18 problems

Let $X$ be a smooth complex projective surface, and let $C$ be a reduced curve on $X$. Bounded Negativity Conjecture. For each such $X$, there exists a number $B_X\geq0$, depending…

Let $X$ be a smooth projective surface, and let $C$ be a curve on $X$. Write $h^i(\mathcal O_X(C))=\dim H^i(X,\mathcal O_X(C))$. The bounded cohomology conjecture asserts that ther…

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

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

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…

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

Effective bounded negativity conjecture. One has

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

Bounded negativity conjecture for generic blow-ups. Then $C^2\ge -1$ for every prime divisor $C$, with equality if and only if $C$ is a smooth rational curve with $K_X\cdot C=-1$.