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The weighted bounded negativity conjecture for smooth projective surfaces

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 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 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).

Sources & referencesView supporting material

Primary source

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

algebraic-surfacesbounded-negativityintersection-theory

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