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  1. I. Cheltsov, O. Li, S. Ma’u and A. Pinardin, K-stability and space sextic curves of genus three, arXiv:2404.07803 (2024), https://arxiv.org/abs/2404.07803.

  2. O. Li, J. Malbon and A. Pinardin, Equivariant rationality of Fano threefolds in the family No. 2.12, arXiv:2508.19471 (2025), https://arxiv.org/abs/2508.19471.

  3. C. Araujo, A.-M. Castravet, I. Cheltsov, K. Fujita, A.-S. Kaloghiros, J. Martinez-Garcia, C. Shramov, H. Süß and N. Viswanathan, The Calabi Problem for Fano Threefolds, London Mathematical Society Lecture Note Series 485, Cambridge University Press, 2023. Authors’ manuscript. Section numbers used here refer to this manuscript.

  4. H. Abban and Z. Zhuang, K-stability of Fano varieties via admissible flags, Forum of Mathematics, Pi 10 (2022), e15. arXiv:2003.13788v3. Numbered references to this paper use this arXiv version.

  5. S. Boucksom, C. Favre and M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier, Journal of Algebraic Geometry 18 (2009), 279–308. arXiv:math/0608260.

  6. H. Blum and M. Jonsson, Thresholds, valuations, and K-stability, Advances in Mathematics 365 (2020), 107062. arXiv:1706.04548.

  7. J. W. Bruce and C. T. C. Wall, On the classification of cubic surfaces, Journal of the London Mathematical Society (2) 19 (1979), 245–256. doi:10.1112/jlms/s2-19.2.245.

  8. I. Cheltsov, E. Denisova and K. Fujita, K-stable smooth Fano threefolds of Picard rank two, Forum of Mathematics, Sigma 12 (2024), e41. arXiv:2210.14770.

  9. I. Cheltsov and Yu. Prokhorov, Del Pezzo surfaces with infinite automorphism groups, Algebraic Geometry 8 (2021), 319–357. doi:10.14231/AG-2021-008.

  10. U. Derenthal and C. Frei, Counting imaginary quadratic points via universal torsors, II, Math. Proc. Cambridge Philos. Soc. 156 (2014), 383–407. arXiv:1304.3352.

  11. J.-P. Demailly and J. Kollár, Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Annales scientifiques de l’École Normale Supérieure (4) 34 (2001), 525–556. arXiv:math/9910118.

  12. E. Denisova, δ\delta-invariants of Du Val del Pezzo surfaces of degree ≥4\geq4, arXiv:2304.11412v2, 2025.

  13. U. Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. 108 (2014), 638–681. arXiv:math/0604194.

  14. E. Denisova, K-stability of Fano 3-folds of Picard rank 3 and degree 20, Annali dell’Università di Ferrara 70 (2024), 987–1022. Journal version. arXiv:2403.09002.

  15. E. Denisova and T. Papazachariou, K-moduli of Fano threefold family 3.5, announced work in preparation; status listed at https://www.tpapazachariou.net/research, consulted 30 September 2026; PDF listed as available upon request.

  16. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995. doi:10.1007/978-1-4612-5350-1.

  17. K. Fujita, On K-stability for Fano threefolds of rank 3 and degree 28, International Mathematics Research Notices 2023 (2023), 12601–12784. arXiv:2107.04820v1.

  18. K. Fujita, A valuative criterion for uniform K-stability of ℚ\mathbb Q-Fano varieties, Journal für die reine und angewandte Mathematik 751 (2019), 309–338. arXiv:1602.00901.

  19. G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proceedings of the London Mathematical Society (3) 14 (1964), 689–713. doi:10.1112/plms/s3-14.4.689.

  20. A.-S. Kaloghiros and A. Petracci, On toric geometry and K-stability of Fano varieties, Trans. Amer. Math. Soc. Ser. B 8 (2021), 548–577. arXiv:2009.02271.

  21. Y. Liu, C. Xu and Z. Zhuang, Finite generation for valuations computing stability thresholds and applications to K-stability, Annals of Mathematics (2) 196 (2022), 507–566. arXiv:2102.09405.

  22. C. Li, K-semistability is equivariant volume minimization, Duke Mathematical Journal 166 (2017), 3147–3218. arXiv:1512.07205.

  23. H. Abban, I. Cheltsov, T. Kishimoto and F. Mangolte, K-stability of Fano 3-folds in the World of Null-A, Transformation Groups (2026). doi:10.1007/s00031-026-09977-9.

  24. H. Abban, I. Cheltsov, T. Kishimoto and F. Mangolte, K-stability of pointless del Pezzo surfaces and Fano 3-folds, arXiv:2411.00767v1, 2024.

  25. I. Cheltsov, K. Fujita, T. Kishimoto and T. Okada, K-stable divisors in ℙ1×ℙ1×ℙ2\mathbb P^1\times\mathbb P^1\times\mathbb P^2 of degree (1,1,2)(1,1,2), Nagoya Mathematical Journal 251 (2023), 686–714. doi:10.1017/nmj.2023.5. Numbered references use the published version.

