This thesis concerns several connections between commutative and noncommutative algebraic geometry. These are abstract areas in fundamental mathematics that have been the object of study since acient times: algebraic geometry was originally introduced to study and solve (systems of) polynomial equations, and study the properties of these solutions. In noncommutative algebraic geometry we extend the type of solutions, and we obtain a more flexible framework in which we can do geometry. But with this extended flexibility comes an increased complexity, explaining the need for quite some technical background to understand the results.

The noncommutative algebraic geometry which appears in this thesis concerns different incarnations of the subject: noncommutative projective geometry à la Artin-Zhang, the study of derived categories of smooth projective varieties à la Bondal-Orlov, and finally this culminates in the study of (smooth and proper) dg categories à la Kontsevich. The connections being made aid us in understanding complicated objects from noncommutative algebraic geometry, by studying them using more familiar techniques from commutative algebraic geometry.

Chapter 1 gives an introduction to "the geometry of derived categories". In this way one of the most important themes of this thesis is introduced by giving an overview of the literature, and explaining how these tools and examples make an appearance in this thesis.

In chapter 2 we study whether the derived category of a finite-dimensional algebra can be embedded in the derived category of a smooth projective surface. As the derived category of a curve is indecomposable, this is the first interesting case, and we give important obstructions to the existence of such an embedding.

In chapter 3 we study when the derived category of a noncommutative quadric can be embedded in the derived category of a deformation of the Hilbert scheme of the quadric. This is a special case of a conjecture of Orlov, and we also formulate an infinitesimal version of this conjecture. Based on this conjecture it becomes interesting to study the Hochschild cohomology of noncommutative planes and quadrics, which we do in chapter 4.

Based on the numerical classification of noncommutative surfaces of rank 4 we construct examples for all types in chapter 5. For this we use a previously unknown construction of the blowup of a noncommutative surface. In chapter 6 we compare this construction in a special case to an earlier construction using noncommutative P^1-bundles. In this way we obtain a noncommutative version of the classical isomorphism Bl_x P^2=F_1, but in this case it is not a deformation of the commutative setup.

In chapter 7 we compute the point variety of a skew polynomial ring. This is a moduli space which contains important information about this noncommutative object, and we can describe these in arbitary dimension, whilst in low dimension we can give a complete classification.

In chapter 8 we study a noncommutative version of Chow groups. These are introduced using tensor triangulated geometry, and it is shown that these coincide with classical invariants which were defined in an ad hoc way.