EAN: 9783319267647

ON THE GEOMETRY OF SOME SPECIAL PROJECTIVE VARI...

Providing an introduction to both classical and modern techniques in projective algebraic
geometry  this monograph treats the geometrical properties of varieties embedded in projective
spaces  their secant and tangent lines  the behavior of tangent linear spaces  the
algebro-geometric and topological obstructions to their embedding into smaller projective
spaces  and the classification of extremal cases. It also provides a solution of Hartshorne's
Conjecture on Complete Intersections for the class of quadratic manifolds and new short proofs
of previously known results  using the modern tools of Mori Theory and of rationally connected
manifolds.The new approach to some of the problems considered can be resumed in the principle
that  instead of studying a special embedded manifold uniruled by lines  one passes to analyze
the original geometrical property on the manifold of lines passing through a general point and
contained in the manifold. Once this embedded manifold  usually of lower codimension  is
classified  one tries to reconstruct the original manifold  following a principle appearing
also in other areas of geometry such as projective differential geometry or complex geometry.

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