4 problems

Let ${\mathbb{X}}\subseteq {\mathbb{P}}^n$ be a set of points having $\mathop{\rm CBP}\nolimits(r)$. A plane configuration $\mathcal{P}$ is a union of distinct positive-dimensional…

Let $\Gamma$ be a complete intersection of $n$ quadrics in $\mathbb{P}^n$. Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ containing a subscheme $\Omega\subset\Gamma$ w…

Plane configuration conjecture. If

Let $\Gamma$ be any subscheme of a zero-dimensional complete intersection of hypersurfaces of degrees $d_1\leq\dots\leq d_c$ in a projective space $P^c$. Say that $\Gamma$ fails to…