Rational points on surfaces

The talk will be devoted to the proof of the following result. Let $X$ be a smooth proper geometrically rational surface over a perfect field $\mathbf{k}$, suppose that $X$ has no rational points and degree of $X$ is at least 6. Also let $\mathbf{L} / \mathbf{k}$ be a field extention whose degree is not divisible by 2 and 3. Then the surface $X \times \mathrm{Spec} (\mathbf{L})$ has no rational points. The proof consists of the case of del Pezzo surfaces and the case of conic bundles. We will pay much attention to both cases.