Let $X^n\subset \mathbb{P}^N$ be an embedding of a non-singular projective toric variety (where variety stands for a reduced irreducible scheme over $\mathbb{C}$, and toric means normal variety admitting the algebraic torus action with dense open orbit). Let $x\in Pic X$ be a class of hyperplane section of $X$ (being a linear equivalence class of divisors). Pick any two non-singular irreducible subvarieties $D_1, D_2$ of $X$ representing $x$. Are $D_1$ and $D_2$ isomorphic as algebraic varieties? In case of arbitrary $x\in Pic X$ I'm aware of the moduli of K3 surfaces in $\mathbb{P}^3$ so that different divisors in the corresponding complete linear system to $x=\mathcal{O}(4)$ are most likely to be non-isomorphic. Also I keep in mind the results on Hodge numbers of generic hypersurfaces in toric varieties by Danilov and Khovanskii, as well as by Batyrev (however I'm primarily interested in non-generic smooth hyperplane sections).
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1$\begingroup$ I do not completely understand your question. The answer is clearly no in general, as your example shows. I'm missing something? $\endgroup$– Francesco PolizziJul 18, 2019 at 13:19
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$\begingroup$ In the general setting, there are some results on the variety having all hyperplane sections isomorphic (or projectively equivalent), by Beauville, by McKernan and by R. Pardini, and possibly more. However, I'm interested in the toric case. $\endgroup$– Grigory SolomadinJul 18, 2019 at 14:06
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$\begingroup$ Sections of $\mathcal{O}(4)$ are hyperplane sections in the quadruple Veronese embedding. $\endgroup$– SashaJul 18, 2019 at 14:42
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$\begingroup$ In you want an example where the answer is yes, take a $2$-dimensional quadric $Q \subset \mathbb{P}^3$, with a polarization of type $(1, \,1)$. Then all smooth divisors in this linear system are plane conics, hence they are projectively equivalent. $\endgroup$– Francesco PolizziJul 18, 2019 at 14:53
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$\begingroup$ I'm thinking of a non-singular hypersurface $D=\lbrace f=0\rbrace$ in the algebraic torus $(\mathbb{C}^{\times})^n$ being compactified in a projective non-singular toric variety $X$. Additionally require the closure $\overline{D}\subset X$ to be smooth. Such a $\overline{D}$ is a hyperplane section of $X$. Are there non-isomorphic varieties among them with a fixed $X$? I've met the non-degeneracy conditions on $f$ (w.r.t. the Newton polytope of $X$), however, they are too restrictive for me. $\endgroup$– Grigory SolomadinJul 19, 2019 at 10:10
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