gpt 6 astra pro attempt on Math-170

Manins Conjecture on the Asymptotic Density of Rational Points on Fano Varieties

Definitions and mathematical statement

A smooth projective variety $X/K$ is Fano when $-K_X$ is ample. Fix an adelic metric on this line bundle, its multiplicative height $H$, and let $\rho(X)$ denote the rank of the Picard group over $K$. For a nonempty Zariski open $U\subseteq X$, put \[ N(U,B)=\#\{x\in U(K):H(x)\le B\}. \] The catalogue requests $N(U,B)\sim cB(\log B)^{\rho(X)-1}$ on a suitable open subset, with $c>0$ when that asymptotic is intended. Formulation warning: this unrestricted closed-exceptional-set version has counterexamples; a primary discussion is [E1], Introduction and Proposition 1.4. Modern formulations allow a thin exceptional set and additional hypotheses. Those are different statements, and are not silently substituted here. The original target is retained for traceability, not certified as an open universal conjecture.

Formulation and concept references: [S1], [E1].

Short English statement

Does deleting a proper algebraic exceptional subset make rational-point counts on every Fano variety follow the proposed height asymptotic? The unrestricted version printed in the catalogue has counterexamples.

Research attempt

Current frontier.

Editorial status note. The universal assertion actually printed in this section is false. This is not a new discovery: Batyrev--Tschinkel's cubic bundles already contradict it, and Gao's Proposition 1.4 explicitly records the failure of every proper closed exceptional set [E1], [E2]. The modern thin-set conjecture is a different problem. Gao's Theorem 1.8 identifies the geometric exceptional maps for this example; it does not prove the proposed counting asymptotic after removing them [E1]. Below the established surface-counting input is separated from a direct verification that it defeats the exact quantifiers of the supplied target.

Attack route.

There are two possible ways to refute the printed assertion: absence of rational points, or excessive accumulation despite Zariski density. The second is more informative and does not exploit the missing rational-point hypothesis. One might try to sum estimates over infinitely many exceptional fibres, but that creates an unnecessary uniformity problem. The route here uses a single fibre, chosen after the open set but before the height bound. Its logarithmic growth already exceeds the ambient prediction.

Attempt: a fibre-amplification lemma.

Let $q:V\to T$ be a dominant morphism of geometrically integral varieties over a number field, and let $H$ be a height on $V$. Suppose there is a Zariski-dense set $D\subset T(K)$ with the following property: for every $t\in D$ and every nonempty open $W\subset V_t$, there are constants $c_{t,W}>0$ and $B_{t,W}$ such that \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-30-3a-66-69-62-72-65-2d-62-6f-75-6e-64} \#\{z\in W(K):H(z)\le B\}\ge c_{t,W}B^a(\log B)^b \qquad(B\ge B_{t,W}). \end{equation} Assume additionally that every nonempty open $U\subset V$ has image containing a nonempty open subset of $T$. Then for every such $U$, \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-30-3a-61-6d-70-6c-69-66-69-63-61-74-69-6f-6e} N(U,B)\gg_U B^a(\log B)^b. \end{equation} Indeed, choose $t\in D\cap q(U)$ and use the nonempty open $W=U\cap V_t$. Its points are distinct points of $U$, so (30006760:fibre-bound) supplies (30006760:amplification). There is no interchange of an infinite sum and a limit, and no uniformity in $t$ is asserted or needed. The image condition holds, for example, for a smooth surjective family; more generally one can restrict to a nonempty open where the dominant family is smooth.

Consequently, if $b>\rho(V)-1$ and $a=1$, no open $U$ can have the asymptotic required by this section. This lemma is an elementary abstraction of the accumulation mechanism, not a claimed new counting theorem.

Attempt: verifying the ambient geometry.

Put $K={\mathbb{Q}}(\sqrt{-3})$ and \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-30-3a-76-61-72-69-65-74-79} X=\left\{([x],[y])\in{\mathbb{P}}^3_K\times{\mathbb{P}}^3_K: \sum_{i=0}^3x_i y_i^3=0\right\}. \end{equation} The four derivatives with respect to the $x_i$ are $y_i^3$. They cannot vanish simultaneously at a projective point. Thus the hypersurface is smooth. More intrinsically, projection to ${\mathbb{P}}^3_y$ identifies $X$ with the projective bundle of one-dimensional subspaces of the rank-three kernel of the everywhere-surjective map \[ \mathcal O_{{\mathbb{P}}^3_y}^{\oplus4}\longrightarrow \mathcal O_{{\mathbb{P}}^3_y}(3),\qquad (u_i)\longmapsto\sum_i u_i y_i^3. \] This also proves geometric integrality and gives $\operatorname{Pic}(X)\cong{\mathbb{Z}}^2$, with generators the restrictions of $\mathcal O(1,0)$ and $\mathcal O(0,1)$. In particular $\rho(X)=2$ over this fixed field, not just after extending it. Adjunction gives \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-30-3a-61-6e-74-69-63-61-6e-6f-6e-69-63-61-6c} -K_X=\mathcal O_X(3,1). \end{equation} It is ample as the restriction of an ample bundle on the product. Hence $X$ meets the smooth projective Fano hypotheses and the proposed ambient order is $B\log B$.

