Geodesic
Stringy $E$-functions of canonical toric Fano threefolds and their applications
Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 676-697.

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Let $\Delta$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_{\Delta}$ associated with $\Delta$. Using the stringy Libgober–Wood identity and our formula, we generalize the well-known combinatorial identity $\sum_{\substack{\theta \preceq \Delta\\ \dim (\theta) =1}}v(\theta) \cdot v(\theta^*) = 24$ which holds for $3$-dimensional reflexive polytopes $\Delta$.
Keywords: Fano varieties, $K3$-surfaces, lattice polytopes, toric varieties. 
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V. V. Batyrev; K. Schaller. Stringy $E$-functions of canonical toric Fano threefolds and their applications. Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 676-697. http://geodesic.mathdoc.fr/item/IM2_2019_83_4_a2/

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