Singular del Pezzo surfaces over finite fields

Abstract

If $X$ is a singular del Pezzo surface of degree $d$ over a finite field ${\mathbb{F}}_q$ with only rational double point singularities, does there always exist a smooth ${\mathbb{F}}_q$-point on $X$? We show that this is true for $d\ge 3$ and give counterexamples in the case of $d=2$.