Arithmetically Gorenstein sets of points on general surfaces in P3

This dissertation examines two questions. In the first two chapters, we study the minimal free resolution of a general set of points on a surface of degree d in P3. Our main result for this problem, contained in Chapter 2, is to give the form of the minimal free resolution of a general set of points on a cubic surface that has at most finitely many double points. We use liaison techniques and count syzygies. In the third and fourth chapters, we study the arithmetically Gorenstein sets of points on a general surface in P3. Our main result is a complete list of the h-vectors of arithmetically Gorenstein sets of points on a general sextic surface in P3. We use a connection between such sets of points and rank two arithmetically Cohen-Macaulay vector bundles on the surface, as well as liaison techniques and Terracini's lemma.