Integral points over number fields: a Clemens complex jigsaw puzzle

TL;DR

This paper analyzes integral points on a singular quartic del Pezzo surface over an arbitrary number field, with respect to a boundary divisor of maximal admissibility. It develops a unified counting framework via twisted universal torsors and o-minimal geometry, connecting the analytic Clemens complex to a single, glue-able polytope that encodes the leading term. The authors establish an explicit asymptotic $N_{\mathcal U,V,H}(B) = c B (\log B)^{2+2q} + O(B (\log B)^{1+2q} \log\log B)$, where $c$ factors into Tamagawa-type measures and archimedean densities, and they prove $\alpha = \frac{1}{q!(q+2)!}$ by solving a jigsaw-puzzle of polytopes arising from the Clemens complex. The result confirms the predicted formula of Wil24/Santens23 for this setting and demonstrates how multiple face contributions cohere into a single leading constant, providing a robust method for counting integral points via universal torsors over general number fields.

Abstract

We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with respect to the largest admissible boundary divisor. The resulting Clemens complex is more complicated than usual, and leads to particularly interesting effective cone constants, associated with exponentially many polytopes whose volumes appear in the expected formula. Like a jigsaw puzzle, these polytopes fit together to one large polytope. The volume of this polytope appears in the asymptotic formula that we obtain using the universal torsor method via o-minimal structures.

Figures (5)

• Figure 1: The jigsaw puzzle for a number field with two archimedean places.
• Figure 2: The $K_v$-analytic Clemens complex of $D$ is a path on five vertices and does not depend on $v$. Each vertex corresponds to an irreducible component of $D$ on a minimal desingularization, labeled as in Section \ref{['sec:torsor']}.
• Figure 3: The extended Dynkin diagram of the weak del Pezzo surface ${\widetilde{S}}$. The $(-1)$-curves are shown in squares and the $(-2)$-curves in circles.
• Figure 4: The cones in the variables $a_{n,1}$ and $a_{n,2}$ described by the four sets of inequalities \ref{['eq:cone-ineqs']}, ordered counterclockwise. The actual polytopes are finite and described by additionally delimiting lines that depend on the other variables but not on the face $B_n$.
• Figure 5: The product $\mathscr{C}(D) = \prod_v \mathscr{C}_v(D)$ if $q=1$.

Theorems & Definitions (21)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof : Deduction of Theorem \ref{['thm:counting-expected-formula']} from Theorem \ref{['thm:main_concrete']}
• Proposition 4.1