text:
\section{Introduction}

One of the most important problems in complex geometry is to understand the relationship between the existence of certain canonical metrics and algebro-geometric notions of stability. For vector bundles, the fundamental result is that a vector bundle admits a Hermite-Einstein metric if and only if it is slope polystable. Moreover, when the vector bundle is unstable, it admits a unique Harder-Narasimhan filtration of subsheaves such that each quotient is stable.

The analogous question for polarised varieties $(X,L)$ is the question of existence of constant scalar curvature K\"ahler (cscK) in the first Chern class of $L$. Here the notion of stability is K-polystability, and the Yau-Tian-Donaldson conjecture states that $(X,L)$ admits a cscK metric if and only if it is K-polystable \cite{SD3}. Loosely speaking, one assigns a certain weight (called the Donaldson-Futaki invariant) to each flat degeneration (called a test configuration) of $(X,L)$; K-semistability requires that this weight is non-negative for every test configuration. This is a weaker notion than K-polystability, and it is known, for example, that when $X$ is smooth that the existence of a cscK metric implies K-semistability \cite{SD}. The Yau-Tian-Donaldson conjecture has recently been proven when $X$ is Fano with $L=-K_X$, so that the metric is K\"ahler-Einstein \cite{RB,CDS,GT}.

Through work of Witt Nystr\"om \cite{DWN} and Sz\'ekelyhidi \cite{GS,ICM}, one can interpret test configurations as filtrations of the co-ordinate ring. This is reviewed in Section \ref{Preliminars}. The approach to K-polystability via filtrations has at least two advantages. The first is conceptual: by including filtrations with non finitely generated Rees algebra the notion of K-polystability is enhanced; this is discussed in \cite{ICM}. A classical example where non-finitely generated filtrations naturally occur is Zariski's example, as presented in \cite[Section 5.5]{Z}. The second is more practical: in some situations, it is easier to produce and describe examples of filtrations rather than of test configurations.

A natural geometric situation in which $(X,L)$ admits no cscK metric is when $X$ is smooth and $\Aut(X,L)$ is non-reductive \cite{YM}. Therefore, in this case, one would expect that such $(X,L)$ is not K-polystable. The goal of the present work is to associate a unique filtration to the co-ordinate ring of a variety with non-reductive automorphism group. In all examples, this filtration has strictly negative Donaldson-Futaki invariant, hence it destabilises the variety.

\begin{theorem} Let $(X,L)$ be a polarised variety. Then there exists a canonical $\Aut(X,L)$-equivariant filtration of its  co-ordinate ring, called the Loewy filtration, which is trivial if and only if $\Aut(X,L)$ is reductive. \end{theorem}

This leads us to the following conjecture.

\begin{conjecture}\label{introconjecture} Suppose $(X,L)$ is a polarised variety with non-reductive automorphism group. Then the Loewy filtration destabilises $(X,L)$. \end{conjecture}

A motivation for studying Conjecture \ref{introconjecture} is its relation to the proof of the Yau-Tian-Donaldson conjecture. One of the main technical steps in Chen-Donaldson-Sun's proof that a K-polystable Fano manifold admits a K\"ahler-Einstein metric was to show, using analytic methods, that if a Kawamata log terminal Fano variety admits a K\"ahler-Einstein metric, then its automorphism group is reductive \cite[Theorem 4]{CDS3}. It is known, however, that the existence of a K\"ahler-Einsten metrics on such singular Fano varieties implies K-polystability \cite{RB}. A proof of Conjecture \ref{introconjecture} would therefore give an algebraic proof of Chen-Donladson-Sun's result. It is natural to expect that this would be important more generally in any attempt to show that K-polystability of a polarised variety implies the existence of a constant scalar curvature K\"ahler metric.

Another motivation for this work is to give a new method of destabilising varieties. All previous destabilising test configurations have either arisen from holomorphic vector fields, or used a particularly simple flat degeneration of the variety, namely deformation to the normal cone \cite{RT}. Stability with respect to such test configurations is called slope stability, and it is known that slope stability is not equivalent to K-polystability. For example, the blow-up of $\mathbb{P}^2$ at two points is slope stable, but admits no cscK metric \cite{PR}. Our method gives an algebro-geometric proof that the blow-up $\mathbb{P}^2$ at two points is K-unstable.

The Donaldson-Futaki invariant associated to the filtration we construct can be calculated in many explicit examples, when the geometry of the variety is well understood.

\begin{theorem}\label{intro_examples}The Donaldson-Futaki invariant of the Loewy filtration is strictly negative in the following cases:
\begin{itemize}
\item[(i)] $\mathbb{P}^2$ blown up at one point with respect to all polarisations.
\item[(ii)] $\mathbb{P}^2$ blown up at n points on a line, with $L=a H - b(E_1 \hdots + E_n)$.
\item[(iii)] The Hirzebruch surfaces with respect to all polarisations.
\item[(iv)] Some projective bundles over $\mathbb{P}^1$ with respect to all polarisations.
\item[(v)] A projective bundle over $\mathbb{P}^2$ with respect to the anti-canonical polarisation.
\item[(vi)] An orbifold del Pezzo surface with respect to the anti-canonical polarisation.
\end{itemize}
\end{theorem}
These varieties are described in details in Section \ref{examples}, where the computation of the Donaldson-Futaki invariant is carried out. It follows that none of the above polarised varieties admit constant scalar curvature K\"ahler metrics. While we conjecture that our filtration is destabilising in general, the proof of this seems out of reach at present.

\

\noindent {\bf Notation and conventions:} We often use the same letter to denote a divisor and the associated line bundle, and mix multiplicative and additive notation for line bundles. A polarised variety $(X,L)$ is a normal projective variety $X$ together with an ample line bundle $L$. Our results are independent of scaling $L$, as such we sometimes assume that $L$ is very ample and projectively normal.

\

\noindent {\bf Acknowledgements:} This project started during the school ``Minicourses on Stability'' at the University of Coimbra; we thank the organisers for the stimulating environment. Part of the work was done during a visit of both authors to the Department of Mathematics of the University of Pavia, funded by the ERC StG 307119 - StabAGDG; we thank Jacopo Stoppa for the invitation. We would like to thank Giovanni Cerulli Irelli, Jesus Martinez Garcia, Julius Ross, Roberto Svaldi and Filippo Viviani for useful discussions. Both authors would especially like to thank David Witt Nystr\"om and Jacopo Stoppa for several discussions on the present work.

GC was funded by the grant FIRB 2012 ``Moduli Spaces and Their Applications''. RD was funded by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1).
\section{Preliminaries on K-polystability}\label{Preliminars}

\begin{subsection}{Filtrations and K-semistability}
Let $(X,L)$ be a polarised variety. We are interested in the algebro-geometric concept of K-polystability; to define it, we associate to each filtration of the co-ordinate ring of $(X,L)$ a weight called the Donaldson-Futaki invariant. In this section we proceed in a purely algebraic way. In Section \ref{testconfigurations}, we will explain, following \cite{GS,ICM}, how to describe these concepts in a more geometric, and perhaps more familiar, language.

\begin{definition}\cite{DWN}\label{decreasingfiltration}
Denote the  co-ordinate ring of $(X,L)$ by $$R(X,L) = \oplus_{k \geq 0} R_k = \oplus_{k \geq 0} H^0(X,L^k).$$  We define a \emph{decreasing filtration} $\mathcal{F}$ of $R$ to be sequence of vector subspaces
$$\cdots \supset \mathcal{F}_i R \supset \mathcal{F}_{i+1} R \supset \cdots$$
which is
\begin{itemize}
\item[(i)] \emph{linearly right bounded}: there exists a constant $C$ such that $\mathcal{F}_{Ck} R_k = \{0\}$ for every $k$,
\item[(ii)] \emph{pointwise left bounded}: for every $k$ there exists a $j=j(k)$ such that $\mathcal{F}_j R_k = H^0(X,L^k)$,
\item[(iii)] \emph{multiplicative}: $(\mathcal{F}_i R_l) (\mathcal{F}_j R_m) \subset \mathcal{F}_{i+j} R_{l+m}$,
\item[(iv)] \emph{homogeneous}: if $f\in \mathcal{F}_iR$ then each homogeneous piece of $f$ is in $\mathcal{F}_iR$; in other words $\mathcal{F}_iR=\bigoplus_k\mathcal{F}_iR_k$.
\end{itemize}
Here we have denoted $\mathcal{F}_iR_k=\mathcal{F}_iR \cap R_k$.
\end{definition}

