scholarly journals Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

2021 ◽  
pp. 231-238
Author(s):  
Edoardo Ballico
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Projective Geometry ◽  
General Position ◽  
Maximum Degree ◽  
Normal Curve ◽  
Dimensional Projective Space ◽  
Rational Normal Curve

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.

The geometry and algebra of the representations of the Lorentz group

1968 ◽  
Vol 303 (1474) ◽  
pp. 381-396 ◽  
Keyword(s):  
Projective Space ◽  
Lorentz Group ◽  
Point Of View ◽  
Normal Curve ◽  
Geometrical Representation ◽  
Spinor Space ◽  
Dimensional Projective Space ◽  
And Algebra ◽  
Rational Normal Curve ◽  
Formal Point

In this paper a (2j + l)-spinor analysis is developed along the lines of the 2-spinor and 3-spinor ones. We define generalized connecting quantities A μv (j) which transform like (j, 0) ⊗ (j -1, 0) in spinor space and like second rank tensors under transformations in space-time. The general properties of the A uv are investigated together with algebraic relations involving the Lorentz group generators, J μv . The connexion with 3j symbols is discussed. From a purely formal point of view we introduce a geometrical representation of a (2j +1)-spinor as a point in a 2j dimensional projective space. Then, for example, the charge con­jugate of a (2j + l)-spinor is just the polar of the corresponding point with respect to a certain rational, normal curve in the projective space. It is suggested that this representation will prove useful.

A Specialised Net of Quadrics Having Selfpolar Polyhedra, with Details of the Fivedimensional Example

1981 ◽  
Vol 33 (4) ◽  
pp. 885-892
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Complex Number ◽  
Roots Of Unity ◽  
Parametric Form ◽  
Normal Curve ◽  
Homogeneous Coordinates ◽  
Image Position ◽  
Rational Normal Curve

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number andso that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics(1.1)in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

scholarly journals On Absolute Points of Correlations of $\mathrm{PG}(2,q^n)$

10.37236/8920 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Jozefien D'haeseleer ◽  
Nicola Durante
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Final Section ◽  
Dimensional Vector ◽  
Sesquilinear Forms ◽  
Dimensional Vector Space ◽  
Sesquilinear Form ◽  
The Absolute ◽  
Dimensional Projective Space

Let $V$ be a  $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of  the $d$-dimensional projective space $\mathrm{PG}(V)$.  Everything is known in this case for both degenerate and non-degenerate reflexive forms if  $\mathbb{F}$  is either  ${\mathbb R}$, ${\mathbb C}$ or a finite field  ${\mathbb F}_q$.   In this paper we consider  degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones,  the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. In the final section we collect some  results from the huge work of B.C. Kestenband  regarding what is known for the set of  the absolute  points  of correlations in $\mathrm{PG}(2,q^n)$ induced  by a  non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.

On the uniqueness of (q + 1)-arcs of PG (5, q), q = 2h, h ≥ 4

1991 ◽  
Vol 110 (1) ◽  
pp. 91-94 ◽  
Author(s):  
Tatsuya Maruta ◽  
Hitoshi Kaneta
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Dimensional Projective Space

Throughout this paper q = 2h with h ≥ 4, and PG(r, q) stands for the r-dimensional projective space over the finite field GF(q) with q elements.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

scholarly journals Equations for point configurations to lie on a rational normal curve

2018 ◽  
Vol 340 ◽  
pp. 653-683
Author(s):  
Alessio Caminata ◽  
Noah Giansiracusa ◽  
Han-Bom Moon ◽  
Luca Schaffler
Keyword(s):  
Normal Curve ◽  
Point Configurations ◽  
Rational Normal Curve

scholarly journals MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350010 ◽  
Author(s):  
E. BALLICO ◽  
A. BERNARDI
Keyword(s):  
Normal Curve ◽  
Binary Forms ◽  
Minimal Decomposition ◽  
Tangential Projection ◽  
Rational Normal Curve

Let C ⊂ ℙn+1 be a rational normal curve and let X ⊂ ℙn be one of its tangential projection. We describe the X-rank of a point P ∈ ℙn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.

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