3-nets realizing a diassociative loop in a projective plane

A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around $3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$ of characteristic $p$, arose from algebraic geometry. It is not difficult to find $3$-nets in $PG(2,K)$ as far as $0<p\le n$. However, only a few infinite families of $3$-nets in $PG(2,K)$ are known to exist whenever $p=0$, or $p>n$. Under this condition, the known families are characterized as the only $3$-nets in $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with $3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$ but not by a group. We prove two structural theorems on $G$. As a corollary, if $G$ is commutative then every non-trivial element of $G$ has the same order, and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such $3$-nets

Light dual multinets of order six in the projective plane

The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures

Group-labeled light dual multinets in the projective plane (with Appendix)

In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length least $9$

On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in P^5

We study the degeneracy loci of general bundle morphisms from the direct sum of m copies of the structural sheaf on $P^n$ to $\Omega(2)$, also from the point of view of the classical geometrical interpretation of the sections of $\Omega(2)$ as linear line complexes. We consider in particular the case of $P^5$ with m=2, 3. For n=5 and m=3 we give an explicit description of the Hilbert scheme H of elliptic normal scrolls in $P^5$, by defining a natural rational map from the Grassmannian G(2,14) to H, which results to be dominant with general fibre of degree four.Comment: 17 pages, 1 figure, to be published in Geometriae Dedicat