TITLE: Isomorphisms of complete intersections
QUESTION [5 upvotes]: Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so that we can conclude that $f$ is induced by an action of the Automorphism group of $\mathbb{P}^n$, $\operatorname{PGL}(n+1)$?
I was reading the "Algebraic Hypersurfaces" note by J. Kollár (https://www.ams.org/journals/bull/2019-56-04/S0273-0979-2019-01663-2/S0273-0979-2019-01663-2.pdf) and I was wondering if there was a generalisation of Theorem $30$ for complete intersections.

REPLY [7 votes]: This always holds for $\dim(X)\geq 3$. The point is that in this case the Picard group of $X$ is cyclic, generated by  the line bundle $\mathscr{O}_X(1):= \mathscr{O}_{\mathbb{P}^n}(1)_{|X}$; this is SGA 2, Exp. 12, Cor. 3.7. Therefore any isomorphism $f:X\rightarrow Y$ induces an isomorphism $f^*\mathscr{O}_Y(1)\cong \mathscr{O}_X(1)$.
Since the embedding in $\mathbb{P}^n$ is given by the global sections of this line bundle, this implies that $f$ is the restriction of an automorphism of $\mathbb{P}^n$.