On the secant varieties to the osculating variety of a Veronese surface

2003 ◽  
Vol 1 (3) ◽  
pp. 315-326 ◽  
Author(s):  
E. Ballico ◽  
C. Fontanari
Keyword(s):  
Secant Varieties ◽  
Veronese Surface

scholarly journals Duality, Bridgeland wall-crossing and flips of secant varieties

2017 ◽  
Vol 28 (02) ◽  
pp. 1750011 ◽  
Author(s):  
Cristian Martinez
Keyword(s):  
Moduli Space ◽  
Blow Up ◽  
Stability Conditions ◽  
Secant Varieties ◽  
Veronese Surface ◽  
Wall Crossing ◽  
General Aspects

Let [Formula: see text] denote the [Formula: see text]-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of [Formula: see text] by embedding the blow-up [Formula: see text] into a suitable moduli space of Bridgeland semistable objects on [Formula: see text].

scholarly journals A lower bound for $$\chi ({\mathcal {O}}_S)$$

2021 ◽  
Author(s):  
Vincenzo Di Gennaro
Keyword(s):  
Lower Bound ◽  
Linear System ◽  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Complex Surface ◽  
Veronese Surface ◽  
Very Ample Line Bundle ◽  
Rational Normal Scroll

AbstractLet $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$ L of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ ( O S ) ≥ - 1 8 d ( d - 6 ) . The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ χ ( O S ) = - 1 8 d ( d - 6 ) if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 ( S , L ) | embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$ d 2 Q , where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$ H ∈ | H 0 ( S , L ) | of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$ V ⊆ P 5 , from a point $$x\in V\backslash C$$ x ∈ V \ C .

scholarly journals Singular Poisson–Kähler geometry of Scorza varieties and their secant varieties

2005 ◽  
Vol 23 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Johannes Huebschmann
Keyword(s):  
Secant Varieties ◽  
Kahler Geometry

scholarly journals Non-defectivity of Grassmannian bundles over a curve

2016 ◽  
Vol 27 (07) ◽  
pp. 1640002 ◽  
Author(s):  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundles ◽  
Manuscripta Math ◽  
Maximal Degree ◽  
Geometric Proof ◽  
Secant Varieties ◽  
Grassmann Bundle ◽  
Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

XI—The Identification of Klein's Quartic

1944 ◽  
Vol 62 (1) ◽  
pp. 83-91
Author(s):  
W. L. Edge
Keyword(s):  
Standard Form ◽  
Geometrical Interpretation ◽  
The Other ◽  
Veronese Surface ◽  
Quartic Polynomial ◽  
New Form ◽  
Quartic Curves ◽  
Quartic Polynomials

SummaryThere is a mode of specialising a quartic polynomial which causes a binary quartic to become equianharmonic and a ternary quartic to become a Klein quartic, admitting a group of 168 linear self-transformations. The six relations which must be satisfied by the coefficients of the ternary quartic were given by Coble forty years ago, but their true significance was never suspected and they have remained until now an isolated curiosity. In § 2 we give, in terms of a quadric and a Veronese surface, the geometrical interpretation of the six relations; we also give, in terms of the adjugate of a certain matrix, their algebraical interpretation. Both these interpretations make it abundantly clear that this set of relations specialising a ternary quartic has analogues for quartic polynomials in any number of variables, and point unmistakably to what these analogues are.That a ternary quartic is, when so specialised, a Klein quartic is proved in §§ 4–6. The proof bifurcates after (5.3); one branch leads instantly to the standard form of the Klein quartic while the other leads to another form which, on applying a known test, is found also to represent a Klein quartic. One or two properties of the curve follow from this new form of its equation. In §§ 8–10 some properties of a Veronese surface are established which are related to known properties of plane quartic curves; and these considerations lead to a discussion, in § 11, of certain hexads of points associated with a Klein curve.

scholarly journals The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 314 ◽  
Author(s):  
Alessandra Bernardi ◽  
Enrico Carlini ◽  
Maria Catalisano ◽  
Alessandro Gimigliano ◽  
Alessandro Oneto
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Tensor Decomposition ◽  
Tensor Rank ◽  
Segre Variety ◽  
Secant Varieties ◽  
Open Problems ◽  
Veronese Variety ◽  
Strong Motivation

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

scholarly journals Singularities and syzygies of secant varieties of nonsingular projective curves

2020 ◽  
Vol 222 (2) ◽  
pp. 615-665
Author(s):  
Lawrence Ein ◽  
Wenbo Niu ◽  
Jinhyung Park
Keyword(s):  
Secant Varieties ◽  
Projective Curves

scholarly journals Higher secant varieties of the minimal adjoint orbit

2004 ◽  
Vol 280 (2) ◽  
pp. 743-761 ◽  
Author(s):  
Karin Baur ◽  
Jan Draisma
Keyword(s):  
Secant Varieties ◽  
Adjoint Orbit
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