Hermitian Surface Research Articles

Two projectively generated subsets of the Hermitian surface

Two projectively generated subsets of the Hermitian surface

The weight distribution of the functional codes defined by forms of degree 2 on Hermitian surfaces

On etudie le code fonctionnel C 2 (X) defini sur une variete algebrique projective X, dans le cas ou X ⊂ P 3 (F q ) est une surface Hermitienne non-degeneree. Nous donnons d'abord des bornes pour #X z(Q) (F q ) meilleures que celles connues. Ensuite nous calculons le nombre de mots de code atteignant le second poids. Nous donnons aussi une estimation exacte du troisieme poids, une description de la structure geometrique des mots correspondant, ainsi que leur nombre. L'article s'acheve par une conjecture formulee sur les quatrieme et cinquieme poids du code C 2 (X).

Some constructions on the Hermitian surface

In the geometric setting of commuting orthogonal and unitary polarities we construct an infinite family of complete (q + 1)2---spans of the Hermitian surface $${\mathcal {H}(3, q^2)}$$ , q odd. A construction of an infinite family of minimal blocking sets of $${\mathcal {H}(3, q^2)}$$ , q odd, admitting PSL 2(q), is also provided.

Orthogonal complex structures on domains in $${\mathbb {R}^4}$$

An orthogonal complex structure on a domain in \({\mathbb {R}^4}\) is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in \({\mathbb {CP}^3}\) under the action of the conformal group SO°(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in \({\mathbb {R}^4}\) .

Computing the 2-distribution of points on Hermitian surfaces (abstract only)

Computing the 2-distribution of points on Hermitian surfaces (abstract only)

On the intersection of Hermitian surfaces

On the intersection of Hermitian surfaces

On the intersection of Hermitian curves and of Hermitian surfaces

Kestenband [Unital intersections in finite projective planes, Geom. Dedicata 11(1) (1981) 107–117; Degenerate unital intersections in finite projective planes, Geom. Dedicata 13(1) (1982) 101–106] determines the structure of the intersection of two Hermitian curves of PG ( 2 , q 2 ) , degenerate or not. In this paper we give a new proof of Kestenband's results. Giuzzi [Hermitian varieties over finite field, Ph.D. Thesis, University of Sussex, 2001] determines the structure of the intersection of two non-degenerate Hermitian surfaces H and H ′ of PG ( 3 , q 2 ) when the Hermitian pencil defined by H and H ′ contains at least one degenerate Hermitian surface. We give a new proof of Giuzzi's results and we obtain some new results in the open case when all the Hermitian surfaces of the Hermitian pencil are non-degenerate.

Hermitian indicator sets

Abstract Indicator sets arising from symplectic line-spreads of a 3-dimensional projective space are characterized. Moreover, locally Hermitian ovoids of a Hermitian surface (3, q 2) are studied in terms of “non-equivalent” associated indicator sets. Translation locally Hermitian ovoids of (3, q 2) arising from regular spreads of PG(3, q), Kantor semifield spreads of PG(3, q) and semifield spreads of PG(3, q) with center are also investigated.

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a Kahler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not Kahler.

Special sets of the Hermitian surface and indicator sets

AbstractAn interesting connection between special sets of the Hermitian surface of PG(3,q2), q odd, (after Shult 13) and indicator sets of line‐spreads of the three‐dimensional projective space is provided. Also, the CP‐type special sets are characterized. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 18–24, 2008

Spreads of PG(3, q) and ovoids of polar spaces

To any spread 𝒮 of PG (3, q ) corresponds a family of locally hermitian ovoids of the Hermitian surface H (3, q 2 ), and conversely; if in addition 𝒮 is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H (3, q 2 ) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed in [Cossidente A., Ebert G., Marino G., Siciliano A.: Shult Sets and Translation Ovoids of the Hermitian Surface. Adv. Geom. 6 (2006), 475–494] as the only translation ovoid of H (3, q 2 ) whose translation group is abelian. If 𝒮 is a spread of PG (3, q ) and 𝒪(𝒮) is one of the associated ovoids of H (3, q 2 ), then using the duality between H (3, q 2 ) and Q − (5, q ), another spread of PG (3, q ), say 𝒮 2 , can be constructed. On the other hand, using the Barlotti-Cofman representation of H (3, q 2 ), one more spread of a 3-dimensional projective space, say 𝒮 1 , arises from the ovoid 𝒪(𝒮). In [Lunardon G.: Blocking sets and semifields. J. Comb. Theory Ser. A to appear] some questions are posed on the relations among 𝒮, 𝒮 1 and 𝒮 2 ; here we prove that 𝒮 and 𝒮 2 are isomorphic and the ovoids 𝒪(𝒮) and 𝒪(𝒮 1 ), corresponding to 𝒮 and 𝒮 1 respectively under the Plücker map, are isomorphic.

