Cheltsov, Ivan

Martens, Johan

Pinardin, Antoine

2026-03-26T15:50:30Z

2026-03-26

It is nowadays well known that the finite subgroups of the Cremona group are isomorphic to the finite groups that act biregularly on rational varieties. While the question of classifying these subgroups of Cremona up to isomorphism would be answered by finding all the groups that can act on rational varieties, one can ask for a finer classification, namely up to conjugation. This leads to the beautiful theory of G-equivariant birational geometry of rational varieties. We can outline two particularly important questions. The first is the linearization problem, that is, determining whether a subgroup G of Crn(C) is conjugate to a subgroup of Aut(Pn), or equivalently, the G-equivariant rationality problem. The second is the G-solidity problem. Roughly speaking, a variety is G-solid if G is not conjugate in Crn(C) to a group that can be decomposed into subgroups of Cremona groups of smaller rank. In dimension two, both questions remained open after the seminal works of Blanc and Dolgachev– Iskovskikh, and we completely answer them in the present thesis. In dimension three, we further the investigations of Cheltsov–Shramov and Prokhorov on actions of the icosahedral group A5 on Fano threefolds, and we treat the surprisingly open case of smooth quadrics. We provide all the G-birational models for the fixed-point-free actions of the icosahedral group A5 on these varieties, and we answer the questions of linearizability and solidity for all such actions. Lastly, we became interested in the Fano threefolds obtained by blowing up a non-hyperelliptic curve of degree six and genus three in the projective space, because their automorphism groups beautifully arise from actions on smooth plane quartics. We construct all the possible isomorphism classes of groups acting faithfully on such a variety. These threefolds also provided an opportunity to dive into the world of K-stability. In the final part of this thesis, we present how we linked their G-equivariant geometry to the existence of a Kähler–Einstein metric and produced many K-stable examples. We will essentially consider the actions of finite groups, and, unless stated otherwise, all the work will be done over the field of complex numbers.

https://era.ed.ac.uk/handle/1842/44530

https://doi.org/10.7488/era/7047

en

The University of Edinburgh

Cheltsov, I., Li, O., Ma’u, S., & Pinardin, A. (2024). K-stability and space sextic curves of genus three. arXiv preprint arXiv:2404.07803

Loginov, K., Pinardin, A., & Zhang, Z. (2025). Finite abelian groups acting on rationally connected threefolds: groups of k3 type. To appear

Pinardin, A. (2024). Antoine Pinardin’s GitHub repository. https://github.com/antoinepinardin. Retrieved from https://github.com/AntoinePinardin (Accessed: 2024-12-12)

Pinardin, A. (2024). G-solid rational surfaces. European Journal of Mathematics, 10(2), 33

Pinardin, A. (2025). Invariant-quadric-threefolds-for-degree-3-A5-representations. https://github.com/AntoinePinardin/Invariant-quadric-threefolds -for-degree-3-A5-representation

Pinardin, A., Sarikyan, A., & Yasinsky, E. (2024). Linearization problem for finite subgroups of the plane cremona group. arXiv preprint arXiv:2412.12022

Pinardin, A., & Zhang, Z. (2025a). https://zhijiazhangz.github.io/quada5

Pinardin, A., & Zhang, Z. (2025b). a5-equivariant geometry of quadric threefolds. Retrieved from https://arxiv.org/abs/2508.11496

G-equivariant

birational geometry

finite groups

rational varieties

K-stability

Equivariant birational geometry of rational varieties for finite group actions

Thesis

Doctoral

PhD Doctor of Philosophy