Integral points on singular del Pezzo surfaces

TL;DR

The work extends Manin-type conjectures to integral points on a non-toric, singular del Pezzo surface by developing a universal torsor framework for integral points. It classifies admissible boundaries on a low-degree weak del Pezzo surface, constructs an explicit Cox-ring torsor with a defining relation, and reduces counting to lattice-point problems that separate into finite (p-adic) and archimedean contributions. The main results give explicit asymptotics N_i(B) ~ c_{i, ext{fin}} c_{i, ext{∞}} B (\\log B)^{b_i-1}, with b_i determined by Picard ranks and Clemens complex, and lead constants computed via polytope volumes and Tamagawa-type densities. This yields the first comprehensive integral-point analogue of Manin’s conjecture in a non-toric setting, highlighting new phenomena near boundary strata and providing precise constants, including obstructions like alpha_{4,A1}=0.

Abstract

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $\mathbf{A}_1+\mathbf{A}_3$ and prove an analogue of Manin's conjecture for integral points with respect to its singularities and its lines.

Figures (4)

• Figure 1: The Clemens complex of $D_3$ is the disjoint union of those of $D_1$ (left) and $D_2$ (right). It is the Dynkin diagram of the $\mathbf{A}_1$- and $\mathbf{A}_3$-singularities $Q_1,Q_2$.
• Figure 2: Integral points on $\widetilde{\mathcal{U}}_1$ of height $\le 90$. The boundary divisor is the central vertical line. Some horizontal and diagonal lines look accumulating, but in fact are not: They contain $\sim c^\prime B$ points, which is less than the $c B(\log B)^5$ points on $U$; the constants $c^\prime$ can however be up to $2$, while the constant $c$ in our main theorem is numerically $\approx .0003$.
• Figure 3: Integral points on $\widetilde{\mathcal{U}}_2$ of height $\le 60$, in neighborhoods of $D_{A_1}$ (left) and $D_{A_2}$ (right). Most points are close to the three boundary divisors, which are the central horizontal line and two vertical lines here.
• Figure 4: Configuration of the divisors $E_i$ and the faces $A_i$ of the Clemens complexes. The $(-1)$-curves are represented by squares and the $(-2)$-curves by circles.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Remark 3
• Remark 4
• Lemma 5
• proof
• Remark 6
• Lemma 7
• proof
• Proposition 8