Let $G(k, n)$ be the Grassmannian of $k$-dimensional subspaces of $K^{n}$, $K$ a field, embedded in $\mathbb{P}^{N}$ by the Plücker embedding. In Harris' Algebraic Geometry, A First Course, Theorem 10.19 states that


$$\mathrm{Aut}(G(k, n)) = \mathrm{Aut}(G(k, n), \mathbb{P}^{N}),$$
where $\mathrm{Aut}(G(k, n), \mathbb{P}^{N}) := \{ T \in \mathrm{Aut}(\mathbb{P}^{N}) \mid T(G(k, n)) = G(k, n) \}$. In his proof, Harris says that it comes down to the assertion that every codimension $1$ subvariety of $G(k, n)$ is the intersection of $G(k, n)$ with a hypersurface in $\mathbb{P}^{N}$. In other words, $\mathrm{Pic}(G(k, n)) \cong \mathrm{Pic}(\mathbb{P}^{N})$, right? How such information can say about the automorphism group of these varieties? Is it true in general? That is, if $Y$ is a subvariety of a variety $X$ and $\mathbb{Pic}(Y) = \mathbb{Pic}(X)$, then every automorphism of $Y$ extends to an automorphism of $X$?


Thank you!



Picard group of $G(k, n)$ saying about automorphisms
In fact, one has the following result.



> 
> **Proposition 1.** Let $X \subset \mathbb{P}^n$ be a smooth subvariety such that $\textrm{Pic}(X)$ is generated by the hyperplane section $\mathscr{O}\_X(1)$. Then every automorphism of $X$ is induced by an automorphism of $\mathbb{P}^n.$
> 
> 
> 


The proof is very easy. Any automorphism of $X$ gives an automorphism of the Picard group $\textrm{Pic}(X)=\mathbb{Z} \cdot \mathscr{O}\_X(1)$, and since effective curves go into effective curves it follows that such a automorphism is the identity. In other words, the hyperplane class is preserved and any automorphism sends hyperplane sections into hyperplane sections. This precisely means that $\textrm{Aut}(X)$ is induced by the projective linear group $\textrm{Aut}(\mathbb{P}^n)$. 


Examples of the situation given in the Proposition 1 are Grasmannians and hypersurfaces $X \subset \mathbb{P}^n$ with $n \geq 4$ (in the last case the condition on the Picard group follows from Lefschetz Hyperplane Theorem).


Another situation where the automorphism group is induced by the ambient space is the following.



> 
> **Proposition 2.** Let $X \subset \mathbb{P}^n$ be a smooth subvariety which is either canonically or anti-canonically embedded (i.e., the embedding is the morphism associated with either $|K\_X|$ or $|-K\_X|$). Then every automorphism of $X$ is induced by an automorphism of $\mathbb{P}^n$.
> 
> 
> 


In fact, the canonical (or anti-canonical) class is preserved by any automorphism. But this class is by assumption nothing but $\mathscr{O}\_X(1)$, so we can conclude as before.


