New lower bounds for the Tur\'{a}n density of $PG_{m}(q)$

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Tur\'{a}n number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Tur\'{a}n density of $\mathcal{H}$ is defined by $$\pi(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.$$ In this paper, we consider the Tur\'{a}n density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q)$, we give a new general lower bound for the Tur\'{a}n density of $PG_{m}(q)$. By a detailed analysis of the structures of complete arcs in $PG_2(q)$, we also get better lower bounds for the Tur\'{a}n density of $PG_2(q)$ with $q=3,\ 4,\ 5,\ 7,\ 8$.