On K-moduli of Fano threefolds  with degree 28 and Picard rank 4

Liana Heuberger; Andrea Petracci

doi:10.5802/afst.1829

On K-moduli of Fano threefolds with degree 28 and Picard rank 4

Liana Heuberger ¹; Andrea Petracci ²

¹ Institut de Mathématiques de Marseille (I2M), 3 place Victor Hugo, 13331 Marseille cedex 3, France
² Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, 40126, Italy

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 1159-1184

Abstract (Original language)
Résumé

We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano $3$-fold, which deforms to smooth Fano $3$-folds with anticanonical volume $28$ and Picard rank $4$. In particular, by constructing an algebraic deformation of this toric singular Fano, we show that the irreducible component of K-moduli parametrising these smooth Fano $3$-folds is a rational surface.

On étudie la structure locale de l’espace de K-modules des variétés de Fano autour d’une variété torique singulière K-polystable, qui se déforme en une variété de Fano lisse de dimension $3$, volume anticanonique $28$ et rang de Picard $4$. En particulier, par la construction d’une déformation algébrique de cette variété torique singulière on montre que la composante irréductible des K-modules correspondante est une surface rationnelle.

Received: 2024-03-28
Accepted: 2024-10-08
Published online: 2025-10-02

DOI: 10.5802/afst.1829

Classification: 14J45, 14M25
Keywords: Fano varieties, K-moduli, K-stability, deformation theory, toric geometry
Mots-clés : Variétés de Fano, K-modules, K-stabilité, théorie des déformations, géométrie torique

Author's affiliations:

Liana Heuberger ¹; Andrea Petracci ²

¹ Institut de Mathématiques de Marseille (I2M), 3 place Victor Hugo, 13331 Marseille cedex 3, France
² Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, 40126, Italy

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Liana Heuberger and Andrea Petracci},
     title = {On {K-moduli} of {Fano} threefolds  with degree~28 and {Picard} rank 4},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Liana Heuberger; Andrea Petracci. On K-moduli of Fano threefolds  with degree 28 and Picard rank 4. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 1159-1184. doi: 10.5802/afst.1829

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