Calabi-Yau complete intersections in fake weighted projective spaces
Marco Ghirlanda
TL;DR
The paper develops a combinatorial framework for classifying Calabi–Yau complete intersections arising from nef-partitions in fake weighted projective spaces, enabling a complete listing up to dimension $5$. It introduces a two-stage algorithm that separates free weight data from torsion constraints, extending Ghi's nef-partition method to fwps, and uses PALP to derive Hodge numbers for $3$-fold families, revealing $20$ new Hodge pairs not realizable by toric hypersurfaces. The work provides explicit results across dimensions and codimensions and gives a sharp characterization of maximal codimension cases in terms of degree matrices $Q=\mathbb{1}_{s}\mathbb{1}_{s}AA$ with $A$ over $\mathbb{Z}/2\mathbb{Z}$, linked to orbits of multisets under the affine linear group. These findings expand the catalog of Calabi–Yau geometries beyond hypersurfaces in toric settings, with potential implications for mirror symmetry and broader toric CY classifications.
Abstract
We present a classification algorithm for Calabi-Yau complete intersections arising from nef-partitions in fake weighted projective spaces, allowing us to determine all such complete intersections up to dimension five. Furthermore, we compute the Hodge pairs of the $3$-dimensional families obtained, and find twenty new Hodge pairs not realized by any toric Calabi-Yau hypersurface. Finally, we provide an explicit characterization for the families of maximal codimension.