The Loewy filtration defined in section \ref{loewy}, which is the main object of study in the present work, is an example of a decreasing filtration. Associated to each decreasing filtration are the following algebraic objects.

\begin{definition}Given a decreasing filtration $\mathcal{F}$ on $(X,L)$ we define its
\begin{itemize}
\item[(i)] \emph{Rees algebra} as $\Rees(\mathcal{F}) = \oplus_{i}(\mathcal{F}_iR)t^i \subset R[t],$
\item[(ii)] \emph{graded algebra} as $\gr(\mathcal{F}) = \oplus_{i} (\mathcal{F}_{i-1}R)/(\mathcal{F}_iR).$
\end{itemize}
We say a filtration is \emph{finitely generated} if its Rees algebra is finitely generated as a $\mathbb{C}[t]$ module. Note that, in the definition of the graded algebra, the grade of $(\mathcal{F}_{i-1}R)/(\mathcal{F}_iR)$ is $i$.
\end{definition}

To each decreasing filtration, one can associate corresponding Hilbert and weight functions.

\begin{definition}\label{polynomials} Given a filtration $\mathcal{F}$, define corresponding functions
\begin{align*}
 w(k) &= \sum_{i } i(\dim \mathcal{F}_{i-1}R_k - \dim \mathcal{F}_iR_k ), \\
 d(k) &= \sum_{i } i^2(\dim \mathcal{F}_{i-1}R_k - \dim\mathcal{F}_iR_k ).
 \end{align*}
We call $w(k)$ the \emph{weight function} and $d(k)$ the \emph{trace squared} function. We say that a filtration is \emph{polynomial} if $w(k)$ and $d(k)$ are polynomials of degree $n+1$ and $n+2$ respectively, where $n$ is the dimension of $X$. \end{definition}

Another invariant, which is associated just to the pair $(X,L)$ and is independent of the filtration is the \emph{Hilbert function}
\begin{align*}h(k)= \dim R_k.\end{align*}
For $k$ sufficiently large this is always a polynomial of degree $n$.

Let $\mathcal{F}$ be a polynomial filtration. Expand the associated Hilbert, weight and trace squared polynomials of $\mathcal{F}$ respectively as \begin{align*}h(k) &=a_0k^n+a_1k^{n-1}+O(k^{n-2}), \\ w(k) &= b_0k^{n+1}+b_1k^n+O(k^{n-1}), \\ d(k) &= d_0 k^{n+2} + O(k^{n+1}).\end{align*}

\begin{definition}\label{donaldsonfutaki} We define the \emph{Donaldson-Futaki invariant} of a polynomial filtration to be $$\DF(\mathcal{F}) = \frac{b_0a_1 - b_1 a_0}{a_0}.$$ We define the \emph{norm} of a polynomial filtration to be $$\|\mathcal{F}\|_2 = \frac{d_0a_0 - b_0^2}{a_0}.$$\end{definition}

\begin{definition}\label{Ksemistable} We say that $(X,L)$ is \emph{K-semistable} if for all polynomial filtrations $\mathcal{F}$, the Donaldson-Futaki invariant $\DF(\mathcal{F})$ is non-negative. If $(X,L)$ is not K-semistable we say that it is \emph{K-unstable}.
\end{definition}

As proven in \cite{DWN,GS} and reviewed in Section \ref{testconfigurations}, finitely generated filtrations are equivalent to test configurations and the respective Donaldson-Futaki invariants are equal. In particular, all finitely generated filtrations are polynomial. On the other hand,  \cite[Example 4]{GS} is an example of a polynomial filtration which is not finitely generated. The coherence of our definition of the Donaldson-Futaki invariant and K-semistability for polynomials filtrations with the convention used in the literature is proved in Section \ref{SectionApproximation}, see in particular Theorem \ref{ApproximationTheorem}.

The conjecture that motivates this definition is the following. We give a precise definition of K-polystability in Section \ref{testconfigurations}, for the moment all we need is that K-polystability implies K-semistability.

\begin{conjecture}[Yau-Tian-Donaldson]\cite{SD2}\label{YTD} A smooth polarised variety $(X,L)$ is K-polystable if and only if $X$ admits a cscK metric in $c_1(L)$. \end{conjecture}

\begin{remark}\label{orbifold} By work of Donaldson, it is known that the existence of a cscK metric implies K-semistability \cite{SD}. Stoppa has strengthened this to K-polystability, provided the automorphism group of $(X,L)$ is discrete \cite{JS}. Therefore, the existence of a polynomial filtration with negative Donaldson-Futaki invariant implies that no cscK metric exists in $c_1(L)$. \end{remark}

\begin{remark}\label{singular} In fact it is expected that Conjecture \ref{YTD} should be true even when $X$ has mild singularities, so that the definition of a cscK metric still makes sense. For instance, when $X$ is a $\mathbb{Q}$-smoothable Kawamata log terminal Fano variety with the anti-canonical polarisation, the above conjecture is known to be true \cite{RB, SSY}. \end{remark}

While the filtration that we study in this paper naturally fits into the definition of a decreasing filtration, there is a another equivalent definition of filtrations which is more suitable to be translated into a geometric language. We will use this second notion to discuss the link between filtrations and test configurations, which are the more familiar object introduced by Donaldson \cite{SD3}.

\begin{definition}\cite{GS}\label{increasingfiltration} Denote by $R(X,L)$ the  co-ordinate ring of $(X,L)$. We define an \emph{increasing filtration} $\mathcal{G}$ of $R$ be sequence of vector subspaces
$$\mathbb{C}=\mathcal{G}_0R \subset \mathcal{G}_1 R \subset \cdots$$
which is
\begin{itemize}
\item[(i)] \emph{pointwise right bounded}: for every $k$ there exists a $j=j(k)$ such that $\mathcal{G}_j R_k = H^0(X,L^k)$,
\item[(ii)] \emph{multiplicative}: $(\mathcal{G}_i R_l) (\mathcal{G}_j R_m) \subset \mathcal{G}_{i+j} R_{l+m}$,
\item[(iii)] \emph{homogeneous}: $f\in \mathcal{G}_iR$ then each homogeneous piece of $f$ is in $\mathcal{G}_iR$.
\end{itemize}
\end{definition}
Remark that the linear right bound in the decreasing case corresponds to the fact that the filtration starts at $0$ and $\mathcal{G}_0=\mathbb{C}$. Reversing the indices, we can similarly define all the previous invariants for an increasing filtration; for instance, the definition of the weight function has to be modified in this way:
$$ w_{\mathcal{G}}(k)= \sum_{i } i(\dim \mathcal{G}_iR_k  - \dim \mathcal{G}_{i-1}R_k).$$

\begin{lemma} \label{swap}
There is a (non-canonical) way to pass from a decreasing to an increasing filtration preserving both the Donaldson-Futaki invariant and the norm.
\end{lemma}

