Galois group of exceptional curves on the generic del Pezzo surface

Abstract: We prove that the Galois action on the exceptional curves on the generic del Pezzo surface of degree $d$ is maximal for all degrees $d$ and over any field $k$. As a consequence of the case $d=3$, we deduce that over $\mathbb{F}_q(u)$, 100% of cubic surfaces have no Brauer-Manin obstruction.

• The paper proves that the Galois action on exceptional curves of generic del Pezzo surfaces attains a maximal image, matching the full automorphism group of their incidence graph.
• It constructs moduli and incidence spaces that are irreducible and smooth, using monodromy arguments to establish a surjective Galois representation.
• The study demonstrates that almost all cubic surfaces over F_q(u) avoid the Brauer-Manin obstruction, linking the geometric maximality to significant arithmetic implications.

Galois Actions on Exceptional Curves of Generic del Pezzo Surfaces

Introduction and Problem Statement

The paper "Galois group of exceptional curves on the generic del Pezzo surface" (2604.01497) investigates the image of the absolute Galois group acting on the set of exceptional curves on the generic del Pezzo surface of degree $d$, for $1 \leq d \leq 7$, over an arbitrary base field $k$. Exceptional curves, being $(-1)$-curves, play a central role in the geometry and arithmetic of del Pezzo surfaces, underpinning their birational classification and arithmetic obstructions.

The main theorem establishes that for the generic del Pezzo surface of each degree, the Galois action on exceptional curves is as large as possible: the image coincides with the full automorphism group of the incidence graph of exceptional curves, i.e.,

$G_d = \operatorname{Aut}(\Gamma_d)$

where $G_d$ denotes the Galois image and $\Gamma_d$ the incidence graph of exceptional curves. This maximality result is proven uniformly for all fields and all $1\leq d\leq 7$. As a significant consequence, the paper deduces that 100% of cubic surfaces over $F_q(u)$, when ordered by height, have no Brauer-Manin obstruction to the Hasse principle.

Parameter Spaces for del Pezzo Surfaces

The method begins by constructing, for each $d$, a moduli space $1 \leq d \leq 7$0 parameterizing del Pezzo surfaces of degree $1 \leq d \leq 7$1 in their natural (anti)canonical embeddings. For $1 \leq d \leq 7$2, these are realized as open subschemes of Hilbert schemes parameterizing degree $1 \leq d \leq 7$3 surfaces in $1 \leq d \leq 7$4 with prescribed Hilbert polynomial. For $1 \leq d \leq 7$5, the surfaces are embedded as weighted hypersurfaces in the appropriate weighted projective spaces $1 \leq d \leq 7$6 or $1 \leq d \leq 7$7, respectively.

These moduli spaces are shown to be irreducible, smooth, and quasi-projective, ensuring the existence of a well-defined generic fiber—the generic del Pezzo surface.

The Incidence Scheme and Monodromy Formulation

To relate the Galois action to geometric monodromy, the analysis introduces the finite étale cover $1 \leq d \leq 7$8, whose fibers parametrize exceptional curves on each surface. $1 \leq d \leq 7$9 is constructed as an incidence subscheme of $k$0, where $k$1 parametrizes lines (or more generally, $k$2-curves) in the relevant (weighted) projective space.

Key technical results establish that $k$3 is geometrically irreducible for $k$4. Monodromy then yields a surjective homomorphism from the étale fundamental group to the automorphism group $k$5, and hence the image of the generic Galois action is maximal. This transitivity provides a group-theoretic framework for analyzing the possible Galois representations arising from geometry.

Inductive and Uniform Proofs of Maximality

The core of the proof proceeds by descending induction on $k$6. The $k$7 case is handled explicitly: for degree $k$8, the automorphism group is $k$9, and one constructs an explicit Galois extension realizing the full group. The induction step uses blow-up constructions: blowing up a rational point on the generic del Pezzo surface of degree $(-1)$0 yields a surface of degree $(-1)$1, and the Galois action on exceptional curves of the latter contains the stabilizer of a line in $(-1)$2, which is isomorphic to $(-1)$3.

Surjectivity is deduced by combining this induction step with the monodromy transitivity. Thus, for every $(-1)$4 and any base field $(-1)$5, the image of the Galois action is precisely the full automorphism group of the incidence configuration of exceptional curves.

Arithmetic Application: The Brauer-Manin Obstruction for Cubic Surfaces

A strong arithmetic application concerns the Brauer-Manin obstruction for cubic surfaces over the function field $(-1)$6. Prior work (notably Swinnerton-Dyer, Poonen-Voloch) established that if the Galois action on the 27 lines on a cubic surface is as large as $(-1)$7, the Weyl group of type $(-1)$8, then $(-1)$9, so the Brauer-Manin obstruction vanishes.

The maximality of Galois action implies, via an effective Hilbert irreducibility theorem for function fields, that 100% of cubic surfaces over $G_d = \operatorname{Aut}(\Gamma_d)$0 (ordered by coefficient height) possess this property. Consequently, the Brauer-Manin obstruction does not explain any failure of the Hasse principle for the generic cubic surface in this setting.

Theoretical and Practical Implications

These results fully resolve the maximal Galois image question for exceptional curves of generic del Pezzo surfaces in all characteristics and for all degrees $G_d = \operatorname{Aut}(\Gamma_d)$1. This provides a uniform and geometric proof framework, independent of the base field and characteristic. The monodromy approach via irreducibility of the incidence scheme $G_d = \operatorname{Aut}(\Gamma_d)$2 is robust and connects the maximal Galois image to the geometry of the del Pezzo moduli stack.

In the arithmetic direction, by confirming that the Galois group is maximal for the generic surface, the proportion of cubic surfaces over rational function fields with nontrivial arithmetic Brauer-Manin obstruction drops to density zero.

With these tools and methods now established, further refinements are possible in the arithmetic statistics of rational points and zero-cycles on Fano surfaces, as well as a possible extension to related moduli problems (forms of other Fano varieties, higher-dimensional generalizations). The uniformity of the argument indicates potential for similar questions in broader moduli contexts.

Conclusion

This paper establishes the maximality of the Galois action on exceptional curves for the generic del Pezzo surface, constructing the moduli and incidence spaces necessary for a monodromy-theoretic proof, and deriving arithmetic corollaries for rational points on cubic surfaces. The approach is uniform in the field and characteristic, and the implications cover both algebraic geometry and arithmetic statistics, with the generic maximality leading to the vanishing of the Brauer-Manin obstruction for almost all cubic surfaces over function fields (2604.01497).