Campana's Conjecture Characterizing Special Varieties by Potential Density of Rational Points

Campana's Conjecture Characterizing Special Varieties by Potential Density of Rational Points

Definitions and mathematical statement

For a rank-one coherent subsheaf $\mathcal F\subseteq\Omega^p_{X_{\bar K}}$, its Kodaira dimension measures the growth of sections of its reflexive tensor powers, with value $-\infty$ if none occur. Campana specialness requires $\kappa(\mathcal F)<p$ for all $1\le p\le\dim X$ and all such $\mathcal F$. Potential density means density after some finite field extension. The conjecture is the biconditional \[ X_{\bar K}\text{ special}\quad\Longleftrightarrow\quad \exists L/K\text{ finite}:\overline{X(L)}^{\rm Zar}=X. \] Both directions are required, for every smooth projective geometrically integral number-field variety. Weak specialness and integral points on open varieties are different notions.

Formulation and concept references: [S1].

Short English statement

Are precisely Campana's special projective varieties those whose rational points become dense after a finite field extension?

Research attempt

Process and the quantifier being attacked.

This GPT-6 Astra Ultra attempt, dated 27 September 2026, checks the differential-form definition against curves and then proves the stated biconditional for products of smooth projective curves. It uses Faltings's theorem and the elementary prime-to-residue-characteristic torsion-reduction theorem as identified inputs. The deductions are standard partial results, not a claim of novelty or a solution in arbitrary dimension.

Potential density asks for one number field $L$ supporting a dense set. It is much stronger than density of $X(\overline K)$, where the fields of definition of successive points may grow without bound. Our positive constructions explicitly choose finitely many algebraic data and then take their finite compositum. The first remaining gap is to construct such a fixed field and a dense set for an arbitrary special variety; even the positive direction in the full statement is not proved here.

Literature and conventions.

Abramovich states the biconditional as [S1, Conjecture 2.2.19]. Campana's foundational paper [E1] supplies the geometric specialness framework. These are projective, characteristic-zero assertions. The argument below works directly with the rank-one subsheaves in the recorded definition; no replacement by weak specialness is made. The one deep arithmetic input is the finiteness of $C(L)$ for a smooth projective curve of genus at least two over any number field $L$, by Faltings [E2]. The local torsion fact used in genus one is recorded in [E3, Chapter II, Section 4], including its extension to finite extensions of local fields.

Specialness for one curve.

Let $C/\overline K$ be a smooth projective connected curve of genus $g$. A coherent rank-one subsheaf of the line bundle $\Omega_C^1$ is torsion-free, hence a line bundle on a smooth curve. It has the form \[ \mathcal F=\Omega_C^1(-D),\qquad D\text{ effective}. \tag{86.1} \] If $g=0$, its degree is negative, so every positive tensor power has no nonzero section. Thus $\kappa(\mathcal F)=-\infty$. If $g=1$, the differential bundle is trivial and $\deg\mathcal F\le0$. A negative-degree line bundle has no sections, while a degree-zero line bundle has at most one section: a nonzero section has an effective zero divisor of degree zero and therefore trivializes the bundle. Consequently $h^0(\mathcal F^{\otimes m})\le1$ for every $m$, giving $\kappa(\mathcal F)\le0<1$.

For $g\ge2$, choose $\mathcal F=\Omega_C^1$ itself. Its degree is $2g-2>0$; Riemann--Roch gives linear growth of $h^0(C,(\Omega_C^1)^{\otimes m})$ for large $m$, so $\kappa(\mathcal F)=1$. Therefore \[ C\text{ is special}\quad\Longleftrightarrow\quad g(C)\le1. \tag{86.2} \] The only value of $p$ on a curve is $p=1$, so no higher-form condition has been omitted.

Genus zero and the need for a field extension.

A geometrically integral curve over a number field has a closed point of finite degree: on a nonempty affine open, any maximal ideal has a residue field finite over the ground field. Pass to that field. A smooth projective genus-zero curve with a rational point is isomorphic to $\mathbb P^1$, by the degree-one linear system and Riemann--Roch. Its rational points are infinite over a number field, and an infinite subset of an irreducible curve is Zariski dense. This proves potential density without requiring a point over the original field.