\begin{proof} Given a decreasing filtration $\mathcal{F}$, we define $\mathcal{G}_0=\mathbb{C}$ and $\mathcal{G}_i(R_k)=\mathcal{F}_{-i+Ck}(R_k)$, where $C$ is the constant appearing in the definition of decreasing filtration (remark that $C$ is not unique). It is easy to show that $\mathcal{G}$ is point wise right bounded and homogenous; we now show that it is multiplicative. Indeed,
\begin{align*}(\mathcal{G}_{i_1} R_{k_1})( \mathcal{G}_{i_2} R_{k_2} )&= (\mathcal{F}_{Ck_1-i_1} R_{j_1})(\mathcal{F}_{Ck_2-i_2} R_{j_2}), \\  &\subset \mathcal{F}_{C(k_1+k_2)-i_1-i_2} R_{j_1+j_2}, \\ &= \mathcal{G}_{i_1+i_2} R_{k_1+k_2}.\end{align*}
Remark that the linear bound is key in this proof. To calculate the weight polynomials, note that we have added $Ck$ to the weight of each section of weight $i$. The weight polynomials are related by $$w_{\mathcal{G}}(k) +Ckh(k)= w_{\mathcal{F}}(k),$$ in particular $$b_{0,\mathcal{G}} = b_{0,\mathcal{F}} + Ca_0, b_{1,\mathcal{G}} = b_{1,\mathcal{F}} + Ca_1.$$
Similarly
$$d_{0,\mathcal{G}}=d_{0,\mathcal{F}}+2Cb_{0,\mathcal{F}}+C^2a_0.$$
Computing the relevant Donaldson-Futaki invariants and norms we see that they are equal. For the reverse direction, it is enough to define $\mathcal{F}_iR:=\mathcal{G}_{-i}R$; all the verifications are straightforward, the constant $C$ can be taken equal to $0$.
\end{proof}

The procedure defined in the proof also gives an isomorphism between the Rees algebras of $\mathcal{F}$ and $\mathcal{G}$. This isomorphism does not preserve the grading, however finite generation is preserved.

\end{subsection}

\subsection{Approximating filtrations}\label{SectionApproximation} In this section, just to fix the notation, we assume that $\mathcal{F}$ is an increasing filtration; however, in view of Lemma \ref{swap}, this does not really matter. The Donaldson-Futaki invariant of a non-finitely generated filtration $\mathcal{F}$ is defined in \cite[Section 3.2]{GS} by choosing a specific approximation $\chi^{(k)}$ of $\mathcal{F}$. It is not clear to us if this invariant depends on the choice of the approximation. A priori, for polynomial non-finitely generated filtrations, this definition does not coincide with ours. We avoid such issues by using the following result.

\begin{theorem}\label{ApproximationTheorem}
Let $\mathcal{F}$ be a polynomial filtration. Then there exists a finitely generated filtration $\mathcal{G}$ such that
$$ DF(\mathcal{F})=DF(\mathcal{G}),$$
with the definition of the Donaldson-Futaki invariant as in Definition \ref{donaldsonfutaki}.
\end{theorem}

Theorem \ref{ApproximationTheorem} follows from the following two Lemmas.

\begin{lemma}\label{approximation}Let $\mathcal{F}$ be an increasing filtration with Rees algebra $\mathcal{R}$. For every integer $r$, there exists a finitely generated filtration $\mathcal{F}^{(r)}$ of $R$ such that for all $p\leq r$ and all $i$ we have
$$\mathcal{F}^{(r)}_iH^0(X,L^p) = \mathcal{F}_i H^0(X,L^p).$$ \end{lemma}

\begin{proof}This is essentially contained in \cite[Section 3.1]{GS}. We construct the finitely generated filtration through its Rees algebra $\mathcal{R}^{(r)}$, by defining $\mathcal{R}^{(r)}\subset R[t]$ to be the $\mathbb{C}[t]$-subalgebra generated by
$$\oplus^r_{p=1}\left(\oplus^{j(p)}_{i=1} (F_i R_p)t^i\right).$$
 Here $j(p)$ is the bound appearing in Definition \ref{increasingfiltration}. Since the filtration $\mathcal{F}$ is multiplicative, this gives a well defined, finitely generated algebra. The corresponding filtration is
$$\mathcal{F}^{(r)}_i R = \{s\in R : t^i s \in \mathcal{R}\}.$$
\end{proof}

\begin{remark} The geometric version of Proposition \ref{approximation} in terms of test configurations is \cite[Proposition 3.7]{RT}, which states that a test configuration is equivalent to an embedding of $X$ into projective space $\mathbb{P}(H^0(X,L^k))$ for some $k$ and a choice of $\mathbb{C}^*$-action on this projective space. \end{remark}

\begin{lemma}\label{approximation2} Let $\mathcal{F}$ be a polynomial filtration and $\mathcal{F}^{(r)}$ be the filtrations defined in Lemma \ref{approximation}. Then, for $r$ sufficiently large, we have
$$\DF(\mathcal{F}^{(r)}) = \DF(\mathcal{F}).$$ \end{lemma}

\begin{proof} Since $\mathcal{F}$ is polynomial, its weight function is a polynomial of degree $n+1$.  Because of this, the weight polynomial is determined by finitely many values of $w(k)$. Take an approximating sequence as in Lemma \ref{approximation}, with $r$ sufficiently big such that the weight polynomial is determined by the weight values for $p<r$. Then $\mathcal{F}^{(r)}$ has the same weight polynomial as $\mathcal{F}$. Therefore $\DF(\mathcal{F}^{(r)}) = \DF(\mathcal{F}),$ as required. \end{proof}

To prove Theorem \ref{ApproximationTheorem} just remark that we can take as $\mathcal{G}$ the filtration $\mathcal{F}^{(r)}$ constructed in Lemma \ref{approximation} for $r$ sufficiently large and then apply Lemma \ref{approximation2}. We can now state the following Corollary.

\begin{corollary}
The Definition of $K$-semistability \ref{Ksemistable} is equivalent to the usual definition. In particular, we can use Definition \ref{donaldsonfutaki} for the Donaldson-Futaki invariant of a polynomial filtration which is not non-finitely generated.
\end{corollary}
\begin{proof}
Our definition is stronger than the usual one, because in the usual definition one needs to check just test configurations, which correspond to finitely generated filtrations.
On the other hand, if a polynomial filtration has negative Donaldson-Futaki invariant, then Theorem \ref{ApproximationTheorem} guarantees that there exists a finitely generated filtration with negative Donaldson-Futaki invariant, so the variety is not K-semistable in the sense of the usual definition.
\end{proof}

\subsection{Test configurations and filtrations}\label{testconfigurations}We now turn to the more geometric notion of test configurations, and recall how they relate to filtrations.

\begin{definition} A \emph{test configuration} for a $(X,L)$ is a normal polarised variety $(\mathcal{X},\mathcal{L})$ together with
\begin{itemize}
\item[(i)] a proper flat morphism $\pi: \mathcal{X} \to \mathbb{C}$,
\item[(ii)] a $\mathbb{C}^*$-action on $\mathcal{X}$ covering the natural action on $\mathbb{C}$,
\item[(iii)] and an equivariant relatively ample line bundle $\mathcal{L}$ on $\mathcal{X}$
\end{itemize}
such that the fibre $(\mathcal{X}_t,\mathcal{L}_t)$ over $t$ is isomorphic to $(X,L^r)$ for one, and hence all, $t \in \mathbb{C}^*$ and for some $r>0$.
\end{definition}

\begin{remark}One should think of test configuration as geometrisations of the one-parameter subgroups that are considered when applying the Hilbert-Mumford criterion to GIT stability. \end{remark}

As the $\mathbb{C}^*$-action on $(\mathcal{X},\mathcal{L})$ fixes the central fibre $(\mathcal{X}_0,\mathcal{L}_0)$, there is an induced action on $H^0(\mathcal{X}_0,\mathcal{L}^k_0)$ for all $k$. Denote by $A_k$ the infinitesimal generator of this action, so that $\mathbb{C}^*$ acts as $t\to t^{A_k}$ on $H^0(\mathcal{X}_0,\mathcal{L}^k_0)$. The total weight $\tr(A_k)$ of the $\mathbb{C}^*$-action on $H^0(\mathcal{X}_0,\mathcal{L}^k_0)$ is a polynomial of degree $k+1$, expanding the Hilbert and weight polynomials as in Definition \ref{donaldsonfutaki} we can define the \emph{Donaldson-Futaki} invariant of a test configuration, just as we did for polynomial filtrations. Similarly, using  $\tr(A_k^2)$, one can define the \emph{norm} of a test configuration.

