Minimum size blocking sets of certain line sets with respect to an elliptic quadric in PG(3,q)PG(3,q)

For a given nonempty subset $\mathcal{L}$ of the line set of \PG(3,q), a set $X$ of points of \PG(3,q) is called an $\mathcal{L}$-blocking set if each line in $\mathcal{L}$ contains at least one point of $X$. Consider an elliptic quadric $Q^-(3,q)$ in \PG(3,q). Let $\mathcal{E}$ (respectively, $\mathcal{T}, \mathcal{S}$) denote the set of all lines of \PG(3,q) which meet $Q^-(3,q)$ in $0$ (respectively, $1,2$) points. In this paper, we characterize the minimum size $\mathcal{L}$-blocking sets in \PG(3,q), where $\mathcal{L}$ is one of the line sets $\mathcal{S}$, $\mathcal{E}\cup \mathcal{S}$, and $\mathcal{T}\cup \mathcal{S}$

Blocking sets of external lines to a conic in PG(2,q), q odd

We determine all point-sets of minimum size in $\mathrm{PG}(2,q)$, $q$ odd that meet every external line to a conic in $\mathrm{PG}(2,q)$. The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of $\mathrm{PGL}(2,q)$