I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ and higher that admit small resolutions. Any reference will be most welcome.
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Consider the projective cone $X$ over $\mathbb{P}^m \times \mathbb{P}^n$. Then $$ \tilde{X} := \mathbb{P}_{\mathbb{P}^m}\Big(\mathcal{O} \oplus \mathcal{O}(-1)^{\oplus (n+1)}\Big) $$ is a small resolution of $X$.
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$\begingroup$ Thank you|! Are there also examples of hypersurfaces in dimension 4 or higher that have a small resolution? $\endgroup$ Commented May 8 at 18:33
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1$\begingroup$ If you mean hypersurfaces with isolated singularities, then no. Because any small resolution is the projective spectrum of a graded algebra associated with a reflexive sheaf of rank 1 which is not invertible, and hypersurfaces do not have such sheaves by Samuel's Conjecture (proved by Grothendieck). $\endgroup$– SashaCommented May 8 at 18:38
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$\begingroup$ No, I do not mean isolated singularities. $\endgroup$ Commented May 8 at 19:10
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$\begingroup$ Then the hypersurface $x_1x_2 - x_3x_4 = 0$ in $\mathbb{A}^5$ has a small resolution. $\endgroup$– SashaCommented May 8 at 19:27
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