10 problems

Let $Z=\sum_i m_iP_i$ be a fat point scheme in $\mathbb{P}^2$ for general points $P_i$, and suppose that $h^1(\mathcal I_Z(k))=0$. Let $\mu_k(Z)$ denote the multiplication map in d…

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…

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…

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…

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

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…

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