The Bounded Negativity Conjecture for smooth projective surfaces

Equivalent formulations 10

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth projective surface over $\a0\mathbb{C}$. For a reduced curve $C$ on $X$, write $C^2$ for its self-intersection number. Bounded negativity conjecture. There exists an integer $b(X)$ such that for every reduced curve $C$ on $X$, we have

   $$C^2\geq -b(X).$$

   This conjecture asserts that the self-intersection numbers of reduced curves on any fixed smooth projective surface are bounded below. Its general status is not specified in the supplied source context.

   source: Xavier Roulleau, “Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations”, arXiv:1411.6996 (2015).

2. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth complex projective surface. It has the bounded negativity property if there exists an integer $b(X)$ such that every reduced curve $C\subset X$ satisfies

   $$C^{2}\geqslant -b(X).$$

   Bounded negativity conjecture. Every smooth complex projective surface has the bounded negativity property.

   This conjecture asks whether the self-intersection numbers of reduced curves on any fixed smooth projective surface are bounded below. It is a central open problem concerning negative curves and bounded negativity on algebraic surfaces.

   source: Piotr Pokora and Halszka Tutaj-Gasińska, “Harbourne constants and conic configurations on the projective plane”, arXiv:1506.06097 (2015).

3. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth complex projective surface, and let $C\subset X$ be a reduced curve.

   Bounded Negativity Conjecture. There exists a positive integer $b(X)\in\mathbb{Z}$ such that, for every reduced curve $C\subset X$,

   $$C^{2}\geq -b(X).$$

   This conjecture asks whether the self-intersection numbers of reduced curves on a fixed smooth projective surface are bounded below. It is known for various classes of surfaces, but remains open in general.

   source: Piotr Pokora and Joaquim Roé, “The 21 reducible polars of Klein's quartic”, arXiv:1803.06411 (2018).

4. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth projective surface. A negative curve is a curve $C$ on $X$ with $C^2<0$. Bounded negativity conjecture. There is a nonnegative integer $b_X\geq 0$ such that for every negative curve $C$,

   $$C^2\geq -b_X.$$

   This is a longstanding and widely open problem in the theory of algebraic surfaces; it asks whether the self-intersections of negative curves on any fixed smooth projective surface are uniformly bounded below.

   source: JongHae Keum and Kyoung-Seog Lee, “Examples of Mori dream surfaces of general type with p_g=0”, arXiv:1804.08382 (2018).

5. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth complex projective surface. For every reduced curve $C\subset X$, consider its self-intersection number $C^2$. Bounded negativity conjecture. There exists an integer $b(X)$ such that

   $$C^2\geqslant -b(X)$$

   for every reduced curve $C\subset X$. This is a folklore conjecture concerning uniform lower bounds for self-intersection numbers of reduced curves on a fixed surface; its status is not established by the supplied source.

   source: Piotr Pokora and Tomasz Szemberg, “Conic-line arrangements in the complex projective plane”, arXiv:2002.01760 (2022).

6. The Bounded Negativity Conjecture for smooth projective surfaces

   Let $X$ be any smooth complex projective surface, and let $C\subset X$ be a reduced curve. Bounded Negativity Conjecture. There is a bound $B_X$ such that

   $$C^2\geq B_X.$$

   This folklore conjecture is traced back at least to F. Enriques and remains open in characteristic zero, although bounded negativity is known to fail in positive characteristic.

   source: Alexandru Dimca, Brian Harbourne and Gabriel Sticlaru, “On the Bounded Negativity Conjecture and singular plane curves”, arXiv:2101.07187 (2021).

7. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth projective surface over $\mathbb{C}$, and let $C\subseteq X$ be a curve. The bounded negativity conjecture. There exists an integer $b(X)\ge 0$ such that

   $$C^2\ge -b(X)$$

   for every curve $C\subseteq X$. This is a fundamental open problem in the theory of projective surfaces; it asks whether the self-intersections of curves on each fixed smooth projective surface are uniformly bounded below.

   source: Sichen Li, “Smooth projective surfaces with bounded cohomology property”, arXiv:2306.07830 (2026).

8. The bounded negativity conjecture for smooth projective surfaces

   Let $X$ be a smooth projective surface over the complex numbers, and let $C\subseteq X$ be a curve. Bounded negativity conjecture. There exists an integer $b(X)\ge 0$ such that

   $$C^2\ge -b(X)$$

   for every curve $C\subseteq X$. This is a central open problem in the theory of projective surfaces; it asks for a uniform lower bound on the self-intersection numbers of curves on each fixed smooth projective surface.

   source: Sichen Li, “Integral Zariski decompositions on smooth projective surfaces”, arXiv:2306.11292 (2024).

9. The bounded negativity conjecture for smooth projective surfaces

   Work over the field $\mathbb{C}$ of complex numbers. Let $X$ be a smooth projective surface, and let $C\subseteq X$ be a curve. The bounded negativity conjecture asserts that there exists an integer $b(X)\ge 0$ such that

   $$C^2\ge -b(X)$$

   for every curve $C\subseteq X$. This is a central problem concerning the self-intersection numbers of curves on projective surfaces. The supplied text does not state whether the conjecture has been resolved in general.

   source: Sichen Li, “Bounding cohomology on a smooth projective surface”, arXiv:1805.10741 (2020).

10. The bounded negativity conjecture for smooth projective surfaces

    Let $X$ be a smooth projective surface over an algebraically closed field of characteristic zero, and let $C$ be a reduced and irreducible curve on $X$. Bounded negativity conjecture. There is a non-negative integer $B(X)$, depending only on $X$, such that

    $$C^2\geq -B(X)$$

    for every reduced and irreducible curve $C$ on $X$. The conjecture asks for a uniform lower bound on the self-intersection numbers of curves on a fixed surface and is a central open problem concerning negative curves on algebraic surfaces.

    source: Carlos Galindo, Francisco Monserrat and Carlos-Jesús Moreno-Ávila, “On weighted bounded negativity for rational surfaces”, arXiv:2408.05466 (2024).