On Blocking Sets of the Tangent Lines to a Nonsingular Quadric in $\mathrm{PG}(3,q)$, $q$ Prime

Let $Q^{-}(3,q)$ be an elliptic quadric and $Q^{+}(3,q)$ a hyperbolic quadric in $\mathrm{PG}(3,q)$. For $\epsilon\in\{-,+\}$, let $\mathcal{T}^{\epsilon}$ denote the set of all tangent lines of $\mathrm{PG}(3,q)$ with respect to $Q^{\epsilon}(3,q)$. If $k$ is the minimum size of a $\mathcal{T}^{\epsilon}$-blocking set in $\mathrm{PG}(3,q)$, then it is known that $q^2+1 \leq k \leq q^2+q$. For an odd prime $q$, we prove that there are no $\mathcal{T}^+$-blocking sets of size $q^2+1$ and that the quadric $Q^-(3,q)$ is the only $\mathcal{T}^-$-blocking set of size $q^2 +1$ in $\mathrm{PG}(3,q)$. When $q=3$, we show with the aid of a computer that there are no minimal $\mathcal{T}^-$-blocking sets of size $11$ and that, up to isomorphism, there are eight minimal $\mathcal{T}^-$-blocking sets of size $12$ in $\mathrm{PG}(3,3)$. We also provide geometrical constructions for these eight mutually nonisomorphic minimal $\mathcal{T}^-$-blocking sets of size $12$.