The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$

We undertake a study of topological properties of the real Mordell-Weil group $\operatorname{MW}{\mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the “subordinate” del Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy types of real lines on $Y{\mathbb R}$ and an explicit presentation of $\operatorname{MW}{\mathbb R}$ in the mapping class group $\operatorname{Mod}(X{\mathbb R})$. Combining these results we establish an explicit formula for the action of $\operatorname{MW}{\mathbb R}$ in $H_1(X{\mathbb R})$.

💡 Research Summary

The paper investigates the topological behavior of real sections (called “real lines”) on real rational elliptic surfaces X and on the associated real del Pezzo surfaces Y of degree one, together with the action of the real Mordell–Weil group MWℝ. The authors work under the standing hypothesis (Assumption A) that X is a smooth, relatively minimal rational elliptic surface with all fibers reduced and irreducible, and that X possesses at least one real section. Under this hypothesis a real line on X coincides with a real section of the elliptic fibration f : X → ℙ¹.

The first major component of the work is a detailed classification of real lines on the subordinate del Pezzo surface Y. Y can be described as a double cover π : Y → Q of a real quadratic cone Q⊂ℙ³, branched over the vertex and a real sextic curve C obtained as the transversal intersection of Q with a real cubic surface. The real topology of C is encoded by a combinatorial code ⟨p | q⟩ (p positive ovals, q negative ovals) together with a possible J‑component ⟨|||⟩ that surrounds the cone vertex. For each such code the authors introduce a numerical invariant τ, the number of ovals that meet the tritangent with an odd total multiplicity of tangency points. According to τ they define five types of positive tritangents: T₁, T₂, T₃ (τ = 1, 2, 3), and two types for τ = 0, namely T₀ and T₀*. The distinction between T₀ and T₀* depends on whether the two tangency points on the same oval are separated by the generatrix of the cone through the J‑tangency point. This fine subdivision is crucial for the later isotopy classification.

To count and locate these tritangents the authors develop a new “oval‑bridge decomposition” based on a lattice Λ_Y = ker(1+conj*)∩K_Y^⊥ ⊂ H²(Y,ℤ). This lattice encodes the action of complex conjugation on H² and allows a mod‑2 arithmetic description of the interaction between ovals and bridges. Using this tool they obtain explicit formulas (Table 1) for the number of tritangents of each type for every possible real deformation class of C. For example, when C has code ⟨4 | 0⟩ the numbers are T₁ = 32, T₂ = 48, T₃ = 32, and both T₀ and T₀* equal 4. The authors also give a complete isotopy classification of the tritangents, which translates directly into a classification of real lines on Y.

The second major component concerns the real Mordell–Weil group MWℝ of X. By blowing up a real line L⊂X one obtains Y, and conversely blowing down the base point of the anticanonical pencil on Y recovers X. This establishes a bijection between pairs (X, L) and del Pezzo surfaces Y. Under this correspondence the lattice Λ_X = ker(1+conj*)∩⟨K_X, L⟩^⊥ coincides with Λ_Y. The structure of Λ_X is completely determined by the topology of the real part Xℝ; Table 2 lists the possible root lattices (E₈, E₇, D₆, D₄⊕A₁, …) together with the corresponding real components (e.g. K # p T², K ⊥⊥ S², etc.).

The authors introduce a topological analogue of MWℝ, denoted Mod_s(Xℝ), consisting of isotopy classes of diffeomorphisms of Xℝ that preserve each real fiber and act by translation on the fibers. There is a natural homomorphism  Φ : MWℝ → Mod_s(Xℝ). The image and kernel of Φ depend only on the real topology of Xℝ, and Table 4 gives a complete description. For instance, when Xℝ ≅ K # 4 T² the image is Z₈ ⊕ (Z/2Z)⁶ and the kernel is Z⁰ (i.e. trivial), whereas for Xℝ ≅ K ⊥⊥ 4 S² the image is Z₈ and the kernel is Z⁴.

A further set of results concerns the realization of homology classes by real lines. Let N be the number of distinct classes in H₁(Xℝ,ℤ) represented by real lines. Theorem 1.3.1 (Table 3) shows that N is infinite precisely when Xℝ contains a component K # p T² with p ≥ 1; otherwise N is finite and explicitly listed. Theorem 1.3.2 proves that under the same condition Xℝ also contains infinitely many vanishing cycles (real circles that appear as vanishing cycles in real nodal degenerations).

Finally, the authors give an explicit matrix description of the action of MWℝ on H₁(Xℝ). When Xℝ ≅ K # p T² ⊥⊥ q S² they choose a basis consisting of: