36 problems

Let $S$ be a smooth surface, and let $C$ be a reduced and irreducible curve on $S$. Bounded Negativity Conjecture. There is a constant depending only on $S$ such that … is bounded…

Weak Bounded Negativity Conjecture. There is a constant $b_{X,g(C)}$, depending only on $X$ and $g(C)$, such that

Weighted Bounded Negativity Conjecture. There exists a nonnegative integer $b_w\in\mathbb Z$ such that

Let $X$ be a smooth projective surface over an algebraically closed field of characteristic zero. Let $C$ be a reduced and irreducible curve on $X$, and let $D$ be a big and nef di…

Deformation-invariance conjecture. The Bounded Negativity Conjecture is true for $X$ if and only if it is true for $Y$.

Degree-uniform bounded negativity conjecture. There exists a constant $c_d$, depending only on $d$, such that every smooth surface of degree $d$ in $\mathbb{P}^3$ has negativity bo…

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

Stronger bounded negativity conjecture. If $C$ is an integral curve in $\operatorname{Bl}_{n}\mathbb{P}^2$, then

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

Very-general blow-up conjecture. For every integral (reduced and irreducible) curve $C$ on $Y$,

Let $Z$ be a smooth complex projective surface, and let $g$ be an integer. Weak bounded negativity conjecture. There exists a non-negative integer $b(Z,g)$, depending only on $Z$ a…

Let $Z$ be a smooth projective surface over the complex numbers. Bounded negativity conjecture. There should exist a non-negative integer $b(Z)$, depending only on $Z$, such that e…

Let $C\subset\mathbb{P}^2$ be a reduced plane curve whose singularities are all nodes or triple points, and let $n_2$ denote the number of nodes. Node-existence conjecture. If $C$…

Reduced sigma-four conjecture. For any reduced singular complex curve $C\subset\mathbb{P}^2$, one has

A reduced plane curve has multiplicity three singularities when all its singularities have multiplicity $3$. Degree-nine conjecture. There is no reduced plane curve in any characte…

Let $K$ be a fixed algebraically closed ground field. A multiplicity sequence is a positive integer sequence $(d,m_1,\ldots,m_r)$ arising from a reduced plane curve, with the $m_i$…

Let $K$ be a fixed algebraically closed ground field. A multiplicity sequence is a positive integer sequence $(d,m_1,\ldots,m_r)$ arising from a reduced plane curve of degree $d$,…

Bounding cohomology conjecture. There exists a constant $c_X>0$ such that

Bounded negativity conjecture. There exists an integer $b(X)\geq 0$ such that

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 in…

A smooth projective surface $X$ has bounded negativity if there is an integer $b(X)$ such that every reduced irreducible curve $C\subset X$ satisfies $C^2\geq -b(X)$. Bounded Negat…

Bounded Negativity Conjecture. Every smooth complex projective surface has bounded negativity.

By a negative curve on a smooth projective surface we mean a reduced, irreducible curve $C$ with $C^2<0$. The bounded negativity conjecture. For each smooth complex projective surf…

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

Let $X$ be a smooth complex projective surface, let $b_X$ be a real number, and let $C$ be a reduced curve on $X$. Bounded negativity conjecture. There exists a bound $b_X$ such th…