  26. F. Hidaka and K.-i. Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo Journal of Mathematics 4 (1981), 319–330. doi:10.3836/tjm/1270215157.

  27. A. Fanelli, L. Gruson and N. Perrin, Rational curves on V5V_5 and rational simple connectedness, arXiv:1901.06930.

  28. I. Cheltsov, D. Hwang, A. Küronya, A. Laface, F. Mangolte, A. Massarenti, J. Park and J. Zhao, On K-stability of Fano’s last Fanos, arXiv:2507.08528, version 2 (2026).

  29. Q.-S. Chi, Z. Xie and Y. Xu, Classification of sextic curves in the Fano 33-fold V5V_5 with rational Galois covers in ℙ3\mathbb P^3, Journal of Geometry and Physics 222 (2026), article 105760. doi:10.1016/j.geomphys.2026.105760.

  30. O. Saeki, Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in ℂ3\mathbb C^3, Journal of the Mathematical Society of Japan 52 (2000), 409–431. https://www.jstage.jst.go.jp/article/jmath1948/52/2/52_2_409/_pdf.

  31. R. Giménez Conejero, Lê Dũng Tráng and J. J. Nuño-Ballesteros, Thom condition and monodromy, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117 (2023), article 18. doi:10.1007/s13398-022-01353-y.

  32. A. Corti, M. Haskins, J. Nordström and T. Pacini, Asymptotically cylindrical Calabi–Yau 3-folds from weak Fano 3-folds, Geometry & Topology 17 (2013), 1955–2059. doi:10.2140/gt.2013.17.1955.

  33. Yu. Prokhorov, Singular Fano threefolds of genus twelve, Sbornik: Mathematics 207 (2016), 983–1009. arXiv:1508.04371.

  34. A.-S. Kaloghiros, Y. Liu, A. Petracci and J. Zhao, The boundary of K-moduli of prime Fano threefolds of genus twelve, arXiv:2603.29827, 2026.

  35. C. Xu, K-stability of Fano varieties: an algebro-geometric approach, EMS Surveys in Mathematical Sciences 8 (2021), 265–354. doi:10.4171/EMSS/51. The CM-degree minimization statement is Remark 7.9 in the published version.

  36. Y. Liu, K-stability of Fano threefolds of rank 2 and degree 14 as double covers, Mathematische Zeitschrift 303 (2023), Paper No. 38. arXiv:2204.06709.

  37. Y. Liu and J. Zhao, K-moduli of Fano threefolds and genus four curves, Journal für die reine und angewandte Mathematik 824 (2025), 1–38. arXiv:2403.16747.

  38. Y. Liu, The volume of singular Kähler–Einstein Fano varieties, Compositio Mathematica 154 (2018), 1131–1158. arXiv:1605.01034.

  39. D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, Berlin, 1994.

  40. I. Cheltsov and P. Pokora, On K-stability of ℙ3\mathbb P^3 blown up along a quintic elliptic curve, Ann. Univ. Ferrara 70 (2024), 1165–1192. doi:10.1007/s11565-024-00532-6; arXiv:2309.12528.

  41. C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics 3, Birkhäuser, 1980; reprinted in Modern Birkhäuser Classics, 2011. doi:10.1007/978-3-0348-0151-5.

  42. I. Cheltsov, K-stability of Fano 3-folds of Picard rank 3 and degree 22, São Paulo Journal of Mathematical Sciences 18 (2024), 609–638. doi:10.1007/s40863-024-00460-5.

  43. G. Belousov and K. Loginov, K-stability of Fano threefolds of rank 3 and degree 14, arXiv:2403.03700v3, 27 June 2026. We use the corrected proof of Lemma 6.3 in this version.

  44. H. Abban and Z. Zhuang, Seshadri constants and K-stability of Fano manifolds, Duke Mathematical Journal 172 (2023), 1109–1144, arXiv:2101.09246.

  45. V. Batyrev and O. Popov, The Cox ring of a del Pezzo surface, Arithmetic of Higher-Dimensional Algebraic Varieties, Progress in Mathematics 226, Birkhäuser (2004), 85–103.

  46. A. Kuznetsov and Y. Prokhorov, Rationality of Fano threefolds over non-closed fields, American Journal of Mathematics 145 (2023), 335–411.

  47. A. Kuznetsov, Y. Prokhorov and C. Shramov, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Japanese Journal of Mathematics 13 (2018), 109–185, arXiv:1605.02010.

  48. T. Kishimoto, Y. Prokhorov and M. Zaidenberg, Affine cones over Fano threefolds and additive group actions, Osaka Journal of Mathematics 51 (2014), 1093–1112, doi:10.18910/50981.

  49. M. Reid, Minimal models of canonical 3-folds, Algebraic Varieties and Analytic Varieties, Advanced Studies in Pure Mathematics 1 (1983), 131–180.

