 lawmakers they wanted to work with them on ways to address the issue of gun control, offered little in the way of proposals.
Lawrence Keane, senior vice president and general counsel of the Newtown-based National Shooting Sports Foundation, said that gun manufacturing in Connecticut is a $1.75-billion industry with 7,300 jobs.
Keane said the organization has been working to keep guns out of the hands of people like Lanza.
Kevin Reid, of Oxford, general counsel for Sturm, Ruger & Co., the gun manufacturer with headquarters in Fairfield, said that while the company does not make guns in Connecticut, it use state subcontractors.
"We pump tens of millions of dollars each year into the Connecticut economy," Reid said.
Senate President Pro Tempore Donald E. Williams Jr., D-Brooklyn, replied that Connecticut firearms makers can -- and should -- make weapons safer.
"You may work with us to improve the safety of these firearms so we can prevent the unauthorized access of these firearms," Williams said.
"The reality is the technology is not mature enough," Keane said.
During morning news conferences, gun manufacturers said they want to be part of the discussion leading to new laws, while a state teachers union said 85 percent of its members oppose arming teachers.
On Tuesday at 2 p.m., the legislative working group on mental health issues will hold a public hearing at 2 p.m. in Room 2-C of the Legislative Office Building.This series for pre-schoolers in the Play School tradition, with stories, songs and puppets, aired in the Watch with Mother lunchtime slot.
Presented by Maggie Henderson and Fred Harris, Ragtime featured a number of puppets – including Mr Porridge (a wooden spoon), Mrs Ragamuffin (another wooden spoon) and Uncle Casserole – who all emerged from the green Ragtime Bag.
Other “regulars” included Dax and Sniff (puppet dogs), Humbug the tiger (a sock puppet) and Bubble – a cushion-shaped toy with a “think bubble” above his head who spoke in rhyme.
\section{Introduction}\label{Sec:Intro}
Cluster algebras are commutative $\ZZ$--subalgebras of the field of rational functions in a finite number of indeterminates which have been introduced and studied by S.~Fomin and A.~Zelevinsky in a series of papers \cite{FZI}, \cite{FZII}, \cite{BFZIII} and \cite{FZIV}. To every quiver $Q$ without loops and $2$--cycles it is associated a coefficient--free cluster algebra $\myAA_Q$. In \cite{CC}, \cite{CK1}, \cite{CK2} and \cite{DWZII} the authors describe the cluster variables of $\myAA_Q$ via a map, called the Caldero--Chapoton map,   between the representations of $Q$  and the field of rational functions in $n$ variables  (we address the reader to the survey \cite{keller-2008}). Such map is defined in terms of Euler--Poincar\'e characteristic of some complex projective varieties attached to every representation $M$ of $Q$ and called quiver Grassmannians. By definition the quiver Grassmannian $Gr_{\mathbf{e}}(M)$ consists of all sub--representations of $M$ of dimension vector $\mathbf{e}$. These varieties are considered in several places, e.g. \cite{CK1}, \cite{CK2}, \cite{CR}, \cite[section~12.3]{hernandez}, \cite{Nakajima}, and in this paper we try to add some more geometric information about them, at least in the case of the Kronecker quiver. In \cite{CZ} authors compute the Euler--Poincar\'e characteristic of quiver Grassmannians associated with the Kronecker quiver and they conjecture the existence of a cellular decomposition which we find here. In \cite{String} a torus action on some quiver Grassmannians has been found and this allows to produce a cellular decomposition of them in the case they are smooth.

In this paper we study quiver Grassmannians associated with the Kronecker quiver. A representation of the Kronecker quiver (in the sequel we say a $Q$--representation) is a quadruple $M=(M_1, M_2,m_a,m_b)$ where $M_1$ and $M_2$ are finite dimensional complex vector spaces and $m_a,m_b:M_1\rightarrow M_2$ are two linear maps between them. Given two non--negative integers $e_1$ and $e_2$, the variety $Gr_{(e_1,e_2)}(M)$ is defined as the set
$$
\{(N_1,N_2)\in Gr_{e_1}(M_1)\times Gr_{e_2}(M_2):\; m_a(N_1)\subset N_2,\;m_b(N_1)\subset N_2\}
$$
where $Gr_e(V)$ denotes the Grassmannians of $e$--dimensional vector spaces in a vector space $V$. This is a projective variety which is in general not smooth. When $M_1=M_2=\mathbf{C}^n$ and $m_a=Id$ is the identity and $m_b=J_n(0)$ is an indecomposable nilpotent Jordan block, the representation $M$ is regular indecomposable and we denote it by $R_n$. The corresponding quiver Grassmannians $Gr_{(e_1,e_2)}(R_n)$ are the main subjects of this paper. We briefly write $X$ for one of these quiver Grassmannians. We find that a one--dimensional torus $T$ acts on $X$ (section~\ref{Sec:ActionOnX}). We provide a stratification of $X$ (see section~\ref{Sec:Stratification})
\begin{equation}\label{Eq:StratificationIntro}
X=X_0\supseteq X_1\supseteq\cdots \supseteq X_s
\end{equation}
for $s=\textrm{min}(e_1,n-e_2)$ into closed subvarieties $X_k \simeq Gr_{(e_1-k,e_2-k)}(R_{n-2k})$. Moreover $X_{k+1}$ is the singular locus of $X_k$ and the difference $X_k\setminus X_{k+1}$ is  a smooth quasi--projective variety which is not complete. As a consequence of the stratification \eqref{Eq:StratificationIntro} we get that $X$ is smooth if and only if $s=0$, i.e. either $e_1=0$ or $e_2=n$, in which cases the quiver Grassmannian specializes to an usual Grassmannian of vector subpaces.

In section~\ref{Sec:CellDec} we prove that Bia{\l}ynicki-Birula's theorem on cellular decomposition of smooth projective varieties applies to $X_k\setminus X_{k+1}$ and we can hence prove that $X$ decomposes
$$
X=\cup_{L\in X^T}X_L
$$
into attracting sets of $T$--fixed points
$$
X_L:=\{N\in X| \lim_{\lambda\rightarrow 0}t_\lambda N=L\}
$$
and theses sets are affine spaces. In section~\ref{Sec:Cells} we describe the cell $X_L$: if $L$ is indecomposable then $X_L$ is the orbit of $L$ by the action of a  unipotent group; if $L=L'\oplus L''$ with $L'$ of ``lower weight than'' $L''$ then the following formula holds:
$$
\textrm{dim}X_{L'\oplus L''}=\textrm{dim} X_{L'}+\textrm{dim}X_{L'}-\langle\mathbf{dim}L',\mathbf{dim}L''\rangle
$$
where $\langle\cdot,\cdot\rangle$ denotes the Euler form of the Kronecker quiver. As a consequence of this formula we are able to compute the Poincar\'e polynomials of every indecomposable representation
of the Kronecker quiver: let $P_n$ (resp. $I_n$) be the indecomposable preprojective of dimension $(n,n+1)$ (resp. $(n+1,n)$). We have (see section~\ref{Sec:Betti})
\begin{equation}\label{Eq:IntroPoincare}
\begin{array}{ccc}
P_{Gr_{\mathbf{e}}(R_n)}(t)&=&P_{Gr_{(e_2-e_1)}(e_2)}(t)P_{Gr_{(e_2-e_1)}(n-e_1)}(t)\\
P_{Gr_\mathbf{e}(P_n)}(t)&=&P_{Gr_{e_1}(e_2-1)}(t)P_{Gr_{(e_2-e_1)}(n+1-e_1