Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0\_{X/k}(k)$ act trivially on $NS(X)$?



does $Aut^0$ act trivially on the Neron-Severi group?
Yes, as abx already said. The morphism $\mathrm{Aut}(X) \to \mathrm{Aut}(\mathrm{NS}(X))$ is algebraic. Thus, it maps the connected component of the algebraic group $\mathrm{Aut}(X)$ to the **trivial** connected component of $\mathrm{Aut}(\mathrm{NS}(X))$. QED


