Finiteness of $\Lambda_X$: the canonical degree conjecture for curves on smooth projective surfaces

This task concerns the canonical degree conjecture of Ciliberto–Roulleau ('On finiteness of curves with high canonical degree on a surface', arXiv:1406.7478; Geom. Dedicata 183 (2016)): for a smooth complex projective surface $X$, do there exist constants $A$, $B$ depending only on $X$ such that every integral curve $C\subset X$ satisfies $K_X\cdot C\le A(g(C)-1)+B$, where $g(C)$ is the geometric genus? Equivalently, is the invariant $\Lambda_X$ — the supremum of $\limsup K_X\cdot C_n/(g(C_n)-1)$ over sequences of curves with genus tending to infinity — finite for every $X$? After Hao's 2019 proof of the weak bounded negativity conjecture the conjecture holds for smooth curves, but for singular curves it remains, in Ciliberto–Fontanari's 2024 wording, 'widely open' — open already for surfaces of general type, where $\Lambda_X>0$ is known but no upper bound exists outside special classes. No proof or counterexample is known.

Background

By a surface we mean a smooth complex projective surface $X$; by a curve on it an integral (irreducible and reduced) 1-dimensional subscheme $C$. Intersection theory provides a bilinear pairing on divisors: $K_X$ denotes a canonical divisor (the class of a holomorphic 2-form), $K_X\cdot C$ is the intersection number, and $C^2$ is the self-intersection of $C$. The adjunction formula $C^2+K_X\cdot C=2p_a(C)-2$ defines the arithmetic genus $p_a(C)$, while the geometric genus $g(C)$ is the genus of the normalization of $C$; the difference $\delta=p_a-g\ge 0$ measures how singular $C$ is. The canonical degree of $C$ is $k_C=K_X\cdot C$. The Kodaira dimension $\kappa(X)$ counts how independent the pluricanonical forms (sections of the tensor powers $mK_X$ of the canonical class, $m=1,2,\dots$) on $X$ are; $\kappa(X)=2$ means $X$ is of general type, the class to which 'most' surfaces belong (e.g., smooth complete intersections of high degree in projective space).

Two classical problems frame the question. The Bounded Negativity Conjecture (BNC) predicts that on a fixed smooth complex projective surface the self-intersections of reduced curves are bounded below, $C^2\ge -B(X)$; Ciliberto–Roulleau state it for surfaces of general type over $\mathbb{C}$, and it is open in characteristic zero. In positive characteristic it is false: Cheng–van Dobben de Bruyn constructed blow-ups of the projective plane $\mathbb{P}^2$ carrying smooth rational curves of arbitrarily negative self-intersection. Nagata's conjecture (1960) is a famous instance of the same circle of ideas: after blowing up $\mathbb{P}^2$ at $n\ge 10$ points in very general position (that is, avoiding a countable union of proper closed subsets of the configuration space of the points), the only curves with negative self-intersection should be the $(-1)$-curves (the exceptional curves created by the blow-ups, with $C^2=-1$); it remains open.

Ciliberto and Roulleau studied how large the canonical degree of a curve can be relative to its genus. Using Miyaoka's orbibundle refinement of the Miyaoka–Yau–Sakai inequality (a Chern-number inequality restricting the characteristic classes of the tangent bundle, in an orbifold version that accommodates singular fibers), they proved that a negative curve not isomorphic to $\mathbb{P}^1$ on a surface with $\kappa\ge 0$ satisfies

$$k_C\le 3(g-1)+\tfrac{3}{4}a+\tfrac{1}{4}\sqrt{9a^2+24a(g-1)},\qquad a=3c_2(X)-K_X^2,$$