\begin{remark} The notion of a test configuration gives a geometric interpretation of the test configurations with zero norm. Indeed, a test configuration $(\mathcal{X},\mathcal{L})$ has norm zero if and only if it has normalisation equivariantly isomorphic to the product configuration $X\times\mathbb{C}$ with the trivial action on $X$ \cite[Theorem 1.2]{RD}; this is also proven under a normality assumption in \cite{BHJ}.\end{remark}

One classical source of test configurations are those arising from automorphisms.

\begin{definition} Given a one paramater subgroups of $\Aut(X,L)$, define the corresponding \emph{product test configuration} by $(\mathcal{X},\mathcal{L})=(X\times\mathbb{C},L)$ with the action on the central fibre over $0\in\mathbb{C}$ induced by the automorphism. \end{definition}

Given such an automorphism, Futaki showed that the corresponding Donaldson-Futaki invariant must vanish if $X$ admits a cscK metric in $c_1(L)$; see \cite{SD3} for a discussion of this point.

The relationship between test configurations and filtrations is as follows.

\begin{theorem}\cite{GS,DWN} Given an arbitrary test configuration, there exists a finitely generated filtration with the same Donaldson-Futaki invariant. Conversely, given any finitely generated filtration, one can associate a test configuration with the same Donaldson-Futaki invariant. \end{theorem}

\begin{proof} We recall the strategy of the proof.  Let $(\mathcal{X},\mathcal{L})$ be a test configuration, and let $s\in H^0(X,L^k)$. We think of $s$ as a section of $\mathcal{X}_t$ for all $t \neq 0$ using the $\mathbb{C}^*$-action on $\mathcal{X}$. In particular, $s$ can be thought of as a section defined when $t\neq 0$, so is a meromorphic section, with a pole of some order along $t=0$. Therefore $t^is$ is a holomorphic section for $i\in \mathbb{N}$ sufficiently large. We then define a filtration by saying that $s$ belongs to $\mathcal{F}_i$ if $t^is$ is regular on all of $\mathcal{X}$.

Conversely, given a finitely generated filtration $\mathcal{F}$, the associated test configuration is $\Proj_{\mathbb{C}[t]}(\Rees (\mathcal{F}))$ with its natural $\mathcal{O}(1)$ polarisation. The $\mathbb{C}^*$-action is given by the grading of the Rees algebra. \end{proof}

With all of this in place, we can define K-polystability.

\begin{definition}\cite[Section 8.2]{LX}\cite{JS2} We say that a polarised variety is \emph{K-polystable} if for all test configurations $(\mathcal{X},\mathcal{L})$ (equivalently polynomial filtrations), the corresponding Donaldson-Futaki invariant satisfies $\DF(\mathcal{X},\mathcal{L}) \geq 0$, with equality if and only if $(\mathcal{X},\mathcal{L})$ is isomorphic to a product test configuration away from a closed subscheme of codimension two. Otherwise we say $(\mathcal{X},\mathcal{L})$ \emph{destabilises} $(X,L)$. \end{definition}

\begin{remark} The definition of K-polystability is independent of scaling $L\to L^r$. In particular, it makes sense for pairs $(X,L)$ where $X$ is a variety and $L$ is a $\mathbb{Q}$-line bundle. From another point of view, there is no loss in assuming that $L$ is very ample and projectively normal.\end{remark}

\section{The Loewy filtration}\label{loewy}

In this section we define a canonical decreasing filtration of the co-ordinate ring of any polarised variety $(X,L)$ and prove some of its properties. The construction uses the automorphism group $\Aut(X,L)$ of $(X,L)$; the filtration is non-trivial if and only if $\Aut(X,L)$ is non-reductive. This filtration satisfies the hypotheses of Definition \ref{decreasingfiltration}: it is homogeneous and point-wise left bounded by construction, multiplicative because of Lemma \ref{mult} and linearly right bounded by Lemma \ref{lin_bound}. We assume that $L$ is very ample and projectively normal; in particular $\Aut(X,L)$ is a closed sub-group of $GL (H^0(X,L^k))$.

\medskip

Let $G$ be a linear algebraic group defined over $\mathbb{C}$ and $V$ be a $G$-module. We are interest in the cases where $G$ is either $\Aut(X,L)$ or its unipotent radical and $V=H^0(X,L^k)$; however, our definition makes sense in a more general context. We define the filtration inductively.

\begin{definition}
Let $V$ be a finite dimensional $G$-module. The \emph{Loewy filtration} $\mathcal{F}_{\bullet}V$ is defined as
\begin{itemize}
\item[(i)] $\mathcal{F}_0V=V$;
\item[(ii)] for $i>0$, $\mathcal{F}_iV$ is the minimal $G$-submodule of $\mathcal{F}_{i-1}V$ such that the quotient $\mathcal{F}_{i-1}V/\mathcal{F}_iV$ is semi-simple.
\end{itemize}
\end{definition}

Equivalently, we can define $\mathcal{F}_iV$ to be the intersection of all maximal non-trivial submodules of $\mathcal{F}_{i-1}V$. The Loewy filtration is sometimes called radical filtration or Loewy decreasing filtration. See \cite[Section V.1]{ASS}  or \cite[Section I.2.14 and Chapter D]{JJ} for a general discussion. In what follows, we will give a few other descriptions of the Loewy filtration. Before that let us point out the following.

\begin{remark} The Loewy filtration is $G$-invariant. \end{remark}

We wish to compute the weight of $\mathcal{F}_{\bullet}$, as defined in Definition \ref{polynomials}. In order to do this more easily, we introduce another description of the Loewy filtration.
\begin{lemma}\label{unipotent_reduction}
Let $U$ be the unipotent radical of $G$ and $V$ be a $G$-module. Then the Loewy filtration of $V$ as a $G$-module is equal to the Loewy filtration of $V$ as a $U$-module.
\end{lemma}
\begin{proof}
The unipotent radical of $G$ is its maximal normal connected unipotent subgroup; it is trivial if and only if $G$ is reductive. In characteristic zero, we have the Levi decomposition
$$
G=R\ltimes U
$$
where $R$ is a reductive group. A $U$-module is semi-simple if and only if the action of $U$ is trivial. Because of the normality of the radical, a maximal trivial $U$-submodule is also a $G$ submodule. Moreover it is a semi-simple $G$-submodule, because $U$ acts trivially and $R$ is reductive.
\end{proof}

There is also a natural increasing filtration that $G$ defines on $V$. This filtration is related to the Loewy filtration in a sense that we are going to make precise; we introduce it because it is a useful tool. In general, however, its Donaldson-Futaki invariant is different from the one of the Loewy filtration.

