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.



Isomorphisms of complete intersections
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 $\mathbb{P}^n$.


