 though I have a few scenarios I think are interesting. I think that’s what I’ll do. I keep of thinking of ideas and then settle on making the game on the most interesting one, not necessarily the one I think I can get done on time. How else am I supposed to improve if I don’t push myself? For, now, G’night!
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In recent years, nets realizing a finite group have been investigated in connection with complex line arrangements and resonance theory; see \cite{fy2007,knp_3,knp_k,miq,per,ys2004,ys2007}. The concept of a multinet was introduced by Falk and Yuzvinsky \cite{fy2007} as multi-arrangements of lines in the complex projective plane, with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. Korchmáros and Nagy \cite{KN_multinets} gave a more formal definition for a dual multinet of the projective plane, labeled by a quasigroup $Q$. Let $\mathbb{K}$ be a field, $Q$ a quasigroup and for $i=1,2,3$, let $\alpha_i:Q\to \mathrm{PG}(2,\mathbb{K})$ be maps such that the points $\alpha_1(x)$, $\alpha_2(y)$ and $\alpha_3(x\cdot y)$ are collinear for all $x,y\in Q$. Define the \textit{multisets} $\Lambda_i=\alpha_i(Q)$, $i=1,2,3$. Then $(\Lambda_1,\Lambda_2,\Lambda_3)$ is a \textit{dual multinet,} labeled by $Q$. If the maps $\alpha_i$ are injective and their images $\Lambda_i$ are disjoint, then the dual multinet is called \textit{light.} This terminology is in accordance with the one introduced by Bartz and Yuzvinsky \cite{Bartz2013,BartzYuz2014}. 

Let $(\Lambda_1,\Lambda_2,\Lambda_3)$ be a light dual multinet in $\mathrm{PG}(2,\mathbb{K})$ labeled by the quasigroup $Q$. As we show later in the abstract setting, if the line $\ell$ intersects two components $\Lambda_i,\Lambda_j$ then there is a integer $r$ such that $r=|\ell\cap \Lambda_1|=|\ell\cap \Lambda_2|=|\ell\cap \Lambda_3|$; this integer $r$ is called the \textit{length} of $\ell$ w.r.t. $(\Lambda_1,\Lambda_2,\Lambda_3)$. Examples of group-labeled light dual multinets were given by Bartz and Yuzvinsky \cite{Bartz2013,BartzYuz2014} and by Korchm\'aros and Nagy \cite{KN_multinets}. In \cite{BartzYuz2014}, the examples of light dual multinets are constructed from a three dimensional multinet. They are labeled by the dihedral group of order $n$ and the long lines have order $2$ or $n/2$. In \cite{KN_multinets} the authors define triangular light dual multinets of order $n$ which are contained in the union of three lines of length $n/3$, and tetrahedron and conic-line type light dual multinets, which have a unique line of length $n/2$. 

The motivation of the present paper is the following observation. The known constructions of light dual multinets and the results of \cite{KN_multinets} suggest that the length $r>1$ of lines of the light dual multinet makes a big difference in their geometric structure. While for $r\geq 9$, the light dual multinet is well structured in geometric and algebraic sense, the case of small $r$, especially $r=2$ shows many irregularities. In this paper, we classify all abstract light dual multinets of order $6$ with a unique line of length $r>1$. Moreover, we compute all possible realizations of these abstract light dual multinets in projective planes over fields of characteristic $0$. For this purpose, we present an algebraic framework to handle the occurring system of polynomial equations. The computation was done using computer algebra systems GAP \cite{GAP4}, Singular \cite{Singular} and SageMath \cite{Sage8.2}. 

The paper is organized as follows. In section 2, we give the basic definitions and terminology and present some simple results which are known and more or less folklore. In Section 3, we introduce the concept of an abstract light dual multinet $\Sigma$, labeled by a quasigroup $Q$ and explain the relation between long lines of $\Sigma$ and subsquares of $Q$. In Section 4, we classify all abstract light dual multinets of order $6$ having a unique line of length $r>1$. Section 5 presents the algebraic machinery for the study of weak embeddings of point-line incidence structures in the projective plane $\mathrm{PG}(2,\mathbb{K})$, where $\mathbb{K}$ is a field. In Sections 6 and 7 we apply this machinery to classify the weak projective embeddings of abstract light dual multinets of order $6$. In the appendices we give the SageMath \cite{Sage8.2} codes which compute the results.  





\section{Preliminaries}

\subsection{Quasigroups, isotopes, parastrophes}
A \textit{quasigroup} $(Q,\cdot)$ is a set endowed with a binary operation $x\cdot y$ such that the equation $x\cdot y=z$ can be uniquely solved if any two of the three values $x,y,z\in Q$ are given. One denotes the solutions with left and right division: $y=x\setminus z$, $x=z/y$. The maps $L_a:x\mapsto a\cdot x$, $R_a:x\mapsto x\cdot a$ are the \textit{left} and \textit{right multiplication maps} of $Q$. 

Let $(Q,\cdot)$ and $(R,\circ)$ be quasigroups. An \textit{isostrophism} from $Q$ to $R$ is a quadruple $(\sigma,\gamma_1,\gamma_2,\gamma_3)$, where $\gamma_1,\gamma_2,\gamma_3$ are bijective maps from $Q$ to $R$ and $\sigma \in S_3$ such that for all elements $x_1,x_2,x_3 \in Q$ the following holds: 
\[x_{\sigma(1)}\cdot x_{\sigma(2)}=x_{\sigma(3)} \Longleftrightarrow \gamma_1(x_1)\circ\gamma_2(x_2) = \gamma_3(x_2).\]
It is straightforward to see that being isostrophic is an equivalence
relation. If $\sigma$ is the identity then we speak of the \textit{isotopism} $(\gamma_1,\gamma_2,\gamma_3)$; for all $x,y \in Q$ 
\[\gamma_1(x)\circ\gamma_2(y)=\gamma_3(x\cdot y).\]
If the underlying sets $Q, R$ are equal and $\gamma_1,\gamma_2,\gamma_3$ are the identical maps than we say that the quasigroups $(Q,\cdot)$ and $(Q,\circ)$ are \textit{$\sigma$-conjugate}. We speak of a \textit{principal isotopism} if the underlying sets $Q,R$ are equal, $\sigma=\id$, $\alpha_1=R_v$, $\alpha_2=L_u$ and $\alpha_3=\id$ for some fixed elements $u,v\in Q$. Then for all $x,y\in Q$
\[x\circ y= x/u \cdot v\backslash y.\]
In this case, $(Q,\circ)$ is a loop with unit element $v\cdot u$. 

We remark that some authors use the term \textit{parastrophe} for isostrophe. In the language of Latin squares, one says that isostrophic quasigroups $Q$ and $R$ belong to the same \textit{main class.} 

\subsection{Subsquares in quasigroups}
While subquasigroups are defined in the obvious way, quasigroups have another important substructure which we call \textit{subsquares.} In \cite{DenesKeedwell},  subsquares of order $2$ are called \textit{intercalates. }

\begin{definition}
Let $(Q,\cdot)$ be a quasigroup and $S_1,S_2,S_3\subseteq Q$ such that
\[S_1\