Eva Ferrara Dentice

Bose-Burton type theorems for finite Grassmannians

In this paper both blocking sets with respect to the s-subspaces and covers with t-subspaces in a finite Grassmannian are investigated, especially focusing on geometric descriptions of blocking sets and covers of minimum size. When such a description exists, it is called a Bose-Burton type theorem. The canonical example of a blocking set with respect to the s-subspaces is the intersection of s linear complexes. In some cases such an intersection is a blocking set of minimum size, that can occasionally be characterized by a Bose-Burton type theorem. In particular, this happens for the 1-blocking sets of the Grassmannian of planes of PG(5,q).

A combinatorial characterization of the Klein quadric

We give a combinatorial characterization of the Klein quadric in terms of its incidence structure of points and lines. As an application, we obtain a combinatorial proof of a result of Havlicek.

Classification of Veronesean caps

In this paper all Veronesean caps of projective spaces of finite dimension over skewfields are classified. More precisely, if PG(M,K), K a skewfield, contains a Veronesean cap X, then K is a field and X is either a Veronese variety or a projection of a Veronese variety. This result extends analogous theorems of Mazzocca and Melone [Caps and Veronese varieties in projective Galois spaces. Discrete Math. 48 (1984) 243-252] and Thas and Van Maldeghem [Classification of finite Veronesean caps, European J. Combin. 25(2) (2004) 275-285] for finite projective spaces.

On finite matroids with one more hyperplane than points

In this paper a complete classification of finite matroids with one more hyperplane than points is obtained.

On finite matroids with two more hyperplanes than points

One of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of Basterfield, Kelly and Green (1968/1970) (J.G. Basterfield, L.M. Kelly, A characterization of sets of n points which determine n hyperplanes, in: Proceedings of the Cambridge Philosophical Society, vol. 64, 1968, pp. 585-588; C. Greene, A rank inequality for finite geometric lattices, J. Combin Theory 9 (1970) 357-364) affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented.

On the Incidence Geometry of Grassmann Spaces

In this paper we identify some properties on the point-line structure of Grassmannians which are useful tools to characterize the incidence geometry of Grassmann varieties and of their special quotients.

Embedding the planar structure of 4-dimensional linear spaces into projective spaces

It is known that a linear spaces of dimensiond has at least as many hyperplanes as points with equality if it is a (possibly degenerate) projective space. If there are only a few more hyperplanes than points, then the linear space can still be embedded in a projective space of the same dimension. But even if the difference between the number of hyperplanes and points is too big to ensure an embedding, it seems likely that the linear space is closely related to a projective space. We shall demonstrate this in the cased=4.