New characterizations of the Clifford tori and the Veronese surface

1993 ◽  
Vol 61 (3) ◽  
pp. 277-284 ◽  
Author(s):  
Chuanxi Wu
Keyword(s):  
Veronese Surface ◽  
Clifford Tori

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 .

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 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.

Spectral characterizations of veronese surface in S 4

2001 ◽  
Vol 5 (1) ◽  
pp. 29-30 ◽  
Author(s):  
Yong-ai Zheng ◽  
Feng Yu ◽  
Yu-rong Liu
Keyword(s):  
Veronese Surface ◽  
Spectral Characterizations

Symplectic semifield spreads of PG(5, q) and the veronese surface

2010 ◽  
Vol 60 (1) ◽  
pp. 125-142 ◽  
Author(s):  
G. Lunardon ◽  
G. Marino ◽  
O. Polverino ◽  
R. Trombetti
Keyword(s):  
Veronese Surface

BIFURCATION OF CLIFFORD TORI IN BERGER 3-SPHERES

2014 ◽  
Vol 65 (4) ◽  
pp. 1345-1362 ◽  
Author(s):  
L. L. De Lima ◽  
J. H. De Lira ◽  
P. Piccione
Keyword(s):  
Clifford Tori

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

A property of the Veronese surface

1988 ◽  
Vol 32 (1-2) ◽  
pp. 157-168 ◽  
Author(s):  
Charles Thas
Keyword(s):  
Veronese Surface

scholarly journals Four-band non-Abelian topological insulator and its experimental realization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Tianshu Jiang ◽  
Qinghua Guo ◽  
Ruo-Yang Zhang ◽  
Zhao-Qing Zhang ◽  
Biao Yang ◽  
...  
Keyword(s):  
Topological Insulators ◽  
Edge State ◽  
Topological Classification ◽  
Symmetric System ◽  
Edge States ◽  
Experimental Realization ◽  
Topological Features ◽  
Significant Difference ◽  
Clifford Tori ◽  
Topological Charges

AbstractVery recently, increasing attention has been focused on non-Abelian topological charges, e.g., the quaternion group Q8. Different from Abelian topological band insulators, these systems involve multiple entangled bulk bandgaps and support nontrivial edge states that manifest the non-Abelian topological features. Furthermore, a system with an even or odd number of bands will exhibit a significant difference in non-Abelian topological classification. To date, there has been scant research investigating even-band non-Abelian topological insulators. Here, we both theoretically explore and experimentally realize a four-band PT (inversion and time-reversal) symmetric system, where two new classes of topological charges as well as edge states are comprehensively studied. We illustrate their difference in the four-dimensional (4D) rotation sense on the stereographically projected Clifford tori. We show the evolution of the bulk topology by extending the 1D Hamiltonian onto a 2D plane and provide the accompanying edge state distributions following an analytical method. Our work presents an exhaustive study of four-band non-Abelian topological insulators and paves the way towards other even-band systems.

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