In view of  Lemma \ref{unipotent_reduction}, we give the definition directly for a unipotent group.

\begin{definition}[Socle filtration]
Given a module $V$ for a unipotent group $U$, the Socle filtration $\mathcal{S}_{\bullet}$ is defined inductively as
\begin{itemize}
 \item[(i)] $\mathcal{S}_0V=\{ 0 \}$,
 \item[(ii)] $\mathcal{S}_iV$ is the pre-image of the $U$-invariant part of $V/\mathcal{S}_{i-1}V$.
In particular, $\mathcal{S}_1V=V^U$.
\end{itemize}
The Socle filtration is also called increasing Loewy filtration. Being increasing, we define
$$ \gr_i (\mathcal{S}_{\bullet}V):=\mathcal{S}_iV/\mathcal{S}_{i-1}V. $$
Note the switch of indices in the definition of the associated graded module compared to the corresponding definition for decreasing filtrations.\end{definition}

\begin{lemma}
Let $U$ be a unipotent group and $V$ a $U$-module. Then
$$\dim \gr_i (\mathcal{F}_{\bullet}V) =\dim \gr_i(\mathcal{S}_{\bullet}V^{\vee}). $$
\end{lemma}
\begin{proof}
For clarity, we consider first the case $i=1$. The module $\gr_1(\mathcal{F}_{\bullet}V)$ is the biggest quotient of $V$ which is trivial as a $U$-module. We have an exact sequence
$$ 0\to \mathcal{F}_1V\to V\to \gr_1 \mathcal{F}_{\bullet}V\to 0. $$
Dualising this sequence shows that  $\gr_1 (\mathcal{F}_{\bullet}V)^{\vee}$ is the maximal $U$-invariant part of $V^{\vee}$, so it is isomorphic to  $\gr_1(\mathcal{S}_{\bullet}V^{\vee})$.

Now, take any $i$. We have $\mathcal{S}_iV^{\vee}=\Ann \mathcal{F}_iV$, see for example \cite[Page 162]{ASS} or \cite[Equation 6 page 441]{JJ}. Because of this
\begin{align*}\gr_i(\mathcal{S}_{\bullet}V^{\vee})&=\Ann (\mathcal{F}_iV)/\Ann (\mathcal{F}_{i+1}V), \\ &=(\mathcal{F}_{i+1}V/\mathcal{F}_iV)^{\vee}, \\ &=\gr_i(\mathcal{F}_{\bullet}V)^{\vee}.\end{align*} \end{proof}

We now look at the co-ordinate ring. Let $R$ be a graded ring
$$
R=\bigoplus_{k\geq 0}R_k
$$
where the $R_k$ are finite dimensional $G$-modules and $G$ preserves the multiplication. Recall that we are interested in the case $R_k=H^0(X,L^k)$. Each module $R_k$ has a Loewy filtration $\mathcal{F}_{\bullet}$; we define
$$ \mathcal{F}_iR:=\bigoplus_k\mathcal{F}_iR_k.$$

\begin{lemma}\label{mult} The Loewy filtration is multiplicative. \end{lemma}

\begin{proof}
One possibility is to show that the Socle filtration is co-multiplicative, and then dualise. We instead use a further description of the Loewy filtration. Let $\mathfrak{u}$ be the Lie algebra of $U$ and $A$ be the universal enveloping algebra $\mathfrak{U}(\mathfrak{u})$ of $\mathfrak{u}$. The advantage of this point of view is that $A$ is an associative (non-commutative) algebra. We can thus consider its Jacobson radical $J(A)$, see for example \cite[Section I.3]{ASS}. It can be defined as the intersection of all maximal left ideals and one can show that it is a two-sided ideal. The relationship between the Loewy filtration and the Jacobson radical is
$$ \mathcal{F}_iR_k=J(A)^iR_k,$$
this follows from \cite[Proposition I.3.7 and Corollary I.3.8]{ASS}. From this we can show that the Loewy filtration is multiplicative. Indeed,

\begin{align*}\mathcal{F}_iR_l\cdot \mathcal{F}_jR_s&=J(A)^iR_l\cdot J(A)^jR_s, \\ &\subset J(A)^{i+j}R_{l+s}, \\ &=\mathcal{F}_{i+j}R_{l+s}.\end{align*}
\end{proof}
\begin{definition}
We define the \emph{length} of a filtration to be the maximum $i$ such that $F_iV$ is not trivial. We then define the \emph{Loewy length} $\ll(V)$ of a module $V$ is the length of the Loewy filtration. The Loewy length of a graded ring
$$R=\bigoplus_k R_k $$
is a function of $k$, and we denote by $\ll_R(k)$ the Loewy length of $R_k$. The Loewy length of a polarised variety $(X,L)$ is the length of its co-ordinate ring, seen as an $\Aut(X,L)$-module.
\end{definition}
General discussions about the Loewy length of a fixed module can be found in \cite[Chapter V]{ASS} and \cite[Chapter D]{JJ}.

\begin{lemma}\label{lin_bound}
The Loewy filtration is linearly right bounded. That is, there exists a constant $C$ such that
$$\ll(k)\leq Ck.$$
We can take $C=h^0(L)$.
\end{lemma}
\begin{proof}
Let $V:=H^0(X,L)$. Since we have assumed that $L$ is projectively normal, we have a surjective map
$$\Sym^k V\to H^0(X,L^k).$$
The Loewy length of the domain is bigger than the Loewy length of the codomain (cf. \cite[Proposition I.3.7]{ASS}), so it is enough to prove the statement for $\Sym^k V$. The unipotent radical $U$ of $\Aut(X,L)$ is a subgroup of some maximal unipotent subgroup $T$ of $GL(V)$. The Loewy length of $\Sym^k V$ as $\Aut(X,L)$-module is smaller than its Loewy length as a $T$-module. Below we will show that the Loewy length of $\Sym^k V$ as a $T$ module is $k(\dim V -1)+1$, and this is enough to conclude the proof.

To compute the Loewy length of $\Sym^k V$ as a $T$ module fix a basis $e_i$ for which $T$ is the group of upper triangular matrices. Assign weight $n-i$ to $e_i$, where $n=\dim V$. Each monomial in $\Sym^k V$ now has a weight and there are exactly $k(n-1)+1$ different weights. The point is that this weight corresponds to the grade assigned by the Loewy filtration because $T$ can only increase the weight.
\end{proof}
Remark that in our examples the Loewy length is linear. It is an interesting question to ask for which class of varieties this is true; moreover, when the Loewy length \emph{is} a linear function, we do not know if its slope has a geometric meaning.

\begin{conjecture}\label{main-conjecture}Let $(X,L)$ be a polarised variety with non-reductive automorphism group. Then the Loewy filtration is polynomial, and destabilises $(X,L)$. In particular, $(X,L)$ is not K-polystable.\end{conjecture}
By destabilises we mean either that it has strictly negative Donaldson-Futaki invariant (which is the case in all the examples), or that the Donaldson-Futaki invariant is zero and one of the corresponding finitely generated filtration that we can associated to $\mathcal{F}$ via Theorem \ref{ApproximationTheorem} is not isomorphic to a product test configuration away from a closed subscheme of codimension two.

\medskip

So far, we have just used representation theory and the geometry of $X$ never appeared. Motivated by understanding the geometry of the Loewy filtration, we introduce the another related geometric filtration,  which makes sense just when $R$ is the co-ordinate ring of an algebraic variety and $U$ is a unipotent group acting on $(X,L)$ (at this stage, $U$ does not have to be the unipotent radical of $\Aut(X,L)$).  Let us denote $E:=X^U$ the fixed subscheme of the action of $U$ on $X$. There are a few general results about the geometry of $E$. The key one is Borel's fixed point theorem, which guarantees that $E$ is non-empty. Horrocks showed that $E$ is connected \cite{GH}. For related work, see \cite{Gross} when $X$ is projective and \cite{JL} for $X$ affine. We can define a multiplicative decreasing filtration on $R$ by the vanishing order along $E$, namely
$$\mathcal{V}_iH^0(X,L^k):=H^0(X,L^k(-iE)).$$
We do not have a general bound on the length of this filtration. When $U$ is the unipotent radical of $\Aut(X,L)$, we have the following relation with the Loewy filtration.
\begin{lemma} Let $U$ be the unipotent radical of $\Aut(X,L)$. Assume that the fixed subscheme $E:=X^U$ is a reduced divisor and $E^{sm}\cap X^{sm}$ is non-empty. Denoting by $\mathcal{F}_i$ the Loewy filtration, we have
$$\mathcal{V}_iH^0(X,L^k)\subseteq \mathcal{F}_iH^0(X,L^k)$$
for all $i$. \end{lemma}

\begin{proof}
Let $R_k:=H^0(X,L^k)$. In this proof we will use that $\mathcal{S}_iR_k^{\vee}=\Ann \mathcal{F}_iR_k$ (cf.\cite[Page 162]{ASS} or \cite[Equation 6 page 441]{JJ}).

