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A note on the smooth blowups of 
$\mathbb {P}(1,1,1,k)$ in torus-invariant subvarieties
Published online by Cambridge University Press: 29 March 2023
Abstract
This papers classifies toric Fano threefolds with singular locus 
$\{ \frac {1}{k}(1,1,1) \}$ for 
$k \in \mathbb {Z}_{\geq 1}$ building on the work of Batyrev (1981, Nauk SSSR Ser. Mat. 45, 704–717) and Watanabe–Watanabe (1982, Tokyo J. Math. 5, 37–48). This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore, we identify birational relationships between entries of the classification. For a fixed value 
$k \geq 4$, there are exactly two such toric Fano threefolds linked by a blowup in a torus-invariant line.
MSC classification
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
This work was supported by Evans’ EPSRC Grant EP/P02095X/2.
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