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The Resource On the Geometry of Some Special Projective Varieties

On the Geometry of Some Special Projective Varieties

On the Geometry of Some Special Projective Varieties
On the Geometry of Some Special Projective Varieties
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Lecture Notes of the Unione Matematica Italiana
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On the Geometry of Some Special Projective Varieties
On the Geometry of Some Special Projective Varieties
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  • Preface -- Ringraziamenti -- Contents -- Introduction -- 1 Tangent Cones, Tangent Spaces, Tangent Stars: Secant, Tangent, Tangent Star and Dual Varieties of an Algebraic Variety -- 1.1 Tangent Cones and Tangent Spaces of an Algebraic Variety and Their Associated Varieties -- 1.2 Join of Varieties -- 1.3 Linear Projections -- 1.4 Terracini's Lemma and Its First Applications -- 1.5 Dual Varieties and Contact Loci of General Tangent Linear Spaces -- Exercises -- Hint for Problems of Chap.1 -- 2 The Hilbert Scheme of Lines Contained in a Variety and Passing Through a General Point -- 2.1 Basics of Deformation Theory of (Smooth) Rational Curves on Smooth Projective Varieties -- 2.2 The Hilbert Scheme of Lines Contained in a Projective Variety and Passing Through a Point -- 2.2.1 Notation, Definitions and Preliminary Results -- 2.2.2 Singularities of Lx,X -- 2.3 Equations for Lx,XP((txX)*) -- 2.3.1 Vx Versus TxXX for a Quadratic Variety -- 2.3.2 Tangential Projection and Second Fundamental Form -- 2.3.3 Approach to Bx,X=Lx,X via BEL -- 2.3.4 Lines on Prime Fano Manifolds -- 2.4 A Condition for Non-extendability -- 2.4.1 Extensions of Lx,YPn-1 via Lx,XPn -- 3 The Fulton-Hansen Connectedness Theorem, Scorza's Lemma and Their Applications to Projective Geometry -- 3.1 The Enriques-Zariski Connectedness Principle, the Fulton-Hansen Connectedness Theorem and the Generalizations of Some Classical Results in Algebraic Geometry -- 3.2 Zak's Applications to Projective Geometry -- 3.3 Tangential Invariants of Algebraic Varieties and Scorza's Lemma -- 3.4 Severi's Characterization of the Veronese Surface Versus Mori's Characterization of Projective Spaces -- 4 Local Quadratic Entry Locus Manifolds and Conic Connected Manifolds -- 4.1 Definitions and First Geometrical Properties -- 4.2 Qualitative Properties of CC-Manifolds and of LQEL-Manifolds
  • 4.3 Classification of LQEL-Manifolds with e{u2265}dim(X)/2 -- 4.4 Classification of Conic-Connected Manifolds and of Manifolds with Small Dual -- 4.4.1 Classification of Varieties with Small Dual -- 4.4.2 Bounds for the Dual Defect of a Manifold and for the Secant Defect of an LQEL-Manifold -- 5 Hartshorne Conjectures and Severi Varieties -- 5.1 Hartshorne Conjectures -- 5.2 Proofs of Hartshorne's Conjecture for Quadratic Manifolds and of the Classification of Quadratic Hartshorne Manifolds -- 5.2.1 The Bertram-Ein-Lazarsfeld Criterion for Complete Intersections -- 5.2.2 Faltings' and Netsvetaev's Conditions for Complete Intersections -- 5.2.3 Proofs of the Main Results -- 5.3 Speculations on Hartshorne's Conjecture -- 5.4 A Refined Linear Normality Bound and Severi Varieties -- 5.5 Reconstruction of Severi Varieties of Dimension 2, 4, 8 and 16 -- 6 Varieties n-Covered by Curves of a Fixed Degree and the XJC Correspondence -- 6.1 Preliminaries and Definitions -- 6.1.1 Examples and Reinterpretation of Known Results -- 6.2 Bounding the Embedding Dimension -- 6.2.1 Previously Known Versions -- 6.2.2 Looking for the Function s(r,n,e) via ProjectiveGeometry -- 6.2.3 Relation to the Castelnuovo-Harris Bound -- 6.3 Rationality of Xr+1(n,e) and of the General Curve of the n-Covering Family -- 6.3.1 Bound for the Top Self Intersection of a Nef Divisor -- 6.4 Quadro-Quadric Cremona Transformations and Xn(3,3)P2n+1 -- 6.5 A Digression on Power Associative Algebras and Some Involutive Cremona Transformations -- 6.5.1 Power Associative Algebras, Jordan Algebras and Generalizations of Laplace Formulas -- 6.6 The XJC-Correspondence -- 7 Hypersurfaces with Vanishing Hessian -- 7.1 Preliminaries, Definitions, Statement of the Problem and of the Classical Results -- 7.2 Instances and Relevance of Hesse's Claim in Geometry and in Commutative Algebra -- 7.2.1 The Polar Map
  • 7.2.2 Curvature and h(f) -- 7.2.3 What Does the Condition f Divides h(f) Measure? -- 7.2.4 Weak and Strong Lefschetz Properties for Standard Artinian Gorenstein Graded Algebras -- 7.3 The Gordan-Noether Identity -- 7.3.1 Hesse's Claim for N=2,3 -- 7.3.2 Cremona Equivalence with a Cone -- 7.3.3 Applications of the Gordan-Noether Identity to the Polar Map -- 7.4 The Gordan-Noether-Franchetta Classification in P4 and Examples in Arbitrary Dimension -- 7.4.1 Gordan-Noether, Franchetta, Permutti and Perazzo Examples -- 7.4.2 A Geometrical Proof of the Gordan-Noether and Franchetta Classification of Hypersurfaces in P4 with Vanishing Hessian -- 7.5 The Perazzo Map of Hypersurfaces with Vanishing Hessian -- 7.6 Cubic Hypersurfaces with Vanishing Hessian and Their Classification for N{u2264}6 -- 7.6.1 Classes of Cubic Hypersurfaces with Vanishing Hessian According to Perazzo and Canonical Forms of Special Perazzo Cubic Hypersurfaces -- 7.6.2 Cubics with Vanishing Hessian in PN with N{u2264}6 -- 7.6.3 Examples in Higher Dimension -- References -- Index
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