Non-Thinness of Integral Points on Double Conic Bundles

• The paper establishes explicit geometric and arithmetic criteria ensuring that integral points form a non-thin, Zariski dense set on double conic bundle surfaces.
• The proof employs a double-fibration strategy that integrates topological, intersection-theoretic, and strong approximation methods to verify non-thinness.
• Key examples, including del Pezzo and cubic surfaces, demonstrate that dual conic bundle structures with a 'good' fiber yield the integral Hilbert property.

A double conic bundle surface is a smooth projective algebraic surface equipped with two distinct conic bundle structures—morphisms to $\mathbb{P}^1$ whose generic fibres are geometrically irreducible conics. The paper of integral points on such surfaces probes the distribution of solutions to natural Diophantine equations and connects to fundamental questions in arithmetic geometry. Recent developments have established that, under precise geometric and arithmetic conditions, the set of integral points on open subsets of double conic bundle surfaces is not thin—that is, these points persistently evade containment in any finite union of images of nontrivial covers and proper subvarieties. This property, exceeding Zariski density, is often termed the integral Hilbert Property, and its verification on double conic bundle surfaces unifies and extends classical results on Hilbert's irreducibility, Diophantine conics, and S-unit equations.

1. Definitions and Structure of Double Conic Bundle Surfaces

Let $k$ be a number field. A double conic bundle surface is a smooth, projective surface $X$ together with two distinct flat morphisms $\pi_1, \pi_2: X \to \mathbb{P}^1_k$ such that every geometric fibre of each $\pi_i$ is a (possibly degenerate) plane conic. Formally, for every $t \in \mathbb{P}^1(\bar k)$, $\pi_i^{-1}(t)$ is a plane conic. Typically, an integral model $$\mathcal{X} \to \operatorname{Spec} \mathcal{O}_{k,S}$$ is fixed—with $S$ a finite set of places containing the archimedeans—along with a smooth irreducible divisor $D \subset X$, its integral closure $\mathcal{D} \subset \mathcal{X}$, and the "affine part" $X^\circ := X \setminus D$. Integral points are defined as

$$X^\circ(\mathcal{O}_{k,S}) = (\mathcal{X} \setminus \mathcal{D})(\mathcal{O}_{k,S}).$$

Notable examples include certain del Pezzo surfaces of degrees $d \in \{1,2,4\}$, cubic surfaces with lines or conic bundle structures, and double covers of $\mathbb{P}^2$ branched in a quartic.

2. Main Non-Thinness Theorem: Criteria and Statement

The fundamental result is a set of explicit conditions guaranteeing the non-thinness of integral points on double conic bundle surfaces. This is encapsulated in Theorem 3.5 of (Alessandrì et al., 21 Nov 2025), which gives sufficient conditions under which $X^\circ(\mathcal{O}_{k,S})$ is not thin:

1. Topological Condition: Define $E \subset X$ as the union of all curves that are unions of fibres for both $\pi_1$ and $\pi_2$. The complement $X \setminus (D \cup E)$ must be simply connected over $\bar k$.
2. Boundary Intersection: For each $t \in \mathbb{P}^1(k)$, the intersection numbers satisfy

$$(\pi_1^{-1}(t) \cdot D) = (\pi_2^{-1}(t) \cdot D) = 2.$$

Each conic fibre meets the boundary divisor $D$ transversely in exactly two points.

1. Existence of a "Good" Integral Point: There exists $P_0 \in \mathbb{P}^1(k)$ and $$x \in X^\circ(\mathcal{O}_{k,S}) \cap \pi_1^{-1}(P_0)$$ with the following fiberwise properties:
   • The fiber $C = \pi_1^{-1}(P_0)$ is smooth, and $C^\circ(\mathcal{O}_{k,S})$ is infinite.
   • For $k = \mathbb{Q}$ or an imaginary quadratic field with $|S|=1$, the set $\pi_2(C^\circ(\mathcal{O}_{k,S})) \cap \pi_2(D(k))$ is finite.
   • The map $\pi_2|_D: D \to \mathbb{P}^1$ is unramified at the two intersection points $C \cap D$.

If (1)-(3) are satisfied, then $X^\circ(\mathcal{O}_{k,S})$ is not thin; in particular, it is Zariski dense and satisfies the integral Hilbert Property (Alessandrì et al., 21 Nov 2025).

3. Proof Strategy and Key Techniques

The proof employs the double-fibration method, integrating arguments from topological purity, arithmetic of conics, and Hilbert irreducibility. The pivotal steps are:

• Fiberwise Non-Thinness: For each fibration, define

$$A_i := \{ t \in \mathbb{P}^1(k) : \pi_i^{-1}(t) \cap X^\circ \text{ has infinitely many integral points}\}.$$

Lemma 3.9 establishes $A_i$ to be infinite. For infinitely many $t \in A_1$, the $\pi_2$-fiber over $\pi_2(x)$ is again a conic with infinite integral points.

