Mirror symmetry for gCICY threefolds (I): a [3,−1] block and a non-Gorenstein toric phase
Abstract
We construct and analyze a mirror family for the generalized complete intersection Calabi-Yau threefold X+=[ℙ1 | 3 −1; ℙ4 | 2 3], with (h1,1,h2,1)=(2,46). An auxiliary-variable presentation turns the negative-degree section into a variation of toric GIT. Crossing one wall flops 16 disjoint (−1,−1)-curves and yields a Calabi-Yau threefold X-, a nef complete intersection in a toric ℚ-Fano eightfold, which carries a genus-one fibration of index 4. Because this ambient space is not Gorenstein, the Batyrev-Borisov construction does not apply. We use the Hori-Vafa equations only to select a 2-parameter Laurent family, and construct a smooth projective crepant compactification Y of the Laurent model with (h1,1,h2,1)=(46,2); for very general parameters the Mordell-Weil group is ℤ/4ℤ. The periods of Y satisfy a rank-6 Picard-Fuchs system with two maximally unipotent boundary points, whose indicial algebras are isomorphic to the rational even cohomology rings of X- and X+. The resulting genus-zero predictions agree with independent counts of 176 vertical lines and 100 vertical conics on the genus-one fibration X-→ℙ2.
Related
- Mirror symmetry for gCICY threefolds (II): [3,−1] blocks, VGIT, and toric phases
- Weighted projective spaces admitting ℚ-Gorenstein smoothings to Fano threefolds with Picard number one
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BibTeX
@misc{kanazawa2026mirror,
title = {{Mirror symmetry for gCICY threefolds (I): a $[3,-1]$ block and a non-Gorenstein toric phase}},
author = {Atsushi Kanazawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03719},
eprint = {2610.03719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03719},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mirror symmetry for gCICY threefolds (I): a [3,−1] block and a non-Gorenstein toric phase
AU - Atsushi Kanazawa
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03719
UR - https://arxiv.org/abs/2610.03719
AB - We construct and analyze a mirror family for the generalized complete intersection Calabi-Yau threefold X_+=[ℙ^1 | 3 −1; ℙ^4 | 2 3], with (h^1,1,h^2,1)=(2,46). An auxiliary-variable presentation turns the negative-degree section into a variation of toric GIT. Crossing one wall flops 16 disjoint (−1,−1)-curves and yields a Calabi-Yau threefold X_-, a nef complete intersection in a toric ℚ-Fano eightfold, which carries a genus-one fibration of index 4. Because this ambient space is not Gorenstein, the Batyrev-Borisov construction does not apply. We use the Hori-Vafa equations only to select a 2-parameter Laurent family, and construct a smooth projective crepant compactification Y of the Laurent model with (h^1,1,h^2,1)=(46,2); for very general parameters the Mordell-Weil group is ℤ/4ℤ. The periods of Y satisfy a rank-6 Picard-Fuchs system with two maximally unipotent boundary points, whose indicial algebras are isomorphic to the rational even cohomology rings of X_- and X_+. The resulting genus-zero predictions agree with independent counts of 176 vertical lines and 100 vertical conics on the genus-one fibration X_-→ℙ^2.
ER -