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Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities.

Assume that there is an isomorphism $f:X\rightarrow Y$. Under which hypothesis may we conclude that $\mathbb{P}(a_0,\dots,a_n)\cong \mathbb{P}(b_0,\dots,b_n)$ and $f$ is induced by an automorphism of the ambient weighted projective space $\mathbb{P}(a_0,\dots,a_n)$?

I am particularly interested in the case when $X$ is a Fano $3$-fold hypersurface in a $4$-dimensional weighted projective space.

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  • $\begingroup$ There are trivial counterexamples like $ X=\mathbb{P}(a_0,\cdots, a_{n-1})\subset \mathbb{P}(a_0,\cdots, a_{n-1}, a_n) $ and $ Y=\mathbb{P}(a_0,\cdots, a_{n-1})\subset \mathbb{P}(a_0,\cdots, a_{n-1}, b_n) $ $\endgroup$
    – Chen Jiang
    Commented May 4, 2020 at 8:44

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