}omega_2$.
– Francois Ziegler
yesterday
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
Thank you very much for the answer, I will check the details.
– Francesco Polizzi
21 hours ago
add a comment |
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Let $mathbb{C}^2$ be the vector space associated to $l$. Isn'it that $mathbb{P}(S^n mathbb{C}^2) cap v_n(mathbb{P}(V)) = v_n(l)$?
– Libli
17 hours ago