A fixed-extension genus-one construction.

For a genus-one curve, first make a closed point rational by a finite extension $K'/K$ and use it as origin, obtaining an elliptic curve $E/K'$. We prove that some further extension of degree at most two over $K'$ contains a point of infinite order. This does not assert that $E(K')$ already has positive rank.

Choose two places of good reduction of $E$ with distinct residue characteristics $p$ and $q$, and residue fields of sizes $a$ and $b$. Set \[ A=\#\widetilde E_p(\mathbb F_{a^2}),\qquad B=\#\widetilde E_q(\mathbb F_{b^2}),\qquad M=AB. \] If $L/K'$ has degree at most two, the residue degrees at places above these chosen places are one or two. The corresponding finite-field groups embed in those over $\mathbb F_{a^2}$ and $\mathbb F_{b^2}$, respectively. Good reduction persists under extension.

Let $P\in E(L)$ be torsion of order $N=p^e u$ with $(u,p)=1$. Reduction at the first place is injective on the cyclic subgroup generated by $p^eP$, by the prime-to-$p$ torsion theorem. Therefore $u\mid A$. Reduction at the second place is injective on the subgroup generated by $uP$, whose order is $p^e$ and is prime to $q$. Therefore $p^e\mid B$. It follows that $N\mid M$. In particular every torsion point defined over any extension of degree at most two lies in the single finite group $E[M](\overline K)$.

Write a short Weierstrass equation $y^2=x^3+\alpha x+\beta$ over $K'$. For each $t\in K'$ choose a square root of $t^3+\alpha t+\beta$ in $\overline K$. The resulting point $P_t$ has degree at most two over $K'$, and distinct $t$ give distinct points. There are infinitely many such points and only finitely many in $E[M]$, so one $P_t$ is not torsion. Choose that point and set $L=K'(P_t)$. All multiples $nP_t$ now belong to this one field $L$ and are distinct. They are dense in the curve. $\square$

The finite group $E[M]$ is finite because multiplication by $M$ on an elliptic curve in characteristic zero is a finite isogeny. No uniform torsion theorem across all elliptic curves is needed: the constants above may depend on this particular curve and field.

The other direction for curves.

When $g\ge2$, every finite extension $L/K$ is still a number field and the genus is unchanged. Faltings's theorem makes $C(L)$ finite for every such $L$. A finite subset cannot be dense in a curve. Combined with $86.2$, this proves the full recorded equivalence in dimension one. It does not give an effective algorithm listing those finite sets, which is not needed for this deduction.

Products: checking every differential degree.

Let \[ X=C_1\times\cdots\times C_r \] over a number field, with all $C_i$ smooth projective geometrically integral. Suppose first that every genus is zero or one. Over $\overline K$, a genus-zero factor is $\mathbb P^1$ and a genus-one factor has trivial cotangent bundle. For $1\le p\le r$, \[ \Omega_X^p\cong\bigoplus_{|I|=p}\mathcal L_I, \qquad \mathcal L_I=\bigotimes_{i\in I}\pi_i^*\Omega^1_{C_i}. \tag{86.3} \] Take any rank-one coherent $\mathcal F\subset\Omega_X^p$. Its reflexive hull $\mathcal M=\mathcal F^{**}$ is a line bundle, since $X$ is smooth. Away from a set of codimension at least two the inclusion has a nonzero component into some summand $\mathcal L_I$. The resulting morphism $\mathcal M\to\mathcal L_I$ extends across that set: sections of a line bundle on a normal variety extend over a codimension-two omission. It is generically nonzero, hence injective, as are its tensor powers.