• Thinness Reduction: One posits that $X^\circ(\mathcal{O}_{k,S})$ is thin—contained in a finite union of images of proper subvarieties or nontrivial covers. By distinguishing ramified and unramified types for the fibration $\pi_1$, thin subsets $T_1, T_2$ of $\mathbb{P}^1(k)$ are constructed such that outside them, only finitely many integral points of smooth $\pi_1$-fibers may lift to covers.
• Fundamental Group Arguments: Utilizing the simply-connectedness of $X \setminus (D \cup E)$, any unramified cover is forced to be trivial. Combined with Siegel's Theorem, only finitely many points can lift to a cover with substantial ramification.
• Strong Approximation on Conics: Proposition 2.11 provides for $v$-adic dispersal of integral points—integral points can be made to approach conic-boundary points arbitrarily closely.
• Double Fibration Contradiction: By alternating between $\pi_1$ and $\pi_2$, the covering assumption is contradicted, yielding the non-thinness conclusion.

4. Verification Criteria and Concrete Implementability

Application of the non-thinness theorem requires explicit criteria:

• The intersection $C \cap D$ for a chosen smooth conic fiber must be of degree 2; both points should be defined over a real quadratic subfield of some $k_v$.
• The discriminant of the points at infinity in affine coordinates must be positive in $\mathbb{R}$.
• The boundary map $\pi_2|_D$ must remain unramified at these intersection points—this follows from non-coincidence of tangent directions, monitorable via the determinant of the quadratic forms.
• Simply-connectedness typically reduces to contracting $E$ and appealing to classifications of ruled or del Pezzo surfaces, particularly complements of smooth anticanonical divisors.

These conditions are checkable by concrete computation for explicit surfaces. Detailed examples and further algorithms are given in (Alessandrì et al., 21 Nov 2025).

5. Corollaries and Example Classes

This framework encompasses several prominent Diophantine scenarios:

• Degree 2 Del Pezzo Surfaces: If $X$ is a del Pezzo surface of degree $2$ with dual conic bundle structures, and one smooth fiber $C$ meets $D \in |-K_X|$ in a real quadratic point of unramified type containing an integral point, then non-thinness of $X^\circ(\mathcal{O}_{k,S})$ follows (Alessandrì et al., 21 Nov 2025).
• Cubic Surfaces: For $X \subset \mathbb{P}^3$ a smooth cubic surface with two coplanar lines and $D$ a smooth hyperplane section, non-thinness of integral points on $X \setminus D$ is established, recovering and extending results such as the Fermat cubic $x^3 + y^3 + z^3 = 1$ (Alessandrì et al., 21 Nov 2025).
• Fermat "Near Miss" Quartics: The quartic surfaces $x^4 + y^4 - w^2 = n$ compactify to degree $2$ del Pezzo surfaces admitting conic bundle structures when $n$ or $2n$ or $-2n$ is a square or $-4n$ is a fourth power. Whenever a single integral solution on such a fiber as above is found, non-thinness is immediately deduced (Alessandrì et al., 21 Nov 2025).

A summary table of canonical cases:

Surface Type	Conic Bundles Present	Hypotheses Needed	Non-Thinness Holds?
Degree 2 del Pezzo	Dual	One "good" fiber	Yes (explicit criteria)
Cubic surface w/ 2 lines	Yes	Same	Yes
Fermat quartic near misses	Yes	Same	Yes

6. Relation to Broader Diophantine Geometry

Non-thinness for integral points on double conic bundle surfaces links to the broader question of the abundance and distribution of rational or integral solutions on higher-dimensional varieties. Non-thinness is strictly stronger than Zariski density and is characterized via the integral Hilbert property (IHP). The double-fibration method—initiated by Corvaja–Zannier for the quartic Fermat surface (Streeter, 2018), and generalized by Coccia, Streeter, Demeio, and others—demonstrates that conic bundle structures together with topological and intersection-theoretic constraints can ensure non-thinness in substantial generality (Alessandrì et al., 21 Nov 2025, Coccia, 2023, Streeter, 2018).

Notably, while the IHP requires simply-connected complements, the non-thinness property for rational or integral points may still hold on non-simply-connected log del Pezzo surfaces provided the requisite fibration structures and sufficiently high $S$-rank, as established through the argumentation in (Streeter, 2018, Coccia, 2023). This suggests a robust and flexible toolkit for constructing new infinite, non-thin families of integral points on a broad class of arithmetic surfaces.

7. Impact, Extensions, and Current Frontiers

The theory applies directly to del Pezzo surfaces of degrees $d=1,2,4$, many cubic surfaces, and certain higher-dimensional analogues, provided dual conic bundle structures exist and the explicit geometric hypotheses are met. The non-thinness of integral points has direct implications for the effective approximation of solutions to Diophantine equations, the integral version of Hilbert's irreducibility theorem, and potential density statements for arithmetic surfaces.

Current research seeks to weaken the necessary hypotheses (e.g., requiring only the existence of a single "good" fiber with infinitely many integral points), extend the toolkit to settings with singular boundaries or more general fibration structures, and apply the double fibration paradigm to higher-dimensional uniruled varieties. These developments underscore the centrality of conic bundle geometry in the arithmetic theory of surfaces and solidify the double fibration method as a principal approach to non-thinness phenomena (Alessandrì et al., 21 Nov 2025, Coccia, 2023, Streeter, 2018).