The Cremona group is the group of all invertible transformations of the plane that can be expressed using rational functions (fractions of polynomials). These transformations, called birational maps, are a central object of study in algebraic geometry. A basic but deep question is: what kinds of finite symmetry groups can appear inside the Cremona group? This paper focuses on a specific family of finite groups, namely those built from copies of the cyclic group of prime order p, which can be thought of as finite grids of symmetries where every element, when applied p times, returns to the identity. The question is how large and how varied such groups can be when they act on the plane via birational maps.
The paper provides a complete classification of all such groups up to "conjugacy," meaning two groups are considered the same if one can be transformed into the other by some birational map. The authors prove sharp bounds on how many copies of the cyclic group can appear: at most two copies for primes five and above, at most three copies for the prime three, and at most four copies for the prime two. Beyond these bounds, they give an explicit list of exactly 20 families of examples, drawn from two main sources: subgroups of a structured class of transformations called the de Jonquieres group, and symmetry groups of special algebraic surfaces called del Pezzo surfaces. They also carefully determine when two families on the list are actually equivalent to each other under a birational change of coordinates.
This work matters because understanding the finite subgroups of the Cremona group is a long-standing program in algebraic geometry, with connections to questions about which algebraic varieties are equivalent under birational maps. Complete classifications of this kind are rare and technically demanding, requiring a blend of group theory, surface geometry, and the theory of rational maps. The bounds discovered here are also surprising in their small size, reflecting deep geometric constraints on how symmetry can act birationally on the plane.