Toric Hermitian surfaces and almost Kähler structures

The aim of this paper is to investigate the class of compact Hermitian surfaces $(M,g,J)$ admitting an action of the 2-torus $T^2$ by holomorphic isometries. We prove that if $b_1(M)$ is even and $(M,g,J)$ is locally conformally Kähler and $\chi (M)\not =

Elation groups of the Hermitian surface H(3,q2) over a finite field of characteristic 2

Let S=(P,ℬ,ℐ) be a finite generalized quadrangle having order (s,t). Let p be a point of S. A whorl about p is a collineation of S fixing all the lines through p. An elation about p is a whorl that does not fix any point not collinear with p, or is the identity. If S has an elation group acting regularly on the set of points not collinear with p we say that S is an elation generalized quadrangle with base point p. The following question has been posed: Can there be two non-isomorphic elation groups about the same point p? In this presentation, we show that there are exactly two (up to isomorphism) elation groups of the Hermitian surface H(3,q2) over a finite field of characteristic 2.

A Geometric Construction for Some Ovoids of the Hermitian Surface

Multiple derivation of the classical ovoid of the Hermitian surface \(\mathcal{H}(3,q^2)\) of \(PG(3,q^2)\) is a well known, powerful method for constructing large families of non classical ovoids of \(\mathcal{H}(3,q^2)\). In this paper, we shall provide a geometric costruction of a family of ovoids amenable to multiple derivation.

Derivation techniques on the Hermitian surface

AbstractWe discuss derivation‐like techniques for transforming one locally Hermitian partial ovoid of the Hermitian surface H(3,q2) into another one. These techniques correspond to replacing a regulus by its opposite in some naturally associated projective 3‐space PG(3,q) over a square root subfield. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 478–486, 2007

Shult sets and translation ovoids of the Hermitian surface

Abstract Starting with carefully chosen sets of points in the Desarguesian affine plane AG(2; q 2) and using an idea first formulated by E. Shult, several infinite families of translation ovoids of the Hermitian surface are constructed. Various connections with locally Hermitian 1-spreads of 𝒬−(5, q) and semifield spreads of PG(3, q) are also discussed. Finally, geometric characterization results are developed for the translation ovoids arising in the so-called classical and semiclassical settings.

On the intersection of Hermitian surfaces

We provide a description of the configuration arising from intersection of two Hermitian surfaces in PG(3, q), provided that the linear system they generate contains at least a degenerate variety.

Codes defined by forms of degree 2 on Hermitian surfaces and Sørensen's conjecture

We study the functional codes C h ( X ) defined by Lachaud in [G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in: Arithmetic, Geometry, and Coding Theory, Luminy, France, 1993, de Gruyter, Berlin, 1996, pp. 77–104] where X ⊂ P N is an algebraic projective variety of degree d and dimension m. When X is a Hermitian surface in PG ( 3 , q ) , Sørensen in [A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for h ⩽ t (where q = t 2 ) the following result: # X Z ( f ) ( F q ) ⩽ h ( t 3 + t 2 − t ) + t + 1 which should give the exact value of the minimum distance of the functional code C h ( X ) . In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. h = 2 ), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight.

Special sets of the Hermitian surface and indicator sets

A connection between special sets of the Hermitian surface of PG(3,q2), q odd, and indicator sets of line–spreads of the 3–dimensional projective space is provided. Also, the CP–type special sets are characterized.

Hermitian manifolds with flat associated connection

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kahler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kahler manifold of constant holomorphic sectional curvatures.