| The problem of determining an effective bound on the multiple which makes an ample divisor $D$ on a smooth variety $X$ very ample is natural and many results are known in characteristic zero. In this talk I will discuss this problem on surfaces in positive characteristic, giving a complete solution in this setting. Our strategy requires an ad hoc study of pathological surfaces, on which Kodaira-type theorems can fail. A Fujita-type theorem and a vanishing result for big and nef divisors on pathological surfaces will also be discussed. This is a joint work with Gabriele Di Cerbo. |