On Permutation Trinomials of the type $X^{q^2-q+1}+A{X}^{q^2}+BX$ over ${\mathbb{F}}_{q^3}$

Necessary and sufficient conditions on $A,B\in {\mathbb{F}}_{q^3}^{\ast }$ for $f(X)=X^{q^2-q+1}+A{X}^{q^2}+BX$ being a permutation polynomial of ${\mathbb{F}}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.