48 problems

Let $F(x,y)$ be a polynomial whose zero set is a plane curve consisting of the product of $d$ real lines. Consider the critical points of $F$ having the same non-zero critical valu…

Let a conic-line arrangement be an arrangement $\mathcal{CL}$ of lines and smooth conics in the complex projective plane with ordinary singularities, and let $\gamma(\mathcal{CL})$…

Let a line arrangement be a finite collection of lines in the complex projective plane, and let $\gamma(\mathcal{L})$ denote its characteristic number. For a real number…

Let $C_{1}, C_{2}$ be reduced curves in $\mathbb{P}^{2}_{\mathbb{C}}$ whose irreducible components are smooth and whose singularities are ordinary quasi-homogeneous singularities.…

Numerical Terao's conjecture. If $\mathcal{L}_{1},\mathcal{L}_{2}\subset\mathbb{P}^{2}_{\mathbb{F}}$ are line arrangements satisfying

Maximal Tjurina existence conjecture. For every integer $d\geq 3$ and every integer $r$ with $d/2\leq r\leq d-1$, there are maximal Tjurina curves of type $(d,r)$. Moreover, for…

For a reduced plane curve $C=\{f=0\}$ in $\mathbb{P}^{2}_{\mathbb{C}}$, let $\nu(C)$ denote its defect. A line arrangement is a union of finitely many lines in…

Let $\mathcal A:f=0$ be a line arrangement of degree $d$ in the complex projective plane. Let $\operatorname{mdr}(f)$ denote the minimal degree of a Jacobian relation for $f$, and…

Let $C$ be a reduced plane curve in $cmathbb{P}^2$. A point $p\in C$ is modular if the central projection from $p$ induces a locally trivial fibration of the complement…

Let $C$ be a line arrangement that is not a union of concurrent lines, and let ${\alpha}$ be a root of its Alexander polynomial $\Delta^1_C(t)$. Suppose that the order of…

Diameter upper-bound conjecture. The number of diameters of $\mathcal R$ does not exceed

Let $\mathcal{L}$ be a sporadic simplicial line arrangement in $\mathbb{P}^{2}_{\mathbb{R}}$, meaning a simplicial line arrangement outside the known infinite families, and let $d$…

A simplicial line arrangement is a finite arrangement of lines in $\mathbb{P}^{2}_{\mathbb{R}}$ whose complement has only open triangular connected components. Grünbaum's catalogue…

Let $k\geq 3$ registration marks be points in the plane, and consider the construction lines determined by the arrangement, subject to the two rules in Section 1. Upper-bound conje…

Four-line generation conjecture.

Deletion double-point conjecture. If ${\mathcal A}\setminus\{H\}$ is not free, then

Nonfree line-arrangement double-point conjecture. If ${\mathcal A}$ is not free, then

Supersolvable Sylvester–Gallai conjecture. One has

Supersolvable arrangement double-point conjecture. One has

Let ${\mathcal A}$ and ${\mathcal B}$ be line arrangements with isomorphic intersection lattices … Let $E({\mathcal A})$ and $E({\mathcal B})$ be the associated rank $2$ vector bun…

Let ${\mathcal A}$ and ${\mathcal B}$ be arrangements of $d$ lines with isomorphic intersection lattices … Let $r({\mathcal A})$ denote the minimal degree of a Jacobian relation. S…

Let ${\mathcal A}$ and ${\mathcal B}$ be line arrangements with isomorphic intersection lattices … For an arrangement ${\mathcal A}$, let $r({\mathcal A})$ denote the minimal degre…

Let ${\mathcal A}$ be a supersolvable line arrangement consisting of $d=|{\mathcal A}|$ lines, and suppose that ${\mathcal A}$ is not a pencil. Denote by $n_2$ the number of inters…

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement, and let $s$ be the number of its lines. A non-pencil complex supersolvable line ar…

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement. An arrangement is nontrivial if it is neither a pencil nor a near pencil, and it i…