K3 projective models in scrolls

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The exposition studies projective models of K 3 surfaces whose hyperplane sections are non-Clifford general curves. These models are contained in rational normal scrolls. The exposition supplements standard descriptions of models of general K 3 surfaces in projective spaces of low dimension, and leads to a classification of K 3 surfaces in projective spaces of dimension at most 10. The authors bring further the ideas in Saint-Donat's classical article from 1974, lifting results from canonical curves to K 3 surfaces and incorporating much of the Brill-Noether theory of curves and theory of syzygies developed in the mean time.