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PrefaceThe cover picture shows a smooth quartic surface in space, the simplest example of a projective model of a K3 surface. In the following pages we will encounter many more examples of models of such surfaces. The purpose of this volume is to study and classify projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls.These models are special in moduli in the sense that they do not represent the general member in the countable union of 19-dimensional families of polarized K3 surfaces. However, they are of interest because they fill up the set of models in P g for g ≤ 10 not described as complete intersections in projective space or in a homogeneous space as described by Mukai, with a few classificable exceptions.Thus our study enables us to classify and describe all projective models of K3 surfaces of genus g ≤ 10, which is the main aim of the volume.Acknowledgements. We thank Kristian Ranestad, who suggested to study certain projective models of K3 surfaces in scrolls that had shown up in connection with his work on varieties of sums of powers (see [I-R1], [I-R2] and [R-S]). This idea was the starting point of our work.We are also grateful to M. Coppens, G. Fløystad, S. Ishii, S. Lekaus, R. Piene, J. Stevens, S. A. Strømme, B. Toen and J. E. Vatne for useful conversations, and to G. M. Hana for pointing out several mistakes in an earlier version of the manuscript.We also thank Alessandra Sarti who made us the nice picture we have used on the cover.