If $I$ contains a genus-zero index, $\mathcal L_I^{\otimes m}$ has no global sections for $m>0$, by restricting along that $\mathbb P^1$ factor. Otherwise it is trivial and has exactly one independent global section. The injection therefore gives \[ h^0(X,\mathcal M^{\otimes m})\le1\quad(m\ge1). \] Reflexive tensor powers of $\mathcal F$ identify with these powers of $\mathcal M$, so $\kappa(\mathcal F)\le0<p$. This verifies all the specialness conditions, including unsaturated subsheaves and $p>1$.

If some $C_j$ has genus at least two, the summand $\pi_j^*\Omega^1_{C_j}\subset\Omega_X^1$ has at least linearly many sections in its tensor powers, by pulling back sections from $C_j$. Thus its Kodaira dimension is at least one and $X$ is not special. We have proved \[ X\text{ special}\quad\Longleftrightarrow\quad g(C_i)\le1\text{ for every }i. \tag{86.4} \]

Products: checking density over one field.

In the positive case choose a finite extension for each factor by the genus-zero or genus-one construction, and take their finite compositum $L$. Each $C_i(L)$ contains an infinite, hence dense, subset $D_i$. The Cartesian product of dense subsets is dense in a product of varieties. One can verify this without confusing product topology with Zariski topology: on affine opens, a regular function vanishing on $D_1\times D_2$ vanishes on each entire first-factor fiber over $D_2$, then on every second-factor fiber, and hence everywhere. Induct on the number of factors. Therefore $X(L)$ is dense.

If $g(C_j)\ge2$, then for every finite $L/K$, the projection of $X(L)$ to $C_j$ is contained in the finite set $C_j(L)$. Consequently $X(L)$ lies in finitely many proper closed fibers of that projection and is not dense. This proves the complete biconditional for products of curves, using Faltings precisely for the negative direction.

Stress tests and the failed generalization.

The construction does not take an infinite compositum of the quadratic fields defining all $P_t$: only one non-torsion point is chosen for each of finitely many genus-one factors. A point chosen merely because it is algebraic could be torsion, so the torsion bound is an essential step. In the differential argument a nonzero projection to one summand suffices; an arbitrary rank-one subsheaf need not itself split as an exterior product. Both residue characteristics are needed to control the entire torsion order, since reduction at one prime only controls the prime-to-that-prime part.

A tempting extension is to replace a product by a fibration whose base and fibers have potentially dense points. The proof no longer applies. The finite extension that supplies points on a fiber may depend on the base point, and infinitely many such extensions need not have a finite compositum. Moreover the differential decomposition $86.3$ is specific to a product; multiple fibers and the induced orbifold base are precisely data relevant to Campana specialness. No claim about arbitrary fibrations follows from the product argument.

Verification and outcome.

The curve genus cases, the two-prime divisibility argument, the reflexive-hull extension, and both directions for products were checked separately. The arithmetic finiteness and good-reduction statements remain explicitly cited inputs. No rational-point search is offered as evidence of density, and no new result beyond these standard deductions is claimed.

UNRESOLVED. The recorded equivalence is proved here for products of smooth projective curves. For a general special variety, the first missing step is a construction of a dense set over one fixed finite extension. For a general nonspecial variety, this attempt also lacks a theorem forcing non-density after every finite extension; the projection-to-a-high-genus-curve argument covers only the stated product case.

Sources

• S1 Abramovich, Birational geometry for number theorists. https://arxiv.org/pdf/math/0701105. Formulation source.
• E1 F. Campana, Orbifolds, special varieties and classification theory, Annales de l'Institut Fourier 54 $2004$, 499--630. https://numdam.org/item/AIF_2004__54_3_499_0/.
• E2 G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae 73 $1983$, 349--366. https://math.uchicago.edu/~drinfeld/Deligne's_conjecture_Manin_conf/Faltings_argument/Faltings.pdf. Finiteness theorem for number-field points on curves of genus at least two.
• E3 J. S. Milne, Elliptic Curves, second edition, Chapter II, Section 4, especially Corollary 4.2 and the extension to complete discretely valued fields following it. https://www.jmilne.org/math/Books/EC2.pdf. Prime-to-residue-characteristic torsion under good reduction; references consulted 27 September 2026.