11 problems

An O'Nan configuration in a unital is a collection of four distinct lines meeting in six distinct unital points. Piper's conjecture. Any unital without an O'Nan configuration is th…

Let $k$ be the cardinality of a finite field, let $G=SL(2,\mathbb F_k)$, and let a $G$-space split into one orbit of size $|G|$ and $k+1$ orbits of size $1$. The SL(2) infinite-ser…

Square-parameter conjecture. If $E$ is a point of the unital, then $a$ is a square. The claim concerns the additional intersection point associated with the BM-special Triple O'Nan…

Let $q$ be a prime power and let ${\mathcal U}$ be a non-classical unital in ${\textup{PG}}(2,q^2)$. A Triple O'Nan configuration is a set of six distinct lines pairwise intersecti…

Fano-plane conjecture. Every non-classical unital ${\mathcal U}$ in ${\operatorname{PG}}(2,q^2)$ contains a Fano plane if $q\geq 4$.

Let $\mathcal U$ be a unital of ${\textup{PG}}(2,q^2)$ and let $P$ be a point not on $\mathcal U$. The $q+1$ points of contact of the tangent lines through $P$ form the pedal set…

A unital $\mathcal U$ of order $q$ embedded in a projective plane of order $q^2$ is a set of $q^3+1$ points such that every line meets $\mathcal U$ in exactly $1$ or $q+1$ points.…

Classical-unital uniqueness conjecture. The classical unitals are the only unitals not containing an O'Nan configuration.

Existence conjecture. There should exist an O'Nan configuration containing the point $(\infty)$ in each of these unitals.

Let $U$ be any unital and let $H$ be a Hermitian unital in $\operatorname{PG}(2,q^2)$, where $q$ is a prime power. Baker–Ebert's congruence conjecture. The size of their intersecti…

Let $U$ be a unital in $\operatorname{PG}(2,q^2)$, and suppose that its automorphism group contains a cyclic $2$-point stabiliser of order $q^2-1$. Donati–Durante's conjecture. The…