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The bounded negativity conjecture for smooth complex projective surfaces

Let XX be a smooth projective surface. We say that XX has bounded negativity if there exists an integer b(X)b(X) such that every reduced curve C⊂XC\subset X satisfies

C2⩾−b(X).C^{2}\geqslant-b(X).

Bounded negativity conjecture. An arbitrary smooth complex projective surface has bounded negativity.

This conjecture is a central problem in the theory of algebraic surfaces and concerns uniform lower bounds for self-intersection numbers of reduced curves. It is refuted in positive characteristic by counterexamples arising from the Frobenius morphism; the stated complex case is not resolved by that observation.

Equivalent formulations 1

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 complex projective surfaces

    Let XX be a smooth complex projective surface. Say that XX has bounded negativity if there exists an integer b(X)b(X) such that every reduced curve C⊂XC\subset X satisfies

    C2⩾−b(X).C^2\geqslant-b(X).

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

    The conjecture concerns whether self-intersections of reduced curves on a fixed smooth projective surface are uniformly bounded below. It is a central open problem in the theory of projective surfaces and is related to bounded negativity on birational models.

    source: Piotr Pokora, “Harbourne constants and arrangements of lines on smooth hypersurfaces in P^3_C”, arXiv:1505.03822 (2015).

Sources & referencesView supporting material

Primary source

Adam Czapliński and Piotr Pokora, “Curve configurations in the projective plane and their characteristic numbers”, arXiv:1511.03879 (2017).

algebraic-surfacesbounded-negativity

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