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I would like to understand how to construct a parametization of a flag variety $F(V,n_1,\ldots,n_r)\subseteq \mathbb{P}^N$ in its minimal embedding.

First, I would like to know if there is a closed formula to compute $N$ in term of the numerical parameters of the flag variety $F(V,n_1,\ldots,n_r)$. I've done some examples but I haven't been able to say if the embedding is the minimal one.

For example a local parametrization for the flag variety $F(\mathbb{C}^4,1,2)\subset G(1,4)\times G(2,4)$ around to the point $\{\langle e_1\rangle,\langle e_1,e_2\rangle\}$ can be given by

$$\begin{array}{ccccc} \varphi&:&\mathbb{A}^1\times\mathbb{A}^4&\longrightarrow&\mathbb{A}^8\\ &&\left(\alpha,\left(\begin{array}{cccc} 1&0&a&b\\ 0&1&c&d \end{array}\right)\right)&\longmapsto&(\alpha,a+\alpha c, b+\alpha d,c,d,-a,-b, ad-bc) \end{array}$$

denoting the coordinates of $\mathbb{A}^8$ by $X_2,X_3,X_4,Y_2,Y_3.Y_4,Y_5,Y_6$, the image of $\varphi$ is given by

$V(Y_4-X_2Y_2+X_3,Y_5-X_2Y_3+X_4,X_{34}-X_3Y_3+X_4Y_2,X_2Y_6- X_3Y_5+X_4Y_4,Y_6-Y_2Y_5+ Y_3Y_4 )$

The question is: "Is $\mathbb{P}^8$ the minimal projective space for which we can embedding $F(\mathbb{C}^4,1,2)$?"

Another example is the flag variety $F(\mathbb{C}^4,1,3)\subset G(1,4)\times G(3,4)$. A local parametrization for $F(\mathbb{C}^4,1,3)$ around to the point $\{\langle e_1\rangle,\langle e_1,e_2, e_3\rangle\}$ can be given by

$$\begin{array}{ccccc} \varphi&:&\mathbb{A}^2\times\mathbb{A}^3&\longrightarrow&\mathbb{A}^6\\ &&\left((\alpha,\beta),\left(\begin{array}{cccc} 1&0&0&a\\ 0&1&0&b\\ 0&0&1&c \end{array}\right)\right)&\longmapsto&(\alpha,\beta, a+\alpha b +\beta c,c,-b,a) \end{array}$$

denoting the coordinates of $\mathbb{A}^6$ by $X_2,X_3,X_4,Y_2,Y_3,Y_4$ then the image of $\varphi$ is given by $V(Y_4-X_2Y_3+X_3Y_2-X_4)$. Again the question is: "Is $\mathbb{P}^6$ the minimal projective space for which we can embedding $F(\mathbb{C}^4,1,3)$?"

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    $\begingroup$ You cannot embed $F(\mathbb{C}^4,1,3)$ into $\mathbb{P}^6$ --- if there was an embedding the image would be a hypersurface and by Lefschetz theorem would have Picard number 1, while it actually has Picard number 2. $\endgroup$
    – Sasha
    Commented Mar 25, 2019 at 20:24
  • $\begingroup$ Concerning the dimension, see mathoverflow.net/questions/325605/embeddings-of-flag-manifolds/… If you want to describe rational parametrization you can use the "big cell". Namely, you represent the fllag variety as the orbit of a point $x$ in the projective space, whose stabilizer is the parabolic subgroup, say $P$, and look at $U^{-}x$, where $U^{-}$ is the unipotent radical of the Borel subgroup corresponding to the opposite parabolic subroup $P^-$. $\endgroup$
    – Victor Petrov
    Commented Mar 26, 2019 at 7:18
  • $\begingroup$ @VictorPetrov My interpretation of the OP's question is the following: how do we know that we cannot get a smaller embedding by composing with a linear projection? That definitely will not be a "linearly normal" embedding, but it might still be an embedding. $\endgroup$
    – Jason Starr
    Commented Mar 26, 2019 at 9:31
  • $\begingroup$ In this specific case $F(1,3,V_4)$ can be identified as a linear section of $\mathbb{P}^3 \times (\mathbb{P}^3)^{\vee}$ given by incidence (see Sasha's comment indeed). In general the same can be said for $F(1, n-1, V_n)$. $\endgroup$
    – Enrico
    Commented Mar 26, 2019 at 10:36

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