Given a smooth manifold $X$ with a very-ample line bundle $L$, we have an associated embedding $$ \phi_{|L|}:X \to \mathbb{P}(H^0(X,L)^\vee) .$$ We know that if an automorphism $\alpha:X\to X$ leaves invariant the class of $L$ inside the Picard group of $X$, then $\phi_{|L|}$ is equivariant with respect to the action of $\alpha$ on the right and on the left.
Consider now as a submanifold $X$ inside a Grassmannian $G(k,n)$, and $L$ the line bundle on $X$ given by the restriction of the Schubert class $\sigma_{1,1}$ to $X$. We have then that the map $\phi_{|L|}$ is the inclusion $X\hookrightarrow G(k,n)$.
My question is: if an automorphism $\alpha:X\to X$ leaves invariant the class of $L$ inside the Picard group of $X$, can we say that $\alpha$ is the restriction of an automorphism of $G(k,n)$, as we did for $\mathbb{P}(H^0(X,L)^\vee)$. It seems very reasonable to me, but I struggle to find out why.
Thank you!
Both statements are wrong without extra hypotheses. Indeed, let $X$ be a zero-dimensional subscheme of length $N \gg 0$ (i.e., just $N$ points). Then the map $\phi_{|L|}$ is a map $X \to \mathbb{P}^{N-1}$ which does not factor through $Gr(k,n)$. Similarly, the automorphism group $\mathfrak{S}_N$ of $X$ is not embedded into the authomorphism group of $Gr(k,n)$.