On the geometry of some special projective varieties
The book is an advanced text in Algebraic Geometry, focused on a topic which received a great deal of interest in the last decades. The study of secant varieties of projective varieties has been resumed in the eighties for geometrical purposes, strictly related with the Hartshorne's Conjecture on the structure of smooth varieties with low codimension. In the years, it produced a lot of interesting results which spread beyond Algebraic Geometry itself, with applications to other disciplines like Statistics or Signal Theory, where secant varieties of Segre or Veronese varieties are considered. The author takes a classical point of view, introducing in the first chapter the basic notions of join varieties, secant varieties, entry loci and dual varieties. In the rest of the book, the author illustrates connections between the theory of secant varieties and many aspects of modern Algebraic Geometry. They include the study of Fano schemes of projective varieties covered by lines, and a classification of conic-connected manifolds. More generally, the author reinterpret, from the previous point of view, results on the study of varieties covered by curves of fixed degree. Natural connections with the theory of Associative Algebras are provided. Some chapters contain a description of very recent results, achieved by methods related with the study of secant varieties. For instance, the author illustrates advances in the study of low codimensional manifolds, with the proof of Hartshorne's Conjecture for quadratic varieties. Varieties with special behavior with respect to secant and tangent spaces can be viewed as manifolds described by basic differential equations. Thus, they can be investigated with the tools introduced in the book. Following these lines, in the last chapter the author illustrates results on the classification of varieties with vanishing Hessian. In conclusion, the book under review covers fundamental aspects of the theory of secant spaces to varieties, and contains a careful description of many of its recent applications to Algebraic Geometry. Under this respect, it provides a fundamental advanced introduction to recent results and developments of a topic which experienced a rapid evolution in the last years.
scientific article; zbMATH DE number 6520138
scientific article