Global aspects of the geometry of surfaces

Several open problems related to the behavior of the monoid of effective divisors and the nef cone for smooth projective surfaces over an algebraically closed field are discussed, motivating and putting into historical context concepts such as Mori dream spaces, Seshadri constants and the resurgence of homogeneous ideals in polynomial rings. Some recent work on these topics is discussed along with the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. Exercises, with solutions, are included