Weighted projective spaces admitting ℚ-Gorenstein smoothings to Fano threefolds with Picard number one
Abstract
We study well-formed weighted projective threefolds X=ℙ(w0,w1,w2,w3) that admit a ℚ-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds Vd and prime Fano threefolds Yg, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal A-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for V5, with w0+w1+w2+w3≤ 5000. We construct ℚ-Gorenstein smoothings to V1, V2, Y6, Y10, and Y12 from numerical candidates of weighted projective threefolds, and prove that ℙ(2,5,8,25), although a numerical candidate for V4, is not ℚ-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.
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BibTeX
@misc{chen2026weighted,
title = {{Weighted projective spaces admitting $\mathbb Q$-Gorenstein smoothings to Fano threefolds with Picard number one}},
author = {Jungkai Alfred Chen and Yongnam Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37268},
eprint = {2609.37268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37268},
}
RIS
TY - UNPB
TI - Weighted projective spaces admitting ℚ-Gorenstein smoothings to Fano threefolds with Picard number one
AU - Jungkai Alfred Chen
AU - Yongnam Lee
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37268
UR - https://arxiv.org/abs/2609.37268
AB - We study well-formed weighted projective threefolds X=ℙ(w_0,w_1,w_2,w_3) that admit a ℚ-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds V_d and prime Fano threefolds Y_g, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal A-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for V_5, with w_0+w_1+w_2+w_3≤ 5000. We construct ℚ-Gorenstein smoothings to V_1, V_2, Y_6, Y_10, and Y_12 from numerical candidates of weighted projective threefolds, and prove that ℙ(2,5,8,25), although a numerical candidate for V_4, is not ℚ-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.
ER -