  50. N. Virin, Cylinders in Fano threefolds of genus 9 and 10, arXiv:2605.30875 (2026).

  51. C. Xu and Z. Zhuang, Open problems in K-stability of Fano varieties, arXiv:2601.15576v2, 1 June 2026.

  52. Z. Zhuang, Birational superrigidity and K-stability of Fano complete intersections of index one, Duke Mathematical Journal 169 (2020), 2205–2229, arXiv:1802.08389.

  53. D. Swinarski, Computing equations of the adjoint variety of G2G_2, computational supplement to Invariant polynomials and Mukai models, https://faculty.fordham.edu/dswinarski/InvariantPolynomialsAndMukaiModels/v1/ComputeG2AdjointVariety.htm, consulted 3 October 2026.

  54. I. Cheltsov, J. Park, Y. Prokhorov and M. Zaidenberg, Cylinders in Fano varieties, EMS Surveys in Mathematical Sciences 8 (2021), 39–105. doi:10.4171/EMSS/44.

  55. H. Abban and Z. Zhuang, K-stability of Fano varieties via admissible flags, Forum of Mathematics, Pi 10 (2022), e15. https://doi.org/10.1017/fmp.2022.11 Numbered references use the published version.

  56. A. Bayer, A. Kuznetsov and E. Macrì, Mukai bundles on Fano threefolds, arXiv:2402.07154v3, 2025. https://arxiv.org/abs/2402.07154v3

  57. D. Luna, Slices étales, Mémoires de la Société Mathématique de France 33 (1973), 81–105. https://www.numdam.org/item/MSMF_1973__33__81_0/

  58. A. Bayer, A. Kuznetsov and E. Macrì, Mukai models of Fano varieties, arXiv:2501.16157. https://arxiv.org/abs/2501.16157

  59. S. Boucksom and H. Chen, Okounkov bodies of filtered linear series, Compositio Mathematica 147 (2011), 1205–1229. https://arxiv.org/abs/0911.2923

  60. C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society 23 (2010), 405–468. https://arxiv.org/abs/math/0610203

  61. D. H. J. Epema, Surfaces with canonical hyperplane sections, CWI Tract 1, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. https://ir.cwi.nl/pub/13121/13121D.pdf

  62. H. Blum and Y. Liu, Openness of uniform K-stability in families of ℚ\mathbb Q-Fano varieties, arXiv:1808.09070v2, 2020. https://arxiv.org/abs/1808.09070

  63. H. Blum, Y. Liu and C. Xu, Openness of K-semistability for Fano varieties, Duke Mathematical Journal 171 (2022), 2753–2797. https://arxiv.org/abs/1907.02408

  64. E. Denisova, δ\delta-invariants of Du Val del Pezzo surfaces of degree 33, arXiv:2311.14181v2, 2025. https://arxiv.org/abs/2311.14181v2

  65. S. K. Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, Handbook of Geometric Analysis, vol. 1, Advanced Lectures in Mathematics 7 (2008), 29–75. https://arxiv.org/abs/0803.0985

  66. S. K. Donaldson, Stability of algebraic varieties and Kähler geometry, Proceedings of Symposia in Pure Mathematics 97.1 (2018), 199–221. https://www.claymath.org/wp-content/uploads/2022/03/Donaldson-AG2015.pdf

  67. R. W. Richardson, On orbits of algebraic groups and Lie groups, Bulletin of the Australian Mathematical Society 25 (1982), 1–28. https://doi.org/10.1017/S0004972700005013

  68. G. Székelyhidi, The Kähler–Ricci flow and K-polystability, American Journal of Mathematics 132 (2010), 1077–1090. https://arxiv.org/abs/0803.1613

  69. K. Fujita, Elementary links from prime Fano threefolds along two lines, arXiv:2604.08097v1, 2026. Sections 2, 4 and 8. https://arxiv.org/abs/2604.08097

  70. K. Takeuchi, Some birational maps of Fano 33-folds, Compositio Mathematica 71 (1989), 265–283. https://www.numdam.org/article/CM_1989__71_3_265_0.pdf

  71. A. Dubouloz, K. Fujita and T. Kishimoto, Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups, arXiv:2512.08409. https://arxiv.org/abs/2512.08409

  72. M. Furushima and M. Tada, Non-normal del Pezzo surfaces and Fano threefolds of the first kind, Journal für die reine und angewandte Mathematik 429 (1992), 183–190. https://doi.org/10.1515/crll.1992.429.183

  73. K. Saito, Einfach-elliptische Singularitäten, Inventiones Mathematicae 23 (1974), 289–325.

  74. A. Isaev, Simple Elliptic Hypersurface Singularities: A New Look at the Equivalence Problem, Proceedings of the Centre for Mathematics and its Applications, Australian National University, 43, 9–17. https://maths.anu.edu.au/files/CMAProc43_Isaev.pdf