20 problems

Hirschowitz's conjecture. The only special linear systems are the $(-1)$-special ones.

Let $K$ be a centrally symmetric convex body in $\mathbb{R}^2$, and consider packings by translates of $K$. A packing is diffusively dominant when its associated mass distribution…

A circle packing is a packing of congruent circles in the plane, and a packing is diffusively dominant if its mass distribution diffusively dominates that of every other packing. C…

Wetzel's conjecture. The triangle $T$ covers every unit arc in the plane.

Let $Z_{s,m}$ be a general union of $s$ flat fat points of multiplicity $m$ in $\mathbb P^2$. Assume $m\geq 4$ and $s\geq 2m-3$. Its Hilbert function should satisfy … A flat fat po…

Let a Jordan curve be a simple closed curve $\gamma\subset\mathbb{R}^2$. An isosceles trapezoid is a quadrilateral whose vertices are the intersection set of two parallel lines and…

Let $\gamma$ be a planar billiard table that is not an ellipse. For a light source chosen inside $\gamma$, let $\Gamma_n$ be the caustic by reflection after $n$ reflections, with…

Gerriets–Poole's conjecture. A region is a cage if and only if it can accommodate every polygonal worm.

Let $\mathcal{F}$ be the family of all arcs of unit length, and let a circular sector have angle $30^\circ$ and unit radius. The 30-degree circular-sector covering conjecture. This…

Let $\gamma$ be a Jordan loop. In the trichotomy theorem, the hyperbolic case is the alternative in which there is a connected set $S$ of inscribed rectangles whose members have un…

Let $\xi$ be a sufficiently general valuation parameter and let $t\ge 8+1/36$. Let $\operatorname{ord}_{\xi,t}$ denote the valuation written in the source, and let maximality mean…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points, and let $\operatorname{NE}(X)$ denote its Mori cone. Let $\mathcal{Q}$ be the nonnegative cone and let…

Degree-jumping conic containment conjecture. Then $Z$ is contained in a single conic.

Let $p_1,\ldots,p_r\in{{\bf P}^{2}}$ be generic points, and let $Z=m_1p_1+\cdots+m_rp_r$ be a fat point subscheme with homogeneous ideal $I_Z\subset R=K[{{\bf P}^{2}}]$. Write … fo…

In the planar achievement game, a finite point set $G\subset\mathbb{R}^2$ is used as the goal set: players alternately place points, and Player 1 seeks to build a congruent copy of…

Let $C$ be a smooth simple closed curve in $\mathbb R^2$, and let $Q$ be four points on a single circle. Makeev's inscribed quadrangle conjecture. There is a similarity transform…

Let $X$ be the blow-up of ${\mathbb P}^2$ at $r$ very general points. Let $N_1(X)$ be the space of numerical curve classes, let $Q\subset N_1(X)$ be the quadric cone of classes…

SHGH negative-curve formulation. Every reduced irreducible curve $C\subset X$ with $C^2<0$ is an exceptional curve, and for every nef divisor $F$ either

SHGH conjecture. Failure of $r_k$ to have maximal rank is completely accounted for by curves $C\subset{{\bf P}^{2}}$ whose strict transforms $\widetilde C\subset X$ are exceptional…

SHGH conjecture. The conditions imposed by the points on degree-$k$ forms are independent whenever $h_Z(k)>0$; equivalently, the displayed expected-dimension formula holds.