Let us start with $i=1$. Recall that $L^k$ embeds $X$ in $\P R_k^{\vee}$. The divisor $E$ is the intersection of $X$ and  $\P \mathcal{S}_1R_k^{\vee}$, so the annihilator of $\mathcal{S}_1R_k^{\vee}$ contains the set of sections vanishing along $E$; in symbols $\mathcal{V}_1R_k \subseteq   \mathcal{F}_1R_k$. (This first part works without any assumption on $E$).

We argue by induction, so we assume the statement for $i$ and we prove it for $i+1$. For any point $p$ of $E^{sm}\cap X^{sm}$ consider the exact sequence
$$0\rightarrow T_pE\to T_pX\to N_{E |X}|_p\to 0.$$
The group $U$ acts on the sequence, acting trivially on $T_pE$. The vector space $N_{E|X}|_p$ is one dimensional because $E$ is a divisor, so the action of $U$ must be trivial.  For any point $p_s$ in $E^{sm}\cap X^{sm}$, denote by $D_s$ a non-zero vector in $N_{E |X}|_{p_s}^{\otimes i+1}$. This defines a linear functional on $H^0(X,L^k(-iE))$; by the previous discussion it is $U$ invariant. Since $H^0(X,L^k(-(i+1)E))$ is defined by a finite number of functionals $D_s$, where $p_s$ are general points in $E^{sm}\cap X^{sm}$, we obtain that $\mathcal{V}_{i+1}R_k\subseteq \mathcal{F}_{i+1}R_k$, as required.
\end{proof}
In all our examples $X^U$ is a reduced divisor; we do not know how general this fact is. In some cases, such as the Hirzebruch surfaces, this inclusion is actually an equality; in other cases, such as $\P^2$ blown up at two points, the inclusion is strict.

\begin{section}{Examples}\label{examples}
In this section we describe the Loewy filtration of some varieties, giving a proof of Theorem \ref{intro_examples}. In all examples the filtration is polynomial, so we do not need to take approximations to compute the Donaldson-Futaki invariant. We do not know if this is true in general. In all of our examples, the Donaldson-Futaki invariant is negative, confirming Conjecture \ref{main-conjecture} in these cases. Some examples are special cases of others, we include them for the sake of clarity.

\begin{subsection}{Degree 8 del Pezzo}\label{deg8}
Let $X$ be the blow-up of $\P^2$ at a point $p$. Fix an ample line bundle $ L=aH-bE$, recall that the ampleness is equivalent to $a>b>0$. Fix a basis $x,y,z$ of $H^0(\P^2, \O(1))$ such that $p=[1,0,0]$. In this basis, we have an identification
\begin{align*}R_k&:= H^0(X,kL), \\&= \Span \{ \mbox{degree } ka \mbox{ monomials such that }  \deg y + \deg z \geq bk \}.\end{align*}
The Hilbert polynomial is
\begin{align*}h(k)&=\sum_{i=0}^{ak-bk}(ak-i+1), \\ &=\frac{1}{2}(a^2-b^2)k^2+\frac{1}{2}(3a-b)k+1,\end{align*} which can also be seen from Riemann-Roch. We now describe the Loewy filtration. The automorphism group $G$ is the sub-group of $\P GL(3)$ fixing $p$; its unipotent radical $U$ is
\begin{displaymath}
U=
\left(
\begin{array}{ccc}
1 & * & * \\
0 & 1 & 0 \\
0 & 0  & 1
\end{array}
\right).
\end{displaymath}
On the space of sections we have the dual action, so this group fixes $y$ and $z$. This means that $\mathcal{F}_1R_k$ is generated by all monomials for which $x$ does not appear with maximal degree; indeed, in this way $\mathcal{F}_1R_k$ is a submodule and quotienting gives a trivial $U$-module. Remark that the maximal degree of $x$ in $R_k$ is $ak-bk$. More generally, we have
\begin{align*}\mathcal{F}_i(R_k) = \Span \{& \mbox{degree } ka \mbox{ monomials such that } \\ & \deg y + \deg z \geq bk \mbox{ and } \deg x \leq ak-bk-i-1\}.\end{align*}
For the associated graded modules we have
\begin{align*} \gr_i(R_k)=\Span \{& \mbox{degree } ka \mbox{ monomials such that } \\ & \deg y + \deg z \geq bk \mbox{ and } \deg x=ak-bk-i\}.\end{align*}
As an example, for $a=3, b=1, k=1$, the graded module associated to the Loewy filtration of $H^0(X,-K_X)$ is
$$ \gr_1 = \Span \{x^2y,x^2z\}, \qquad \gr_2 = \Span \{xy^2,xyz,xz^2\}, \qquad\gr_3=\Span \{y^3,z^3\}.$$

We wish to count the dimension of the weight space of weight $i$. Since $\deg x = ak-bk-i$, we have $\deg y + \deg z =bk+ i$. There are $bk+i+1$ polynomials in two variables of degree $bk+i$. Therefore the dimension of the weight space is $bk+i+1$. The weight polynomial is
\begin{align*}w(k)&=\sum_{i=0}^{ak-bk}i(bk+i+1), \\ &= (\frac{1}{3}a^3-\frac{1}{2}a^2b+\frac{1}{6}b^3)k^3+(a^2-\frac{3}{2}ab+\frac{1}{2}b^2)k^2+(\frac{2}{3}a-\frac{2}{3}b)k. \end{align*}

The numerator of the Donaldson-Futaki invariant is
$$\DF_{num}=-\frac{1}{6}b^4(\frac{a}{b}-1)^3.$$
This is negative \emph{exactly} when $a>b$, which is required for ampleness. The Loewy length is $\ll(k) = (a-b)k$. The norm is
$$\|\mathcal{F}\|_2=\frac{a^4}{4}-\frac{2(a^3-b^3)^2}{9(a^2-b^2)},$$
which is positive for $a>b$.  For $a=b$ the Donaldson-Futaki invariant vanishes. This is not surprising because, in this case, we are dealing with a line bundle which is a bull-back from $\P^1$, which is K-polystable.

\end{subsection}

\begin{subsection}{$\mathbb{P}^2$ blown up at $n$ points on a line}

Let $X$ be the blowup of $\P^2$ at $n$ points $p_1,\hdots,p_n$ on a line, with $p_1 = [1:0:0]$ and $p_2=[0:1:0]$. The picard rank is $\rho(X) = n+1$, generated by $H$, which is the pullback of the hyperplane class from $\P^2$, and the $n$ exceptional divisors $E_1,\hdots,E_n$. To check a line bundle $L$ is ample on $X$, it suffices to show it has positive intersection with $H-E_1-\hdots-E_n$ and the exceptional divisors $E_1,\hdots,E_n$. The ample cone is therefore those $L=aH-b_1E_1-\hdots-b_nE_n$ such that $a>b_1+\hdots+b_n$, with $a,b_1,\hdots,b_n>0$. For simplicity for the rest of this calculation we assume $b_1=\hdots=b_n=:b$, so that the condition for ampleness becomes $a>nb$. We also assume that $n\geq 2$, having considered the $n=1$ case in Section \ref{deg8}. Denote $L=aH-bE_1-\hdots-bE_n$, and $R_k = H^0(X,kL)$. For ease of notation denote $E = E_1+\hdots+E_n$. $E$ has the property that $E.E = -n$, $L.E = nb$.

