188 problems

Let $q_1,\ldots,q_n$ be independent generic points in $\mathbb{P}_{\mathbb{C}}^2$, with $n\geq 10$. For a plane curve of degree $d$, suppose that it passes through each $q_i$ with…

Toeplitz's conjecture. Every Jordan curve has an inscribed square.

Coolidge–Nagata conjecture. Every rational cuspidal curve can be mapped onto a line by a Cremona transformation of the plane.

Yoshihara's conjecture. If $\mathbb{P}^2_f$ and $\mathbb{P}^2_g$ are isomorphic, then there is an element of $\operatorname{Aut}(\mathbb{P}^2)$ that sends $\mathbb{P}^2_f$ onto…

Yano's conjecture. For generic curves in some $\Gamma$-constant deformation of $f$, the $b$-exponents are

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.…

Let $X\subset\mathbb P^2$ be a rational cuspidal plane curve, meaning a rational plane curve whose singularities are cusps. Dimca–Sticlaru conjecture. Every rational cuspidal plane…

Let $C$ be a rational cuspidal plane curve of degree $d$. The source identifies three specific series of tricuspidal curves in a table. Piontkowski's conjecture. If $d\geq 6$, then…

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 $f\in\mathbb{C}[[x,y]]$ be an isolated hypersurface singularity, with Milnor number $\mu$ and Tjurina number $\tau$. Dimca–Greuel's conjecture. … The conjecture was proved comp…

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…

Let $C\subset\mathbb{C}P^2$ be a rational cuspidal plane curve. Orevkov's conjecture. The curve $C$ has at most four singular points. Moreover, there is only one curve of degree…

Let $C\subset\mathbf{P}^2$ be an irreducible sextic plane curve with no toric decomposition of type $(2,3)$. Let $\Delta_C(t)$ denote its Alexander polynomial, and let $\ell$ be a…

Let $m$ be the highest multiplicity among the singular points of a rational cuspidal plane curve of degree $d$, and let $\alpha=(3+\sqrt{5})/2$. Orevkov constructed two families of…

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.

Let $D\subset\mathbb{CP}^2$ be an algebraic singular plane curve with cusps and nodes, and let $\theta:\pi_1(\mathbb{CP}^2-D)\to S_N$ be a symmetric-group-valued monodromy morphism…

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…

Smoothness conjecture for $D_k$. The curve $D_k$ is smooth if and only if the polynomial $x^7+kx^5+1$ has no zeros in $\mathbb{F}_q$. This is proposed as the even-characteristic co…

Degeneration conjecture. The spectral sequences degenerate at the $E_3$-term for any reduced plane curve $C:f=0$.

Harbourne-Hirschowitz conjecture. The system $\mathscr{L}(d,n,m)$ is non-special, meaning that it has expected dimension $\max\{-1,v\}$. This predicts a precise dimension for homog…

A linear system $\mathcal{L}$ is (-1) special if there are pairwise disjoint (-1) curves $A_1,\dots,A_r$ such that $\mathcal{L}\cdot A_j=-N_j$ with $N_j\geq 1$ for every $j$ and…

Let $V_d^{\operatorname{irr}}(S_1,\dots,S_r)$ denote the family of reduced irreducible complex plane curves of degree $d$ with isolated singular points of prescribed topological ty…

Let $S$ be a topological singularity type, let $r,d$ be positive integers, let $V_d^{\mathrm{irr}}(r\cdot S)$ denote the family of irreducible degree-$d$ plane curves with $r$ sing…

A multinet in $\mathbb{CP}^2$ is a line arrangement with the multinet structure defined in the paper, and a net is the corresponding structure with all line multiplicities equal to…

A pencil of plane curves is said to be of rank three when it has three completely reducible fibers satisfying the other condition of Definition. Connectedness conjecture for rank-t…