where $c_2$ is the second Chern class (the topological Euler characteristic); hence on a surface of general type there are only finitely many negative curves with $k_C\ge(3+\epsilon)(g-1)$, and their Theorem 4.1 similarly bounds non-negative curves with $\delta/C^2>1/2$. They then posed (their Conjecture 5.1, prompted by a conjecture of Vojta in arithmetic geometry): for every smooth projective surface $X$ there exist constants $A$, $B$ depending only on $X$ such that every integral curve satisfies $K_X\cdot C\le A(g(C)-1)+B$. For smooth curves this is essentially equivalent to BNC, since adjunction gives $k_C=2(g-1)-C^2$, so bounded negativity would imply it with $A=2$. Vojta's conjecture predicts the stronger version with $A=4+\epsilon$ for every $\epsilon>0$, and Autissier–Chambert-Loir–Gasbarri record the expected optimal form $A=2+\epsilon$. In asymptotic language, define

$$\Lambda_X=\sup_{(C_n)}\,\limsup_n\frac{K_X\cdot C_n}{g(C_n)-1},$$

the supremum over all sequences of integral curves with $g(C_n)>1$ and $g(C_n)\to\infty$; Conjecture 5.1 amounts to $\Lambda_X<\infty$ together with control of the curves of genus at most 1. Known cases: $\Lambda_X=0$ when $K_X$ is trivial; Miyaoka's method gives the conjecture with explicit $A$, $B$ when $K_X^2>c_2(X)$, and $k_C\le 3(g-1)$ on ball-quotient surfaces (surfaces whose universal cover is the complex hyperbolic unit ball in $\mathbb{C}^2$); Autissier–Chambert-Loir–Gasbarri prove $k_C\le 4(g-1)$ on surfaces whose universal cover is the bidisk (the product of two complex unit disks), with equality along Shimura curves (totally geodesic curves) on quaternionic Shimura surfaces (certain arithmetic quotients of the bidisk). Ciliberto–Roulleau's Theorem 5.4 shows that on any surface of general type $\Lambda_X\ge K_X\cdot L/(L^2+\gamma-1)>0$ for a very ample divisor $L$ (one whose complete linear system embeds $X$ into projective space), where $\gamma$ is the arithmetic genus of curves in the complete linear system $|L|$ (the family of all divisors linearly equivalent to $L$); when $K=mL$ this gives $\Lambda_X\ge 2m/(m+3)$, which tends to 2 on high-degree complete intersections. So $\Lambda_X$ is genuinely positive on surfaces of general type — the open question is whether it is ever infinite.

Problem Statement

Prove or refute the canonical degree conjecture (Conjecture 5.1 of Ciliberto–Roulleau): for every smooth complex projective surface $X$, do there exist constants $A$, $B$, depending only on $X$, such that every integral curve $C\subset X$ (integral meaning irreducible and reduced) satisfies

$$K_X\cdot C\le A\,(g(C)-1)+B\ ?$$

Here $K_X$ is a canonical divisor on $X$, $K_X\cdot C$ the intersection number, and $g(C)$ the geometric genus of $C$ (the genus of the normalization of $C$). Equivalently, up to the behavior of the curves of geometric genus at most 1 (which the inequality additionally controls): is $\Lambda_X<\infty$ for every $X$, where

$$\Lambda_X=\sup_{(C_n)}\limsup_n\frac{K_X\cdot C_n}{g(C_n)-1}$$

is the supremum over all sequences of integral curves with $g(C_n)>1$ and $g(C_n)\to\infty$? The question is open in this generality and already for surfaces of general type, the case emphasized in the source: there $\Lambda_X>0$ by Theorem 5.4, and no upper bound is known beyond the special classes (trivial canonical bundle, $K_X^2>c_2(X)$, ball- and bidisk-quotient surfaces). After Hao's proof of the weak bounded negativity conjecture, the conjecture is known for smooth curves; the unresolved core is singular curves. A refutation consists of a single smooth complex projective surface carrying an infinite family of integral curves with $K_X\cdot C/(g(C)-1)$ unbounded, or infinitely many curves of geometric genus at most 1 with unbounded canonical degree.

References

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