Let $\Delta \subset \mathbb{R}^4$ be a (reflexive) polytope and $X$ be the hypersurfacedefined by a generic section of the any-canonical bundle of the toric variety $\mathbb{P}_{\Delta}$. Are there any good ways to see what kind of singularity $X$ gets in terms of $\Delta$?

For example, the singular locus of the (singular) mirror quintic $$ \{\sum_{i=0}^4x_i^5+\phi\prod_{i=0}^4x_i=0\}/(\mathbb{Z}_5)^3 \subset \mathbb{P}^4/(\mathbb{Z}_5)^3 $$ is well-known. Is it possible to read these singularities from the dual polytope $\Delta^*$ of $\Delta$? Here $$ \Delta=\text{Conv}(e_1,e_2,e_3,e_4,-\sum_{i=1}^4e_i) $$ and we have $\mathbb{P}_{\Delta}=\mathbb{P}^4$ and $\mathbb{P}_{\Delta^*}=\mathbb{P}^4/(\mathbb{Z}_5)^3$.

The K3 surface case (1 dimension less than above) may be instructive. Any explicit answers and references will be highly appreciated. Thank you in advance.

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    $\begingroup$ "Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori" by Victor Batyrev is the original reference. The bottom line is that X inherits singularities from the ambient space, in the sense that it looks, local analytically, as a disc times a toric singularity. $\endgroup$ Oct 21, 2014 at 2:11

1 Answer 1


In their book Mirror Symmetry and Algebraic Geometry, Cox and Katz prove that X is a "Calabi-Yau orbifold" with at most canonical singularities. You can read the orbifolding group (in your case $(\mathbb{Z}/5\mathbb{Z})^3$) directly from the appropriate polytope using homogeneous coordinates.


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