Birational geometry of actions on del Pezzo surfaces

Abstract: We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify birationally rigid actions on del Pezzo surfaces.

• The paper’s main contribution is the complete classification of regular generically free finite group actions on del Pezzo surfaces up to birational equivalence.
• It applies the G-equivariant minimal model program and equivariant Burnside groups to reduce classifications by conjugacy in Cr₂ and Aut(S), with explicit analysis for degrees 1–8.
• The work extends the Segre–Manin theorem by detailing when birational actions imply biregular equivalence, offering new insights for arithmetic and geometric applications.

Birational Geometry of Actions on Del Pezzo Surfaces

Introduction and Main Results

The article "Birational geometry of actions on del Pezzo surfaces" (2604.20425) establishes a comprehensive classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. This work settles previously open questions in the context of equivariant birational geometry, including the birational rigidity and solidness of group actions, the Segre–Manin phenomenon for birational versus biregular equivalence, and the structure of equivariant birational automorphism groups.

A central focus is the study of finite group actions with the condition $\operatorname{rk} \operatorname{Pic}(S)^G = 1$, i.e., the fixed part of the Picard group under $G$ is of rank one, corresponding to $G$-Mori fiber structures. The classification builds on and completes the foundational treatments of Dolgachev–Iskovskikh and others, and includes detailed analysis for cases not fully resolved, specifically in degrees $4$, $6$, and $8$.

Methods and Reduction Framework

The paper systematically reduces the general problem to classifying conjugacy classes of group actions in $\operatorname{Cr}_2$ via the $G$-equivariant minimal model program. The key reduction is that, for a regular generically free $G$-action, the minimal model is either a del Pezzo surface ($\operatorname{rk} \operatorname{Pic}(S)^G = 1$) or a $G$0-conic bundle. The classification for conic bundles, being infinite in families, is deferred, while the del Pezzo case is handled fully.

A thorough delineation is made between actions up to conjugacy in both $G$1 and the plane Cremona group $G$2, distinguishing between the subtleties of action versus mere group inclusion. The identification of the relevance of the equivariant Burnside group and the homomorphism $G$3 clarifies when birational and biregular conjugacy of actions coincide.

Classification by Degree

Degree 1, 2, 3

For $G$4, group actions are never linearizable, and the groups of equivariant birational and biregular automorphisms coincide: $G$5. All such actions are $G$6-birationally rigid.

Degree 4

The paper completes the classification of actions on degree $G$7 del Pezzo surfaces, which are smooth intersections of two quadrics in $G$8. It establishes that rigidity fails only for a small list of group types (e.g., $G$9, $G$0), with the remainder being $G$1-birationally rigid or solid. The analysis utilizes detailed group-theoretic information about possible automorphism groups, action types, and the structure of possible Sarkisov links.

Degree 5

Degree $G$2 del Pezzo surfaces have automorphism group $G$3. The classification tracks the subgroup lattice and determines precisely which actions are linearizable, which are birationally rigid, and which present nontrivial birational equivalence classes.

Degree 6

For $G$4, which corresponds to the blowup of three non-colinear points in $G$5, both regular and some exceptional cases are detailed. The authors give a case-by-case structural analysis of the finite subgroups of $G$6 and classify the birational equivalence classes of their actions. The exceptional case $G$7 produces nontrivial outer automorphisms influencing the birational automorphism group, a phenomenon not seen in degrees $G$8.

Degree 8: $G$9

Actions on the quadric surface are classified in depth. Fine distinctions between linearizable and nonlinearizable actions are made, extending prior incomplete treatments. The authors prove that for a large family of actions (excluding those with short orbits), birational and biregular classification coincide, and provide explicit generators for the equivariant birational automorphism groups, including involutions arising from Sarkisov links and compositions thereof.

Birational Rigidity, Solidness, and the Segre–Manin Theorem

A primary achievement is the proof of a sharp birational rigidity theorem: A del Pezzo surface with $4$0 and $4$1 is $4$2-birationally rigid if and only if specific group-theoretic and geometric conditions hold (explicitly classified).

The analysis extends to $4$3-solidness and gives new cases where birationality of $4$4-actions forces biregularity. Notably, the authors show that except for well-characterized exceptions, birational $4$5-actions on del Pezzo surfaces imply biregular $4$6-actions, generalizing the Segre–Manin theorem to broader settings (including all degree $4$7 and $4$8 surfaces). The only exceptions occur for transitive non-linearizable actions on $4$9 and certain cases on $6$0 and the dP$6$1 surface.

Moreover, the index of $6$2 is at most $6$3 for non-linearizable actions ($6$4), and the paper lists the explicit cases where the index is exactly $6$5, showing tight control over the possible birational ambiguity.

Burnside Groups and Birational Invariants

The work makes essential use of the equivariant Burnside group $6$6, extending earlier work on birational invariants and their relation to group actions. The fine structure of the possible birational types is clarified, and the relationship between the outer automorphism group of $6$7 and the structure of $6$8 is made entirely explicit.

Broader Implications, Applications, and Future Directions

The complete classification of group actions on del Pezzo surfaces up to birational equivalence will serve as a foundation for future work on equivariant birational geometry, effective Noether–Castelnuovo theory, and equivariant Sarkisov programs in higher dimensions.

These results provide essential data for arithmetic applications, classification problems for rational surfaces, and insights into related rationality questions. The methods may also inform advances in the structure theory of higher Cremona groups and the study of derived categories, as evidenced by connections to recent categorical proofs for quartic del Pezzo surfaces.

Refinements to the birational/outer automorphism correspondences and explicit generators for equivariant birational automorphism groups enable further explicit calculations in equivariant Galois theory, the study of moduli of varieties with group actions, and the arithmetic of del Pezzo surfaces.

Possible future directions include the completion of the analogous classification for conic bundles, which present infinite families, and extend the analysis to higher dimensions and more general Mori fiber spaces. The close interplay of group theory, algebraic geometry, and combinatorics likely offers new phenomena in these more complex cases.

Conclusion

This paper provides a definitive resolution of the birational classification of finite group actions on del Pezzo surfaces, identifies the precise scope of birational rigidity and the relation to biregularity, and gives a complete account of the structure of the equivariant birational automorphism groups. The methods and results substantially enrich the toolkit for equivariant birational geometry and rational surface theory and clarify several long-standing problems in the birational geometry of group actions.