Tangents and secants of algebraic varieties

This book is an introduction to the use and study of secant and tangent varieties to projective algebraic varieties. As mentioned in the Preface, these notes could also be thought of as a natural preparation to parts of the work of F. L. Zak [Tangents and secants of algebraic varieties}, Translated from the Russian manuscript by the author, Amer. Math. Soc., Providence, RI, 1993]. Chapter 1 deals with the basic definitions of tangent and secant varieties, as well as joins and dual varieties. The author also presents the Terracini lemma as well as various characterizations of the Veronese surface in P5. Chapter 2 presents the Fulton-Hansen connectedness theorem [W. Fulton and J. Hansen, Ann. of Math. (2) 110 (1979), no. 1, 159–166; MR0541334 (82i:14010)], as well as some of Zak's work, specifically Zak's theorem on tangencies. Chapter 3 deals with Hartshorne's conjectures, Zak's theorem on linear normality, and the classification of Severi varieties. Chapter 4 discusses the deficiency of higher secant varieties and Scorza varieties. Chapter 5 presents some miscellaneous topics, including Waring's problem and subhomaloidal systems. The book closes with obituaries (in Italian) of Scorza and Terracini. In all, this seems a well-written and enjoyable introduction to the topics at hand. The examples provided are instructive and the background material is presented interestingly.