Minimal Blocking Sets in $PG(2,9)$

Abstract

We classify the minimal blocking sets of size 15 in $\mathrm{P}\mathrm{G}(2,9)$. We show that the only examples are the projective triangle and the sporadic example arising from the secants to the unique complete 6-arc in $\mathrm{P}\mathrm{G}(2,9)$. This classification was used to solve the open problem of the existence of maximal partial spreads of size 76 in $\mathrm{P}\mathrm{G}(3,9)$. No such maximal partial spreads exist $[13]$. In $[14]$, also the non-existence of maximal partial spreads of size 75 in $\mathrm{P}\mathrm{G}(3,9)$ has been proven. So, the result presented here contributes to the proof that the largest maximal partial spreads in $\mathrm{P}\mathrm{G}(3,q=9)$ have size $q^2-q+2=74$.