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…

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

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…

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 $K_X\cdot C=-1$.