109 problems

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.…

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

Yoshihara's conjecture. Then $C$ and $D$ are projectively equivalent.

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

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…

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…

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

Let $\{p_1,\ldots,p_r\}\subset\mathbb{P}^2$ be general points, let $\nu:\widetilde{\mathbb{P}}^2\to\mathbb{P}^2$ be their blow-up, and let…

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…

Fix a degree $d$ and consider the torus $\mathcal H_d$ of hedgehogs whose support functions are trigonometric polynomials of degree $d$, with the amplitudes of each harmonic fixed.…

Let $C$ be a cuspidal plane algebraic curve and let $H_2$ be its 2-Hessian curve. At a point $p\in C$, write $m$ for the multiplicity of $C$ at $p$, $\delta$ for its delta invarian…

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 $\mathcal{L}=\mathcal{L}(d;m_0,m^n)$ 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…

Let $(C,p_i)_{i=1}^\nu$ be a collection of locally irreducible local plane curve singularities satisfying … for some integer $d$. Let $c_l$ be the coefficients associated with the…

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…

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 $B$ be the plane cubic from the preceding setup, let $m_d$ denote the number of degree-$d$ plane curves satisfying (AL) and meeting $B$ at a specified point of order $3d$, and…

Expected intersection-point conjecture. For $d>m$ and $d\ge 3$,

Let $\alpha'_p$ be the number of closed prime self-intersecting curves with $p$ crossings, and let $\alpha”_p$ be the number of prime alternating knots with $p$ crossings. Let…

Let $a_p$ and $\alpha_p$ denote the numbers of open and closed plane curves, respectively, with $p$ self-intersections. Let $\gamma=-\frac{1+\sqrt{13}}6$. Asymptotic enumeration co…

Cactus-shadow extension conjecture. For any embedded shadow whose block-adjacency graph is a cactus and whose cyclic blocks are annular in the sense of the necklace definition, the…

Tree--necklace Gauss-load conjecture. For tree-like and tree--necklace shadows, the minimum Gauss load is obtained by a reduced tangent-angle realization of a coorientation minimiz…

Complexity conjecture. For an appropriate purely combinatorial encoding of embedded shadows, the decision problem $\textnormal{\textsc{MinInflection}}$ for unrestricted shadows is…

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…