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During the last twenty years algebraic geometry has experienced a remarkable shift from development of abstract theories to investigation of concrete properties of projective varieties. Many problems of classical algebraic geometry centered on linear systems, projections, embedded tangent spaces, and so on. Use of modern techniques has made it possible to make progress on some of these problems. Following these themes, this book covers these topics, among others: tangent spaces to subvarieties of projective spaces and complex tori, projections of algebraic varieties, classification of Severi varieties, higher secant varieties, and classification of Scorza varieties over an algebraically closed field of characteristic zero.