The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?
Abstract
We consider a certain pair of families of Picard rank 1 Calabi--Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family 𝒳 of degree 33 threefolds in G(2,6) (constructed by Miura) and the family 𝒴 of arithmetically Gorenstein degree 21 threefolds in ℙ8 (constructed by Schenck--Stillman--Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier--Mukai partners to members of 𝒳, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus--Jockers, in a physical context. In fact, starting from the other family 𝒴, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi--Yau threefolds having Picard rank 1.
Related
- Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces
- All-Genus Open Mirror Symmetry for Toric Calabi-Yau 3-Orbifolds with Multiple Branes
- Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds
- Densities of Picard Rank Jumps for K3 Surfaces Over Global Fields
- Line-filtered ACM bundles and Ulrich defects on abelian varieties of Picard rank one
BibTeX
@misc{kapustka2026last,
title = {{The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?}},
author = {Michał Kapustka and Marco Rampazzo and Prajwal Samal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20704},
eprint = {2609.20704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20704},
}
RIS
TY - UNPB
TI - The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?
AU - Michał Kapustka
AU - Marco Rampazzo
AU - Prajwal Samal
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.20704
UR - https://arxiv.org/abs/2609.20704
AB - We consider a certain pair of families of Picard rank 1 Calabi--Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family 𝒳 of degree 33 threefolds in G(2,6) (constructed by Miura) and the family 𝒴 of arithmetically Gorenstein degree 21 threefolds in ℙ^8 (constructed by Schenck--Stillman--Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier--Mukai partners to members of 𝒳, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus--Jockers, in a physical context. In fact, starting from the other family 𝒴, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi--Yau threefolds having Picard rank 1.
ER -