Smoothness and Very Ampleness in the Ohsugi–Hibi Example

Definitions and mathematical statement

Over the unspecified ground field $\mathbf k$ of the source, set \[ I=(x_2x_8-x_4x_7,\;x_1x_6-x_3x_5,\;x_1x_3-x_2x_4) \subseteq\mathbf k[x_1,\ldots,x_8], \qquad X=\operatorname{Proj}(\mathbf k[x_1,\ldots,x_8]/I)\subseteq{\mathbb{P}}^7_{\mathbf k}. \] The record calls this the projective fourfold from the Ohsugi--Hibi example and calls $I$ its toric ideal. Its two questions are whether $X$ is smooth and whether the divisor corresponding to the embedding is very ample. Smoothness here would mean that the structure morphism $X\to\operatorname{Spec}(\mathbf k)$ is smooth; over an algebraically closed field this is the usual nonsingularity condition.

A line bundle $L$ is very ample if it is the pullback of $\mathcal O_{{\mathbb{P}}^m}(1)$ under some closed immersion into projective space. Consequently, if the second question literally refers to the hyperplane bundle $\mathcal O_X(1)$ of the displayed closed embedding, its very ampleness is already part of that definition. A separate nontrivial very-ampleness question must therefore specify a different divisor or a different toric model, polytope, or morphism. The supplied record does not identify such an alternative object. No alternative divisor, normalization, saturation, or additional defining equation is substituted here.

Short English statement

Is the displayed Ohsugi--Hibi fourfold smooth? The accompanying very-ampleness question needs clarification, since the hyperplane bundle of the stated closed embedding is very ample by definition.

Sources

• Local UnsolvedMath registry, permanent ID $30000816$, source code catalogue entry, "Smoothness and Very Ampleness in the Ohsugi--Hibi Example." The example attribution is supplied; no individual proposer is named.
• Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/39, pp. 2283--2320: https://doi.org/10.4171/owr/2007/39. This is the sole supplied external reference; no separate Ohsugi--Hibi paper citation is provided.
• The literature triage checked 2026-08-21 labels the item open but says the incomplete polytope description prevents identifying the questions with a resolved theorem. That dated assessment has not been independently verified.