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The weighted bounded negativity conjecture for smooth projective surfaces

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Let XX be a smooth projective surface over the complex numbers. Let C⊂XC\subset X be an integral curve, and let HH be a big and nef line bundle such that C⋅H>0C\cdot H>0. Write C2C^2 for the self-intersection of CC and C⋅HC\cdot H for the intersection number.

Weighted bounded negativity conjecture. For every smooth projective surface XX over the complex numbers, there exists a non-negative integer bω(X)∈Zb_{\omega}(X)\in\mathbb{Z} such that

C2≥−bω(X)(C⋅H)2C^2\geq -b_{\omega}(X)(C\cdot H)^2

for all integral curves C⊂XC\subset X and all big and nef line bundles HH for which C⋅H>0C\cdot H>0.

The source introduces this as a weighted version of bounded negativity and mentions recent work on it, but gives no resolution status for the stated conjecture.

Equivalent formulations 1Other wordings

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 weighted bounded negativity conjecture for smooth projective surfaces

    Let XX be a smooth projective surface over an algebraically closed field of characteristic zero. Let CC be an integral curve on XX, let DD be a big and nef divisor on XX, and assume D⋅C>0D\cdot C>0. The quantity Bw(X)B_w(X) denotes a non-negative integer depending only on XX.

    Weighted bounded negativity conjecture. There exists a non-negative integer Bw(X)B_w(X) such that

    C2(D⋅C)2≥−Bw(X)\frac{C^2}{(D\cdot C)^2}\geq -B_w(X)

    for all integral curves CC on XX and all big and nef divisors DD on XX satisfying D⋅C>0D\cdot C>0.

    This is a weighted, asymptotic variant of bounded negativity, seeking a uniform lower bound as both the curve and the big and nef divisor vary. The supplied text gives no resolution, so the conjecture remains open.

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

References

Primary source

Snehajit Misra and Nabanita Ray, “On Weak bounded negativity conjecture”, arXiv:2408.15187 (2024).

algebraic-surfacesbounded-negativityintersection-theory
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