2009/09/15 by Paltin Ionescu, Ionescu, Paltin, Francesco Russo +2 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14J45 #msc:14M07 #msc:14MXX #msc:14NXX
paper · pdf · doi:10.48550/arxiv.0909.2763
21 pages; This paper has been withdrawn because its contents were divided into two papers reconstructing integrally this one and resubmitted to arXiv. This was done due to requirements of referee(s)/Editorial Boards
arxiv created 2012/09/10 · arxiv updated 2012/09/11
Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. We discuss our conjecture that every dual defective manifold with cyclic Picard group should also be secant defective, of a very special type, namely a local quadratic entry locus variety.