Integral points of bounded height on a certain toric variety

TL;DR

This work studies integral points of bounded height on a specific toric variety by transferring the counting problem to a universal torsor and exploiting a Clemens-complex–driven boundary analysis. It constructs a Peyre-like α-constant α_A for maximal boundary faces and identifies obstructions to Zariski density that affect which faces contribute to the leading term. The main result computes an explicit asymptotic N(B) ∼ c B (log B)^2 with c = 4 ∏_p ((1 − 1/p)^2 (1 + 2/p − 1/p^2 − 1/p^3)), and interprets this in terms of a boundary-face obstruction that excludes certain regions from dominating the count. The work also explains a gap in Chambert-Loir–Tschinkel’s toric analysis by showing that only the face corresponding to M contributes, yielding a nonzero leading constant α_M and a reduced log-power exponent, thereby enriching the geometric understanding of integral points on toric varieties.

Abstract

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant $α$ and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.

Figures (4)

• Figure 1: Integral points of height at most $9$ in $\mathfrak{U}(\mathbb{Z})\cap T(\mathbb{Q})$, viewed as a subset of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. The two lines $l_1$ and $l_2$ blown up are in the plane $a_1=0$. ByarXiv:1006.3345, one expects arbitrarily small neighborhoods of the intersection of the two red lines to dominate the counting function---but in fact, any sufficiently small such neighborhood contains no points counted by $N$ at all: as $a_1/a_0$ is an integer for all integral points, all integral points lie on "sheets", and all these sheets have distance $\ge 1$ from the intersection point, which corresponds to the unique maximal dimensional face of the Clemens complex. (The plane $a_1/a_0=0$ defined by both lines contains integral points on $\mathfrak{U}$, which are not shown as they are not in $T(\mathbb{Q})$; in fact, it contains infinitely many points, all of height $1$, hence has to be discarded as an accumulating subvariety to achieve a well-defined counting function.)
• Figure 2: Integral points on $\mathbb{G}_\mathrm{m}$ over $K=\mathbb{Q}(\sqrt{5})$---that is, units of the ring of integers---of small height. Those of norm $1$ are shown in black, those of norm $-1$ in grey. They are embedded into $\mathbb{G}_\mathrm{m}(\mathbb{R})\times \mathbb{G}_\mathrm{m}(\mathbb{R}) = \mathbb{R}^\times \times \mathbb{R}^\times$ along its two places $v_1$ and $v_2$; a chart showing both $0$ and $\infty$ is used, and the complement of the multiplicative group is designated by dashed lines. The maximal faces $(0,0)$ and $(\infty,\infty)$ of $\mathcal{C}^{\mathrm{an}}_\infty(D)$ are obstructed, and no points are near them; the other two maximal faces $(0,\infty)$ and $(\infty, 0)$ are not, and integral points accumulate near them.
• Figure 3: The fan $\Sigma_X$ of $X$, its rays labeled with the corresponding generators of the Cox ring.
• Figure 4: The Clemens complex of $D$.

Theorems & Definitions (59)

• Theorem 1.0.1
• Example 2.1.1
• Lemma 2.1.2
• proof
• Remark 2.1.3
• Remark 2.1.4
• Definition 2.2.1
• Lemma 2.2.3
• proof
• Lemma 2.2.4