The anti-canonical class of $X$ is $-K_X = 3H-E$, which intersects $L$ as $-K_X.L = 3a-nb$. $L$ has self-intersection $L.L = a^2-nb^2>0$. By Riemann-Roch the Hilbert polynomial is given as $$h(k) = \frac{a^2-nb^2}{2}k^2 + \frac{3a-nb}{2}+1.$$

The automorphism group of $X$ is the subgroup of $PGL(3)$ consisting of $3\times 3$ matrices which either fix the line $[a:b:0]$ or permute some of the $n$ points on the line. The permutations do not lie in the connected component of the identity, so the maximal normal unipotent subgroup of $\Aut(X)$, which we denote by $U$, is given by matrices of the form \begin{displaymath}
U=
\left(
\begin{array}{ccc}
1 & 0 & * \\
0 & 1 & * \\
0 & 0  & 1
\end{array}
\right).
\end{displaymath} Note that matrices of this form fix the line which joins the blown-up points.

The space $R_k$ is the space of degree $ak$ polynomials which vanish with order at least $kb$ at each point $p_i$; therefore it contains all polynomials whose degree in $z$ is at least $bk$, and some of the others. The Loewy filtration is given by the degree in $z$, because the action of $U$ on $H^0(\P^2,\O(1))$ fixes exactly $z$.

For $i\geq kb$, the Loewy filtration is $$\mathcal{F}_iR_k = \{ \deg ka \mbox{ monomials with } kb \leq \deg z \leq i \}.$$ Thus $\dim \mathcal{F}_iR_k - \dim \mathcal{F}_{i-1}R_k = ka-i+1$.

We now calculate the dimensions of the spaces $\mathcal{F}_iR_k$ for $i \leq kb$. Fixing some $\deg z=i$, we wish to count the number of degree $ka-i$ polynomials in $2$ variables which vanish along $n$ points on a line to at least $kb-i$. Such polynomials have weight $i$, and vanish to order at least $kb$ along $n$ points. The number of such polynomials is given by Riemann-Roch on $\P^1$. Indeed, let $$M=(ka-i)\O_{\P^1}(1) - (kb-i)p_1 - \hdots - (kb-i)p_n,$$ where we consider those points to be in $\P^1$. By Riemann-Roch for curves we have \begin{align*}\dim H^0(\P^1, M) &= \deg M +1 = ka-i - n(kb-i) +1, \\&= k(a-nb)+(n-1)i+1.\end{align*} So for $i\leq kb$ we have $$\dim \mathcal{F}_iR_k - \dim \mathcal{F}_{i-1}R_k = k(a-nb)+(n-1)i+1.$$ The Loewy length is $\ell \ell(k)=ak$. Using this method, the Hilbert polynomial is $$h(k)=\sum^{kb-1}_{i=0} (k(a-nb)+(n-1)i+1) + \sum^{ka}_{i=kb}(ka-i+1),$$ which agrees with the calculation using Riemann-Roch. Note that when $i=kb$ the two summands are equal.  The weight polynomial is $$w(k)=\sum^{kb-1}_{i=0} i(k(a-nb)+(n-1)i+1) + \sum^{ka}_{i=kb}i(ka-i+1).$$

The weight polynomial has highest terms $$b_0 = \frac{a^3-nb^3}{6},\qquad b_1 = \frac{a^2}{2}.$$ The numerator of the Donaldson-Futaki invariant is \begin{align*}12\DF(\mathcal{F})_{num}&=12(b_0a_1-b_1a_0), \\ & = (a^3-nb^3)(3a-bn) - 3(a^2)(a^2-nb^2).\end{align*} Expanding and setting $c=\frac{a}{b}$ we get $$12\DF(\mathcal{F})_{num}=-nc^3+3nc^2-3nc+n^2.$$ The condition $a>nb$ becomes $c>n\geq 2$. Dividing by $n$ we wish to show $$-c^3+3c^2-3c+n<0.$$ Note that when $n=c$ this is given as $c(c(3-c)-2),$ which is less than or equal to zero when $c\geq 2$. Its derivative is $-3(c-1)^2$, which is negative when $c>1$. In particular, this is a decreasing polynomial in $c$ when $c>1$, which is negative when $c=2$. So it is negative for all $c>2$, therefore this filtration destabilises. The norm is given as $$\|F\|_2 = \frac{1}{36}\left(3(a^4-b^4n) - 2\frac{a^3-nb^3}{a^2-nb^2}\right),$$ setting $c=\frac{a}{b}$ and using $c>n\geq 2$ one sees that this is strictly positive.

\end{subsection}
\begin{subsection}{An orbifold del Pezzo surface}\label{orbifold}
This example is based on the analysis developed in the previous section, hence we will keep the same notations. Let $\mu\colon X\to \P^2$ be the blow up of $\P^2$ at $3$ points $p_i$ on a line $\ell$. Let $\hat{\ell}$ be the proper transform of $L$, its class is $H-E$. The divisor $\hat{\ell}$ is a $-2$ curve so we can consider its blow-down
$$\nu\colon X\to F.$$
The variety $F$ is a singular Fano; the singular point is an $A_1$ singularity so the surface is an orbifold. The singularity is rational, so $\nu^*K_F=K_X$ (this can be shown also by explicit computation). For any line bundle $L$ on $F$, the pull-back defines an isomorphism
$$
\nu^*\colon H^0(F,L)\to H^0(X,\nu^*L).
$$
Since $\hat{\ell}$ is fixed by $U$, the isomorphism is an isomorphism of $U$-modules, so it preserves the Loewy filtration. This means that the Loewy filtration, and its Donaldson-Futaki invariant, can be equivalently computed on $X$ or on $F$. Line bundles on $X$ which are of the form $\nu^*L$ are the ones with zero intersection with $\hat{\ell}$, therefore we are interested in multiples of $3H-E$, which is actually the anti-canonical class of $X$. Plugging $c=n=3$ in the formula for the numerator of the Donaldson-Futaki invariant obtained in the previous section, we see that the Loewy filtration destabilises $(F,-K_F)$.

\end{subsection}

\begin{subsection}{Hirzebruch surfaces}
We consider the Hirzerburch surface
$$X=\P ( \O\oplus \O(n)).$$
This is a $\P^1$-bundle over $\P^1$. Denote by $H$ the pull-back of the hyperplane from $\P^1$ and $\O(1)$ the tautological line bundle. Let $$L=a\O(1)+bH,$$ this is ample if and only if $a,b>0$. Pushing forward to $\P^1$, we have
\begin{align*}R_k:&= H^0(X,kL), \\&=\bigoplus_{i=0}^{ka}H^0(\P^1,\O_{\mathbb{P}^1}(bk+in)).\end{align*}
The Hilbert polynomial is
\begin{align*}h(k)&=\sum_{i=0}^{ak}(bk+in+1),\\ &=k^2(\frac{1}{2}a^2n+ab)+k(\frac{1}{2}an+a+b)+1.\end{align*}

Let us describe the Loewy filtration. The unipotent radical of the automorphism group is $H^0(\P^1,\O(n))$; it acts on the total space of $\O\oplus\O(n)$ as the upper triangular matrices. That is, a section $s$ of $H^0(\P^1,\O(n))$ maps an element $c\oplus 0$ of $\O\oplus \O(n)$ to $c\oplus s$. The induced action on $H_k$ maps $H^0(\P^1,\O(bk+in))$ to $H^0(\P^1,\O(bk+in))\oplus H^0(\P^1,\O(bk+(i+1)n))$. The Loewy filtration is thus
$$\mathcal{F}_iH_k=\bigoplus_{j=i+1}^{ka}H^0(\P^1,\O(bk+jn)).$$
The graded modules are
$$
\gr_i=H^0(\P^1,\O(bk+in)).
$$
The weight polynomial is
$$w(k)=\sum_{i=0}^{ak}i(bk+in+1),$$
so that
$$b_0=\frac{1}{3}a^3n+\frac{1}{2}a^2b,\qquad b_1=\frac{1}{2}a^2n+\frac{1}{2}a^2+\frac{1}{2}ab. $$
The numerator of the Donaldson-Futaki invariant is
$$ \DF_{num}=-\frac{1}{12}a^4n^2+\frac{1}{12}a^4n-\frac{1}{6}a^3bn,   $$
which is negative. The Loewy length is $\ll(k) = ak$. The norm is
$$\|\mathcal{F}\|_2=\frac{a^3(a^2n^2+6abn+6b^2)}{36(an+2b)},$$
which is positive for $a,b>0$.
Remark that for $n=1$ the Hirzebruch surface is isomorphic to the blow up at $\P^2$ at one point, and the Loewy filtrations using both descriptions coincide. This can be checked by an explicit computation. Indeed, to compare the two descriptions take $b=b'$ and $a=a'+b'$, where $a$ and $b$ are the parameters appearing in the del Pezzo description, and $a'$ and $b'$ are the parameters appearing in the projective bundle description.

