76 problems

Classification conjecture. The only numerically elementary $(-1)$-curves satisfying $\delta<2\mu_1$ are the lines through two points.

Specialness criterion. The system $\mathcal L$ is special if and only if at least one of the following holds:

Urbano's conjecture. There exists a very ample divisor $A$ such that $\mathcal{O}_X(mD)\otimes \mathcal{O}_X(A)^{\otimes m}$ is globally generated for every $m\geq 1$.

Let $Z=\sum_i m_iP_i$ be a fat point scheme in $\mathbb{P}^2$ for general points $P_i$, and suppose that $h^1(\mathcal I_Z(k))=0$. Let $\mu_k(Z)$ denote the multiplication map in d…

Let $\mathcal{L}=\mathcal{L}(d;m_0,m^n)$ be a linear system of plane curves with one base point of multiplicity $m_0$ and $n$ base points of equal multiplicity $m$. A system is…

Hirschowitz–Harbourne conjecture. The system $L$ is special if and only if there exists a $-1$-curve $C\subset\mathbb{P}^{2}$ such that

Let $D\subset\mathbb{N}^{2}$ be finite, and consider a system $\mathcal{L}_{D}(m_{1},\dots,m_{r})$. The system admits a reduction algorithm if there is a sequence of sets … that sa…

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

Numerical special effect conjecture. The system ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special if and only if it is numerically special.

Segre's conjecture. If ${\mathcal{L}}_{2,d}(-\sum_{i=1}^hm_iP_i)$ is special, then its general member is non-reduced; equivalently, by Bertini's theorem, the system has a multiple…

Let $X$ be a K3 surface with $H^2=n\in2{\mathbb{Z}}$, and let ${\mathcal{L}}={\mathcal{L}}^n(d,m_1,\dots,m_h)$ be the system of curves in $|dH|$ through general points…

Let ${\mathcal{L}}=|ah+bF-\sum_{i=1}^{h}m_iP_i|$ be a system on the Hirzebruch surface ${\mathbb{F}}_e$, and let ${\mathcal{M}}$ be its residual system after repeatedly subtracting…

Segre's conjecture. If $\mathcal L$ is non-empty and reduced, then it is non-special.

Let $m_1,\ldots,m_r$ be positive integers corresponding to general points $p_1,\ldots,p_r\in\mathbb{P}^2$. Let $\mathcal{L}=\mathcal{L}_d(m_1,\ldots,m_r)$ be the linear system of d…

Let $n\geq 2$, and let $(X,H)$ be a degree-$2$ polarized irreducible symplectic variety, meaning that $X$ is an irreducible symplectic manifold deformation equivalent to…

Let $S$ be a rational surface, let $D$ be an ample, non-special divisor on $S$, and let $p_1,\ldots,p_r$ be $r$ points on $S$. Write … for the corresponding linear system, and call…

Let $S$ and $\mathcal L$ be as above: $S$ is a generic $K3$ surface with ${\rm Pic}(S)=\mathbb{Z}H$, $H^2=2g-2\geq 2$, and $\mathcal L=\mathcal L^n(d,m_1,\ldots,m_r)$ is the system…

Let $S$) be a generic $K3$ surface with ${\rm Pic}(S)=\mathbb{Z}H$, where $H^2=2g-2\geq 2$. Let $p_1,\ldots,p_r$ be points in general position on $S$, with prescribed multiplicitie…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effect…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $C_e$ be the negative section of $\mathbb F_e$ with strict transfor…

Let $X$ be a smooth projective variety of dimension $n\geq 2$, let $K_X$ be its canonical line bundle, and let $L$ be an ample line bundle on $X$. Moraga–Yeong's conjecture. The co…

Converse parametrization conjecture. If a Brunovsky transformation $(T,F,G)$ transforms the staircase linear system into Brunovsky form, then there exists an observation matrix $C$…

Let $p_1,\dots,p_r$ be $r$ very general points in $\mathbb P^2$. For an integral curve $C\subset\mathbb P^2$, write $\operatorname{mult}_{p_i}(C)$ for its multiplicity at $p_i$. Eq…

The common-ground conjecture. The pair $\{E,F\}$ is transformable into strong standard canonical form, and the Jordan normal form of its constant nilpotent matrix $N$ consists exac…

Fujita-type jet-ampleness conjecture. There exist integer-valued polynomial functions $a_\ell(n)$ and $b_\ell(n)$ depending only on $n$ such that