Embeddings of flag manifolds
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Consider the flag manifold $mathbb{F}(a_1,dots,a_k)$ parametrizing flags of type $F^{a_1}subseteqdotssubseteq F^{a_k}subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a sub-vector space of dimension $a_i$.
Then $mathbb{F}(a_1,dots,a_k)$ embeds in the product of Grassmannians $G(a_1,V)timesdotstimes G(a_k,V)$ which in turn embeds in $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ via the product of the Plücker embeddings. Now we can embed $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ in a projective space $mathbb{P}^N$ via the Segre embedding.
Finally, we get an embedding $mathbb{F}(a_1,dots,a_k)hookrightarrowmathbb{P}^N$. Is this embedding the minimal rational homogeneous embedding of $mathbb{F}(a_1,dots,a_k)$?
ag.algebraic-geometry projective-geometry homogeneous-spaces flag-varieties embeddings
edited 9 hours ago
Michael Albanese
7,71655293
asked 14 hours ago
gxggxg
1538
$endgroup$
add a comment |
$begingroup$
Consider the flag manifold $mathbb{F}(a_1,dots,a_k)$ parametrizing flags of type $F^{a_1}subseteqdotssubseteq F^{a_k}subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a sub-vector space of dimension $a_i$.
Then $mathbb{F}(a_1,dots,a_k)$ embeds in the product of Grassmannians $G(a_1,V)timesdotstimes G(a_k,V)$ which in turn embeds in $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ via the product of the Plücker embeddings. Now we can embed $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ in a projective space $mathbb{P}^N$ via the Segre embedding.
Finally, we get an embedding $mathbb{F}(a_1,dots,a_k)hookrightarrowmathbb{P}^N$. Is this embedding the minimal rational homogeneous embedding of $mathbb{F}(a_1,dots,a_k)$?
ag.algebraic-geometry projective-geometry homogeneous-spaces flag-varieties embeddings
edited 9 hours ago
Michael Albanese
7,71655293
asked 14 hours ago
gxggxg
1538
$endgroup$
2
$begingroup$
What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
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– Jason Starr
13 hours ago
add a comment |
$begingroup$
Consider the flag manifold $mathbb{F}(a_1,dots,a_k)$ parametrizing flags of type $F^{a_1}subseteqdotssubseteq F^{a_k}subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a sub-vector space of dimension $a_i$.
Then $mathbb{F}(a_1,dots,a_k)$ embeds in the product of Grassmannians $G(a_1,V)timesdotstimes G(a_k,V)$ which in turn embeds in $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ via the product of the Plücker embeddings. Now we can embed $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ in a projective space $mathbb{P}^N$ via the Segre embedding.
Finally, we get an embedding $mathbb{F}(a_1,dots,a_k)hookrightarrowmathbb{P}^N$. Is this embedding the minimal rational homogeneous embedding of $mathbb{F}(a_1,dots,a_k)$?
ag.algebraic-geometry projective-geometry homogeneous-spaces flag-varieties embeddings
edited 9 hours ago
Michael Albanese
7,71655293
asked 14 hours ago
gxggxg
1538
$endgroup$
Consider the flag manifold $mathbb{F}(a_1,dots,a_k)$ parametrizing flags of type $F^{a_1}subseteqdotssubseteq F^{a_k}subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a sub-vector space of dimension $a_i$.
Then $mathbb{F}(a_1,dots,a_k)$ embeds in the product of Grassmannians $G(a_1,V)timesdotstimes G(a_k,V)$ which in turn embeds in $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ via the product of the Plücker embeddings. Now we can embed $mathbb{P}^{N_1}timesdotstimesmathbb{P}^{N_k}$ in a projective space $mathbb{P}^N$ via the Segre embedding.
Finally, we get an embedding $mathbb{F}(a_1,dots,a_k)hookrightarrowmathbb{P}^N$. Is this embedding the minimal rational homogeneous embedding of $mathbb{F}(a_1,dots,a_k)$?
ag.algebraic-geometry projective-geometry homogeneous-spaces flag-varieties embeddings
ag.algebraic-geometry projective-geometry homogeneous-spaces flag-varieties embeddings
edited 9 hours ago
Michael Albanese
7,71655293
asked 14 hours ago
gxggxg
1538
edited 9 hours ago
Michael Albanese
7,71655293
asked 14 hours ago
gxggxg
1538
edited 9 hours ago
Michael Albanese
7,71655293
edited 9 hours ago
Michael Albanese
7,71655293
edited 9 hours ago
Michael Albanese
7,71655293
7,71655293
asked 14 hours ago
gxggxg
1538
asked 14 hours ago
gxggxg
1538
asked 14 hours ago
gxggxg
1538
1538
2
$begingroup$
What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
$endgroup$
– Jason Starr
13 hours ago
add a comment |
2
$begingroup$
What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
$endgroup$
– Jason Starr
13 hours ago
2
2
$begingroup$
What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
$endgroup$
– Jason Starr
13 hours ago
$begingroup$
What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
$endgroup$
– Jason Starr
13 hours ago
add a comment |
1 Answer
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In general there is a more efficient way: $a_1,ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
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1 Answer
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$begingroup$
In general there is a more efficient way: $a_1,ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
$endgroup$
add a comment |
$begingroup$
In general there is a more efficient way: $a_1,ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
$endgroup$
add a comment |
$begingroup$
In general there is a more efficient way: $a_1,ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
$endgroup$
In general there is a more efficient way: $a_1,ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
answered 14 hours ago
Victor PetrovVictor Petrov
1,23968
1,23968
add a comment |
add a comment |
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What @VictorPetrov writes is completely correct. I just want to clarify one point. When you write "minimal", do you mean that the dimension of the projective space is minimal, or do you mean that the ample cone is the translate of the nef cone by the divisor class of this embedding (the embedding is the "vertex" of the ample cone)?
$endgroup$
– Jason Starr
13 hours ago