109 problems

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…

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 $(C,p_i)_{i=1}^\nu$ 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…

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 $m_1,\ldots,m_r$ 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…

Let $C:f=0$ be a reduced plane curve that is not free, with $t(C)=\sum_{j=1}^{m-2}\epsilon_j\geq 1$, and let $\pi_C$ be the ordered partition of $t(C)$ whose parts are the positive…

Let $F(s,a)$ be the obstruction polynomial, let $\widehat{F}(S,A,Z)$ be its homogenization, and let … be the projective closure. The computation lists the $8$ rational points … Com…

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 $C=C_J\cup C_K$ be a reducible plane curve with the notation used above, where $C_J$ and $C_K$ are the corresponding subcurves and $\textup{I}(C_J,C_K)$ denotes their intersect…

Let $C=V(f)$ be a reduced $3$-syzygy plane curve, meaning that it has a first syzygy matrix $M_f$ that is a square matrix of size $3$. Let $p\in\operatorname{Sing}(C)$ be an isolat…

Existence conjecture for curves of type $(d,r,m)$. For any integers $d,m\geq 3$ and any integer $r$ such that

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…