5 problems

Nonexistence conjecture. The polar space $\mathsf{Q}^+(9,q)$ does not have an ovoid.

Ovoid non-existence conjecture. No ovoid exists in any finite classical dual polar space of rank at least three.

Let $q$ be a prime power greater than $2$, and let $\mathcal{Q}(8,q)$ be the parabolic quadric in projective $8$-space. An $(q^2+q+1)$-ovoid is a set of points meeting every genera…

Let $Q$ be a generalized quadrangle of even prime-power order $q$, and let an ovoid be a set of points meeting every line of $Q$ exactly once. Ovoid-intersection conjecture. Every…

Bamberg et al.'s conjecture. If an $m$-ovoid exists in $\operatorname{W}(2r-1,p^e)$ with $r>2$, then