Let X be an smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete linear systems associated to powers of the canonical class $nK_{X}$. I would like to understand some examples of how intricate these maps can be for small $n$.

For simplicity let's restrict to the case of algebraic surfaces. We assume that $X$ is of general type and minimal. A concrete question that I have in mind is the following. One expects that for large $n$ the complete linear systems become basepoint free, but how large is large? What explicit examples are known where the systems $nK_{X}$ all have a basepoint for $n \leq N$ but become free when $n>N$ ? Is there some absolute bound on $N$, and if so for what surface is the bound saturated?

Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete linear systems associated to powers of the canonical class $nK_{X}$. I would like to understand some examples of how intricate these maps can be for small $n$.

For simplicity let's restrict to the case of algebraic surfaces. We assume that $X$ is of general type and minimal. A concrete question that I have in mind is the following. One expects that for large $n$ the complete linear systems become basepoint free, but how large is large? What explicit examples are known where the systems $nK_{X}$ all have a basepoint for $n \leq N$ but become free when $n>N$ ? Is there some absolute bound on $N$, and if so for what surface is the bound saturated?