Existence of minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields

We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field ${\mathbb{F}}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the size of the field over which they exist, and determine values of $q$ for which certain types cannot exist. In particular, we solve the inverse Galois problem for certain types of minimal del Pezzo surfaces of degree 1 over finite fields with a conic bundle structure. Additionally, we give bounds on the values of $q$ for which del Pezzo surfaces of degree 1 of index 8 exist over ${\mathbb{F}}_q$.