Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Some big sets of mutually disjoint subgeometries of PG(d, q t )

In PG(d, q ^t ) we construct a set ? of mutually disjoint subgeometries isomorphic to PG(d, q) almost partitioning the point set of PG(d, q ^t ) such that there is a group of collineations of PG(d, q ^t ) operating simultaneously as a Singer cycle on all elements of ?. In PG(t?1,q ^t ) we construct big subsets ? of ? whose elements are far away from each other in the following sense:

u

? If P ₁, P ₂ ∈ ? _k , then no point of P ₁ lies on ak-dimensional subspace of P ₂.

For example, we get a set ofq - 1 subplanes of orderq of PG(2,q ³) such that no point of one subplane lies on a line of another subplane, and such that no three points of three different subplanes are collinear.     

Mixed partitions and related designs

We define a mixed partition of Π =  PG(d, q ^r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q ²) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q ²) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q ²ⁿ . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q ^r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r ² n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q ^rn ).     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

Conics and caps

In this article, we begin with arcs in PG(2, q ⁿ ) and show that they correspond to caps in PG(2n, q) via the André/Bruck?CBose representation of PG(2, q ⁿ ) in PG(2n, q). In particular, we show that a conic of PG(2, q ⁿ ) that meets ?_?? in x points corresponds to a (q ⁿ ?+?1 ? x)-cap in PG(2n, q). If x?=?0, this cap is the intersection of n quadrics. If x?=?1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.     

Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

We classify the cohomology classes of Lagrangian 4-planes ?⁴ in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = ?2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = ?γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ? of a line in a smooth Lagrangian n-plane ? ⁿ must satisfy (?,?) = ?(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.     

The embedding of (0, α)-geometries in PG(n, q): part II

In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U_2,3(9) in PG(5, 2).     

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.     

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q ^h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q ^h ).     

On small blocking sets and their linearity

We prove that a small minimal blocking set of PG(2,q) is “very close” to be a linear blocking set over some subfield GF(pe)<GF(q). This implies that (i) a similar result holds in PG(n,q) for small minimal blocking sets with respect to k-dimensional subspaces (0?k?n) and (ii) most of the intervals in the interval-theorems of Sz?nyi and Sz?nyi-Weiner are empty.     

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.     

Intersections of t-reguli,rational curves,and orthogonal Latin squares

The celebrated net-embedding theorem of R.H. Bruck asserts that a net with a large number of parallel classes, i.e., a net with small deficiency, can be embedded in an affine plane. The only known class of examples of unembeddable nets of small deficiency has been constructed by geometry, using partial spreads of PG(3, q). In this note we attempt to generalize that discussion to PG(n, q), with particular emphasis on the case n=5.     

Generalized Veronesean embeddings of projective spaces

We classify all embeddings θ: PG(n, q) → PG(d, q), with $d \geqslant \tfrac{{n(n + 3)}} {2}$d \geqslant \tfrac{{n(n + 3)}} {2}, such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ?n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.     

A Hamada type characterization of the classical geometric designs

The dimension of a combinatorial design ${{\mathcal D}}$ over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of ${{\mathcal D}}$ as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design ${{\mathcal D}}$ that has the same parameters as the complement of a classical point-hyperplane design PG _n-1(n, q) or AG _n-1(n, q) is greater than or equal to n + 1, with equality if and only if ${{\mathcal D}}$ is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PG _d (n, q), where 1 ≤ d ≤ n ? 1, in terms of associated codes defined over some extension field E?=?GF(q ^t ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AG _d (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG _d (n, q) for d = 1 and for d > (n ? 2)/2.     

Rectifier Circuits of Bounded Depth

Asymptotically tight bounds are obtained for the complexity of computation of the classes of (m, n)-matrices with entries from the set {0, 1,..., q ? 1} by rectifier circuits of bounded depth d, under some relations between m, n, and q. In the most important case of q = 2, it is shown that the asymptotics of the complexity of Boolean (m, n)-matrices, log n = o(m), logm = o(n), is achieved for the circuits of depth 3.     

A non‐existence result on Cameron–Liebler line classes

Cameron–Liebler line classes are sets of lines in PG(3, q) that contain a fixed number x of lines of every spread. Cameron and Liebler classified Cameron–Liebler line classes for x ∈ {0, 1, 2, q² ? 1, q², q² + 1} and conjectured that no others exist. This conjecture was disproven by Drudge for q = 3 [8] and his counterexample was generalized to a counterexample for any odd q by Bruen and Drudge [4]. A counterexample for q even was found by Govaerts and Penttila [9]. Non‐existence results on Cameron–Liebler line classes were found for different values of x. In this article, we improve the non‐existence results on Cameron–Liebler line classes of Govaerts and Storme [11], for q not a prime. We prove the non‐existence of Cameron–Liebler line classes for 3 ≤ x < q/2. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 342–349, 2008