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Corrado Zanella

Publications and source records attributed to Corrado Zanella.

29 records · Page 2Linked to original sources
arXiv2014

Subgeometries and linear sets on a projective line

We define the splash of a subgeometry on a projective line, extending the definition of \cite{BaJa13} to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank $r$, generalizing the definition from \cite{FaSz2006}, and show that clubs correspond to tangent splashes. We determine the condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent.

math.CO↗
arXiv2014

Subspaces intersecting each element of a regulus in one point, André-Bruck-Bose representation and clubs

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\cR$ of $(n-1)$-subspaces in $\PG(2n-1,q)$, with respect to the subgroup of $\PGL(2n,q)$ fixing $\cR$. Such results have consequences on several aspects of finite geometry. First of all, a necessary condition for an $(n-1)$-subspace $U$ and a regulus $\cR$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in \cite{BaJa12} on the André-Bruck-Bose representation of a $q$-subline in $\PG(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.

math.CO↗
arXiv2013

Geometry of the inversion in a finite field and partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the projective mapping induced by $x\mapsto x^{-1}$ have been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$, then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves of degree $2^k-1$. For smaller $q$ the same construction gives partitions in $(q+1)$-tuples of independent points.

math.CO↗
arXiv2013

Incidence and Combinatorial Properties of Linear Complexes

In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which maps $(h-1)$-subspaces to hyperplanes, satisfying suitable linearity and reciprocity properties. Generalised polarities with the null property give rise to a linear complexes and vice versa. Given that there exists for $h>1$ a linear complex of $h$-subspaces which contains no star --this seems to be an open problem over an arbitrary ground field --the combinatorial structure of a partition of the line set of the projective space into non-geometric spreads of its hyperplanes can be obtained. This line partition has an additional linearity property which turns out to be characteristic.

math.AG↗
arXiv2013

Lifting of divisible designs

The aim of this paper is to present a construction of $t$-divisible designs for $t>3$, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called $t$-lifting, is developed. It starts from a set $X$ containing a $t$-divisible design and a group $G$ acting on $X$. Then several explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a projective embedding of a Witt design. As a result, for any integer $t\geq 2$ infinitely many non-isomorphic $t$-divisible designs are found.

math.CO↗
arXiv2013

Divisible designs from twisted dual numbers

The generalized chain geometry over the local ring $K(ε;σ)$ of twisted dual numbers, where $K$ is a finite field, is interpreted as a divisible design obtained from an imprimitive group action. Its combinatorial properties as well as a geometric model in 4-space are investigated.

math.CO↗
arXiv2013

On automorphisms of flag spaces

We show that the automorphisms of the flag space associated with a 3-dimensional projective space can be characterized as bijections preserving a certain binary relation on the set of flags in both directions. From this we derive that there are no other automorphisms of the flag space than those coming from collineations and dualities of the underlying projective space. Further, for a commutative ground field, we discuss the corresponding flag variety and characterize its group of automorphic collineations.

math.AG↗
arXiv2013

On linear morphisms of product spaces

Let $χ$ be a linear morphism of the product of two projective spaces $PG(n,F)$ and $PG(m,F)$ into a projective space. Let $γ$ be the Segre embedding of such a product. In this paper we give some sufficient conditions for the existence of an automorphism $α$ of the product space and a linear morphisms of projective spaces $ϕ$, such that $γϕ=αχ$.

math.AG↗
arXiv2013

On embedded products of Grassmannians

Let $Γ'$ and $Γ$ be two Grassmannians. The standard embedding $ϕ:Γ'\timesΓ\to \bar{P}$ is obtained by combining the Plücker and Segre embeddings. Given a further embedding $η: Γ'\timesΓ\to P'$, we find a sufficient condition for the existence of $α\in Aut(Γ)$ and of a collineation $ψ: \bar{P} \to P'$ such that $η=({\rm id}_{Γ'}\timesα)ϕψ$.

math.AG↗
arXiv2012

Quadratic embeddings

The quadratic Veronese embedding $ρ$ maps the point set $P$ of $\PG{n,F)$ into the point set of $PG({n+2 \choose 2}-1, F$ ($F$ a commutative field) and has the following well-known property: If $M\subset P$, then the intersection of all quadrics containing $M$ is the inverse image of the linear closure of $M^ρ$. In other words, $ρ$ transforms the closure from quadratic into inear. In this paper we use this property to define "quadratic embeddings". We shall prove that if $ν$ is a quadratic embedding of $PG{n,F)$ into $PG(n',F')$ ($F$ a commutative field), then $ρ^{-1}ν$ is dimension-preserving. Moreover, up to some exceptional cases, there is an injective homomorphism of $F$ into $F'$. An additional regularity property for quadratic embeddings allows us to give a geometric characterization of the quadratic Veronese embedding.

math.AG↗