Choose the product metric with height $H([x],[y])=H_3([x])^3H_3([y])$, where the projective heights use the product-formula normalizations of the formulation and of [E2], Section 2. The same argument works for any continuous adelic metric on this fixed bundle: on the projective variety the ratio of two such heights is bounded above and below by positive constants. This comparison is used only for a fixed variety and a fixed metric, not uniformly in a family.

Attempt: selecting a split cubic fibre.

Over the open set $T=\{x_0x_1x_2x_3\ne0\}\subset{\mathbb{P}}^3_x$, projection $q:X\to{\mathbb{P}}^3_x$ is a smooth cubic-surface family. The set \[ D=\{[a_0^3:a_1^3:a_2^3:a_3^3]:a_i\in K^\times\} \] is Zariski dense in $T$: it is the image of the dense set of $K$-points of the torus under the dominant coordinatewise cubing map. For $t\in D$, the change $z_i=a_i y_i$ identifies $X_t$ with the Fermat cubic surface $\sum z_i^3=0$. All its lines are defined over $K$. The precise established input is Batyrev--Tschinkel, Corollary 2.4: for every nonempty open $W$ on such a surface, its anticanonical height count is $\gg_W B(\log B)^3$ [E2]. Only this lower bound is used; a full Manin asymptotic for an arbitrary smooth cubic surface is not presumed.

For completeness, the reason the surface theorem applies to the ambient height is an equality of line bundles, not an analogy: \[ (-K_X)|_{X_t}=\mathcal O_{X_t}(1)=-K_{X_t}. \] The $x$-height is a fixed positive multiplier on this fibre, and the coordinate change modifies the $y$-height by bounded factors. Thus the surface lower bound is exactly of the form (30006760:fibre-bound) with $a=1$ and $b=3$.

Now let $U\subset X$ be any nonempty Zariski open subset. It meets $q^{-1}(T)$, and the image of that intersection is nonempty open in $T$ because the restricted morphism is smooth. Choose $t\in D$ in this image. The lemma proves \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-30-3a-63-6f-6e-74-72-61-64-69-63-74-69-6f-6e} N(U,B)\ge c_U B(\log B)^3, \qquad \frac{N(U,B)}{B\log B}\longrightarrow+\infty. \end{equation} The second assertion follows from the lower bound, since $c_U>0$ is fixed. It excludes every finite proposed constant $c$, even before the requirement $c>0$ is considered.

Stress test.

An open set may remove many special fibres, but cannot remove all fibres above the dense set $D$: the smooth-image argument checks precisely this point. Conversely, density of $D$ alone would not imply density of an arbitrary selection of one point per fibre; no such unsupported implication is used. The surface estimate applies after removing any proper closed subset of the chosen surface, so removing its 27 lines does not repair the argument for the conjecture. All height-comparison constants are fixed before $B\to\infty$.

The height on a cubic fibre is not the height induced from a power of its anticanonical bundle: (30006760:anticanonical) verifies that the power is one. Confusing this point would change the exponent of $B$. Also, this proof says nothing against a thin exceptional set. The cubing map itself illustrates why a dense collection of exceptional arithmetic fibres need not survive a thin-set deletion. Finally, the construction is known; the research contribution of this entry is its quantifier and normalization audit, together with the isolated amplification lemma.

Outcome.

Outcome: Counterexample candidate.

Equation (30006760:variety), over the single fixed field ${\mathbb{Q}}(\sqrt{-3})$, defeats the literal statement for every nonempty open $U$, by (30006760:contradiction). Here the prescribed candidate label refers to the reconstruction in this notebook: the underlying Batyrev--Tschinkel counterexample is an established result, not a new unverified disproof. No replacement conjecture has been substituted for the target, and no solution of its modern thin-set refinement is claimed.

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Sources

• S1 J. Franke, Y. I. Manin, Y. Tschinkel, Invent. Math. 95$2$:421-435, 1989. https://doi.org/10.1007/BF01393829. Catalogue formulation source.
• E1 Runxuan Gao, The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example, arXiv:2311.04138v3, Introduction and Proposition 1.4. https://arxiv.org/html/2311.04138v3. Added primary source: the failure of the closed-exceptional-set formulation and the distinction from thin-set refinements. Consulted 24 September 2026.
• E2 Victor V. Batyrev and Yuri Tschinkel, "Rational points on some Fano cubic bundles," C. R. Acad. Sci. Paris Sér. I Math. 323 $1996$, 41--46; arXiv:alg-geom/9602013. Section 2, especially Theorem 2.3 and Corollary 2.4, and Theorem 3.1. https://arxiv.org/pdf/alg-geom/9602013. Established counterexample and the split-cubic-surface lower bound; the notebook reconstructs the geometry and quantifiers.