On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P⁶³ where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.     

Partial Flocks of Non-Singular Quadrics in PG(2r + 1, q)

We generalise the definition and many properties of partial flocks of non-singular quadrics in PG(3, q) to partial flocks of non-singular quadrics in PG(2r + 1, q).     

The small weight codewords of the functional codes associated to non-singular Hermitian varieties

This article studies the small weight codewords of the functional code C _Herm (X), with X a non-singular Hermitian variety of PG(N, q ²). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q ²) consisting of q + 1 hyperplanes through a common (N ? 2)-dimensional space Π, forming a Baer subline in the quotient space of Π. The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C ₂(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729–1739, 2010), and C ₂(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27–35, 2010).     

Higher Weights of Anticodes and the Generalized Griesmer Bound

We show that the minimum r-weight d_r of an anticode can be expressed in terms of the maximum r-weight of the corresponding code. As examples, we consider anticodes from homogeneous hypersurfaces (quadrics and Hermitian varieties). In a number of cases, all differences (except for one) of the weight hierarchy of such an anticode meet an analog of the generalized Griesmer bound.     

A Matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real Associative Quadrics

In this paper we describe the holomorphic automorphisms for two infinite series of Hermitian quadrics: quadrics of real co-dimension 2 in ℂⁿ⁺² and a special class of quadrics of co-dimension n in ℂ²ⁿ with large automorphism groups (Real Associative Quadrics). We give explicit formulas of the automorphisms. They are rational maps of degree not exceeding the co-dimension.     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

A characterization of quadrics by intersection numbers

This work is inspired by a paper of Hertel and Pott on maximum non-linear functions (Hertel and Pott, A characterization of a class of maximum non-linear functions. Preprint, 2006). Geometrically, these functions correspond with quasi-quadrics; objects introduced in De Clerck et al. (Australas J Combin 22:151–166, 2000). Hertel and Pott obtain a characterization of some binary quasi-quadrics in affine spaces by their intersection numbers with hyperplanes and spaces of codimension 2. We obtain a similar characterization for quadrics in projective spaces by intersection numbers with low-dimensional spaces. Ferri and Tallini (Rend Mat Appl 11(1): 15–21, 1991) characterized the non-singular quadric Q(4,q) by its intersection numbers with planes and solids. We prove a corollary of this theorem for Q(4,q) and then extend this corollary to all quadrics in PG(n,q),n ≥ 4. The only exceptions occur for q even, where we can have an oval or an ovoid as intersection with our point set in the non-singular part.      

Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

Partitions and their stabilizers for line complexes and quadrics

Summary It is shown that PG(2Nr −1, q) can be partitioned by totally isotropic PG(r −1, q) of a non-singular line complex. The stabilizer in PSp_2Nr(q) of the spread given is identified, and its geometric action is discussed. Using this partition and the various inter-relations of quadrics, line complexes and their groups when q is even we obtain various orbits of partitions of quadrics over GF(2^α) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell's heptagons for the Klein quadric in PG(5, 2). Entrata in Redazione il 14 giugno 1976.     

Maximal sets of non-polar points of quadrics and symplectic polarities over GF(2)

A cap on a non-singular quadric over GF(2) is a set of points that are pairwise non-polar; equivalently the join of any two of the points is a chord. A non-secant set of the quadric is a set of points off the quadric that are pairwise non-polar; equivalently the join of any two of the points is skew to the quadric. We determine all the maximal caps and all the maximal non-secant sets of all non-singular quadrics over GF(2); and also all the maximal sets of non-polar points for symplectic polarities over GF(2). The classification is in terms of caps of greatest size on elliptic quadrics Q _– ^8k+3 (2), hyperbolic quadrics Q ₊ ^8k+7 (2) and on quadrics Q _4k+2(2), and of non-secant sets of greatest size of Q _– ^8k+1 (2), Q ₊ ^8k+5 (2) and Q _4k (2), for all quadrics of these types that occur as sections of the parent quadric or belong to the symplectic polarity. The sets of greatest size for these types of quadrics are larger than for other types. The results have implications about the non-existence of ovoids and the exterior sets of Thas. Only one part of the simple geometric inductive argument extends to larger ground fields.     

Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

To characterize Hermitian varieties in projective space PG(d, q) of d dimensions over the Galois field GF(q), it is necessary to find those subsets K for which there exists a fixed integer n satisfying (i) 3 ? n ? q ? 1, (ii) every line meets K in 1, n or q + 1 points. K is called singular or non-singular as there does or does not exist a point P for which every line through P meets K in 1 or q + 1 points. For q odd, a non-singular K is a non-singular Hermitian variety (M. Tallini Scafati “Caratterizzazione grafica delle forme hermitiane di un S_{r, q}” Rend. Mat. Appl.26 (1967), 273–303). For q even, q > 4 and d = 3, a non-singular K is a Hermitian surface or “looks like” the projection of a non-singular quadric in PG(4, q) (J.W.P. Hirschfeld and J.A. Thas “Sets of type (1, n, q + 1) in PG(d, q)” to appear). The case q = 4 is quite exceptional, since the complements of these sets K form a projective geometry code, a (21, 11) code for d = 2 and an (85, 24) code for d = 3. The full list of these sets is given.     

Structure of functional codes defined on non-degenerate Hermitian varieties

We study the functional codes of order h defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by Edoukou (2009) [8]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first 2h+1 weights.     

Calabi's diastasis function for Hermitian symmetric spaces

In this paper we study the Calabi diastasis function of Hermitian symmetric spaces. This allows us to prove that if a complete Hermitian locally symmetric space (M,g) admits a Kähler immersion into a globally symmetric space (S,G) then it is globally symmetric and the immersion is injective. Moreover, if (S,G) is symmetric of a specified type (Euclidean, noncompact, compact), then (M,g) is of the same type. We also give a characterization of Hermitian globally symmetric spaces in terms of their diastasis function. Finally, we apply our analysis to study the balanced metrics, introduced by Donaldson, in the case of locally Hermitian symmetric spaces.     

Quadriques hermitiennes et ensembles de classe (0, 1, n,q+1) de PG(d,q)

This paper is a first step in the characterization of finite Hermitian quadrics as sets of class (0, 1, n, q+1) in PG(d, q).     

A characterization of the non-Baer lines of a Hermitian surface

We give a characterization of the set of the lines which either belong to or are tangent to a non-singular Hermitian surface in the projective space of dimension 3 and order q ².     

Real four-dimensional M-triquadrics

Nonsingular maximal intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensionalM-triquadrics. The dimensions of their cohomology spaces with coefficients in the field of two elements are calculated.     

Gröbner bases for spaces of quadrics of codimension 3

Let R=⊕_i≥0R_i be an Artinian standard graded K-algebra defined by quadrics. Assume that dimR₂≤3 and that K is algebraically closed, of characteristic ≠2. We show that R is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by K[x,y,z]/I where I is a complete intersection of three quadrics not containing a square of a linear form.     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .