Flexibility and computational methods

The project in the framework of the competition ''International Academic Cooperation of HSE University''.

Leaders of the teams: Ivan Arzhantsev, Hoang Le Truong.

Implementation period: 2024-2026.

About the project

Algebraic geometry and commutative algebra can be rightfully called one of the most actively developing areas that interest a significant part of modern mathematicians. A joint project of the Laboratory on Algebraic Transformation Groups HSE University and scientific team from Institute of Mathematics Hanoi lies at the intersection of these two fields. We will investigate the flexibility of various classes of algebraic varieties, focusing on affine cones over projective varieties, the flexibility of affine varieties equipped with an action of some algebraic groups. We plan to find geometric interpretations of results in commutative algebra concerning local Artinian algebras to investigate algebraic varieties and their automorphism groups and develop freely distributed software products that will be useful for solving project tasks.

Obtained results may be applied both for the study of geometry of algebraic varieties and for obtaining new results in commutative algebra. We believe that the free software developed by us will become the basis for obtaining new mathematical results both in the project's theme and in related fields. Additionally, working on individual modules of this software project will serve as an excellent platform for collaboration among interested mathematical groups worldwide.

The current research of the Russian team and the Partner team has significant overlap. Colleagues from Hanoi are leading experts in commutative algebra and the flexibility of affine cones over Fano-Mukai varieties, while the Moscow team has a strong background in questions of infinite transitivity and work with affine algebraic varieties. We look forward to fruitful collaboration and research with our colleagues from Vietnam. Collaboration and participation in joint conferences will allow for close interaction between research groups, which will contribute to mutual development and the emergence of joint publications.

The tasks of scientific research: 

-  investigate the flexibility of affine cones over projective varieties such as Fano–Mukai fourfolds and fivefolds of certain genus and certain Verra fourfolds and Fano fourfolds of low Picard rank;

-  study flexibility of quotients with respect to actions of algebraic groups;

-  compute automorphism groups of classes of local Artinian algebras and investigate the modality of additive actions;

-  develop a computational framework for certain manipulations with tuples of roots in exceptional root lattices;

-  apply computational methods for manipulating with polyhedral cones and Picard lattices to study the flexibility of Fano varieties.

The project involves participation of young researchers, including undergraduate and doctoral students. The Russian team includes 2 undergraduate and 4 doctoral students, all of them have experience on the project's topic, and almost all have publications in international journals. During the project, we plan to organize 2 schools, 2 student seminars, and 3 annual student tutorials. In addition, students and graduate students participate in scientific events both as listeners and as speakers.

The project team was divided into four intersecting groups:

1). Flexibility of Affine Cones: Alexander Perepechko (leader of the group) Sergey Gaifullin, Anton Shafarevich, Kirill Shakhmatov and Ivan Beldiev;

2). Flexibility of Affine G-varieties: Sergey Gaifullin (leader of the group), Mikhail Ignatev, Anton Shafarevich,  Yulia Zaitseva, Veronika Kikteva and Timofey Vilkin;

3). Additive Actions and Hassett-Tschinkel Correspondence: Yulia Zaitseva (leader of the group), Roman Stasenko, Ivan Beldiev, Veronika Kikteva, Ekaterina Presnova  and Timofey Vilkin;

4). Computational Methods for Transformation Groups: Mikhail Ignatev (leader of the group), Alexander Perepechko, Roman Stasenko, Kirill Shakhmatov and Ekaterina Presnova.

The scientific team of the project:

Instute of Mathematics, VAST

Le Tuan Hoa (Professor, Doctor Institute of Mathematics, VAST)

Hoang Le Truong (Associate Professor, Doctor Institute of Mathematics, VAST)

Tran Nam Trung (Associate Professor, Doctor Institute of Mathematics, VAST)

Nguyen Dang Hop (Researcher, Doctor Institute of Mathematics, VAST)

Hoang Ngoc Yen (PhD Institute of Mathematics, VAST)

Nguyen Thi Anh Hang (PhD Institute of Mathematics, VAST)