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On the Geometry of Some Special Projective Varieties (Paperback) (Francesco Russo)
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Providing an introduction to some classical and recent results and techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behaviour of tangent linear spaces, the algebro-geometric and topological obstructions to their embedding into smaller projective spaces and the classification in the extremal cases.
These are classical themes in algebraic geometry and the renewed interest at the beginning of the '80 of the last century came from some conjectures posed by Hartshorne, from an important connectedness theorem of Fulton and Hansen, and from its new and deep applications to the geometry of algebraic varieties, as shown by Fulton, Hansen, Deligne,Lazarsfeld and Zak.
The novelty of this treatment resides in the use of modern tools usually applied in a more abstract context, like the theory of deformations of rational curves on a manifold (Mori Theory), the theory of rationally connected varieties, birational geometry.