Orbifold Degenerations of Hirzebruch Surfaces
Abstract
We study orbifold degenerations of Hirzebruch surfaces. Our main theorem shows that every such degeneration X arises as a partial smoothing of a toric surface. We then give a combinatorial description of the singularities that arise when −KX is not nef. This complements previous joint work with G. Urzúa, which treated the case in which −KX is ample. For Hirzebruch surfaces 𝔽k with k≥ 3, we obtain an explicit description of all possible central fibers. For k≤ 1, we use the threefold minimal model program to reduce the problem to the case of central fibers whose anticanonical divisor is nef. The remaining surfaces are toric del Pezzo surfaces of degree $8$ with T-singularities. We classify these by studying the birational geometry arising from mutations of Fano polygons.
Related
- Phantom subcategories on blow-ups of Hirzebruch surfaces
- The Segre invariant and Brill-Noether theory on Hirzebruch surfaces
- A Note On the moduli space of rank two vector bundles on Hirzebruch surfaces
- Intrinsic mirror symmetry for orbifold cubic surfaces and the A1 spherical DAHA
- The moduli space of cubic surfaces and its orbifold fundamental group
BibTeX
@misc{zuniga2026orbifold,
title = {{Orbifold Degenerations of Hirzebruch Surfaces}},
author = {Juan Pablo Zúñiga},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27652},
eprint = {2608.27652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27652},
}
RIS
TY - UNPB
TI - Orbifold Degenerations of Hirzebruch Surfaces
AU - Juan Pablo Zúñiga
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27652
UR - https://arxiv.org/abs/2608.27652
AB - We study orbifold degenerations of Hirzebruch surfaces. Our main theorem shows that every such degeneration X arises as a partial smoothing of a toric surface. We then give a combinatorial description of the singularities that arise when −K_X is not nef. This complements previous joint work with G. Urzúa, which treated the case in which −K_X is ample. For Hirzebruch surfaces 𝔽_k with k≥ 3, we obtain an explicit description of all possible central fibers. For k≤ 1, we use the threefold minimal model program to reduce the problem to the case of central fibers whose anticanonical divisor is nef. The remaining surfaces are toric del Pezzo surfaces of degree $8$ with T-singularities. We classify these by studying the birational geometry arising from mutations of Fano polygons.
ER -