\begin{proposition}
The Loewy filtration of a Hirzebruch  surface equals the filtration by vanishing order along the fixed locus. Moreover, it is finitely generated.
\end{proposition}

\begin{proof}
The fixed locus of the action of $U$ on $X$ is the $-n$ curve $E$. This curve has class $E=\O(1)-nH$. We have
\begin{align*} \mathcal{V}_iR_k&=H^0(X,ka\O(1)+kbH-iE), \\&=H^0(X,(ak-i)\O(1)+(kb+ni)H).\end{align*}
Pushing forward to $\P^1$ we get
\begin{align*} H^0(\P^1,\O(kb+in)\otimes \Sym^{ak-i}(\O\oplus\O(n)))&=\bigoplus_{j=0}^{ak-i}H^0(\P^1,\O(kb+(i+j)n)), \\ &=\bigoplus_{j=i}^{ak}H^0(\P^1,\O(kb+jn)).\end{align*}
This coincides with the Loewy filtration. The Rees algebra associate to this filtration is isomorphic to
$$\bigoplus_{d,m}H^0(X,dL-mE).$$
This is finitely generated because the Hirzebruch surfaces are toric varieties.
\end{proof}
Following \cite[Section 8]{DWN}, the test configuration associated to this filtration is the deformation to the normal cone of the $-n$-curve with parameter $a$. To identify the two filtrations we used that Loewy length is linear.

\end{subsection}

\begin{subsection}{Projective bundles $\P (\O^{\oplus r } \oplus \O(n))$ over $\P^s$ - partial computation}\label{GeneralProjectiveBundle}

This is a generalisation of the Hirzerbrch surface example. Let $X=\P (\O^{\oplus r} \oplus \O(n))$ over $\P^s$. Take as polarisation $L=a \O(1)+bH$, where $H$ is the pull-back of the hyperplane section of $\P^s$. We take $a$,$b$ and $n$ strictly positive. Pushing-forward we have
$$H^0(X,kL)=H^0(\P^s,\O(bk)\otimes \Sym^{ka}(\O^{\oplus r}\oplus \O(n))).$$
The right hand side is isomorphic to
$$  \bigoplus _{J}H^0(\P^s,\O(bk+J_{r+1}n)),$$
where the sum runs over all partitions $J=(J_1,\cdots , J_{r+1})$ of $ak$ into $r+1$ non-negative numbers. Here we are just writing out monomials in $r+1$ variables, $J_{r+1}$ is the exponent of $\O(n)$. The unipotent radical of the automorphism group is $H^0(\P^1,\O(n))^{\oplus r}$. It maps $\O^{\oplus r}$ to $\O^{\oplus r}\oplus \O(n)$. The graded module associated filtration induced by the action is
$$ \gr_i=H^0(\P^s,\O(bk+in))^{\oplus P(ak-i,r)},$$
where $P(ak-i,r)$ is the number of partition of $ak-i$ into $r$ non-negative numbers. The index $i$ ranges from $0$ to $ak$. That is, $\gr_i$ is isomorphic to the sub-vector space where $\O(n)$ appears with multiplicity exactly $i$.  The Loewy length is $\ell \ell(k)=ak$. The Hilbert polynomial is
$$ h(k)=\sum_{i=0}^{ak}\binom{bk+in+s}{s}P(ak-i,r).$$
The weight polynomial is
$$ w(k)=\sum_{i=0}^{ak}i\binom{bk+in+s}{s}P(ak-i,r).$$
\begin{proposition}
The Loewy filtration is finitely generated. Moreover, it equals to the filtration by vanishing order along the fixed locus $E$ of the action of $U$.
\end{proposition}
\begin{proof}
The fixed locus $E$ of the action of $H^0(\P^1,\O(n))^{\oplus r}$ is the reduced divisor corresponding to the quotient
$$
\pi \colon \O^{\oplus r}\oplus \O(n)\to \O^{\oplus r}
$$
because $\O(n)$ is the maximal sub-vector bundle of $\O^{\oplus r}\oplus \O(n)$ which is $H^0(\P^1,\O(n))^{\oplus r}$ invariant. A section of $kL$ vanishes along $E$ if and only if it is in the kernel of $\Sym^k (\pi)$, so the Loewy filtration equals the filtration by vanishing order along $E$. The Rees algebra of this filtration is isomorphic to
$$ \bigoplus_{m,d}H^0(X,dL-mE).$$
This is finitely generated because $X$ is a toric variety.
\end{proof}
Counting the number of partitions of an integer is well-known to be a difficult problem, in the sequel we will carry out the computation in some special cases.
\end{subsection}

\medskip
\begin{subsection}{Projective bundles $\P (\O\oplus \O \oplus \O(n))$ over $\P^1$}
Keeping the notation of the previous section, take $s=1$ and $r=2$. In particular, $P(ak-i,2)=ak-i+1$. The variety $X$ now has dimension 3. In this set up, the Hilbert polynomial is
$$ h(k)= \sum_{i=0}^{ak}(ak-i+1)(bk+in+1),$$
and  so
$$a_0=\frac{1}{6}a^3n+\frac{1}{2}a^2b, \qquad a_1=\frac{1}{2}a^2n+\frac{1}{2}a^2+\frac{3}{2}ab.$$
The weight polynomial is
$$ w(k)= \sum_{i=0}^{ak}i(ak+1)(bk+in+1),$$
this gives
$$b_0=\frac{1}{12}a^4n+\frac{1}{6}a^3b \qquad b_1=\frac{1}{3}a^3n+\frac{1}{6}a^3+\frac{1}{2}a^2b.$$

The numerator of Donaldson-Futaki invariant is
 $$\DF_{num}=-\frac{1}{72}a^6n^2+\frac{1}{72}a^6n-\frac{1}{24}a^5bn.$$
This is negative since $n^2$ is bigger than $n$. The norm is
$$\|\mathcal{F}\|_2=\frac{1}{144}a^3(18a^4bn+6a^4n^2+24a^3b^2+8a^3bn-an-2b),$$
which is positive as $a$,$b$ and $n$ are positive integers.
\end{subsection}

\begin{subsection}{Projective bundle $\P( \O\oplus \O(1))$ over $\P^2$}
Let $$X=\P( \O_{\P^2}\oplus \O_{\P^2}(1)).$$
We have
$$-K_X=2(\O(1)+H).$$
To ease notation we denote $L= -\frac{1}{2}K_X$. Using the formulae of Section \ref{GeneralProjectiveBundle}, the Hilbert polynomial is
$$h(k)=\sum_{i=0}^k\binom{2+i+k}{i+k},$$
giving
$$a_0= \frac{7}{6}, \qquad a_1=\frac{7}{2}.$$

The unipotent radical of the automorphism group is $H^0(\P^2,\O(1))$. The graded modules associated to the Loewy filtration are
$$
\gr_i=H^0(\P^2,\O(i+k)).
$$
The weight polynomial is
$$w(k)=\sum_{i=0}^ki\binom{2+i+k}{i+k},$$
so
$$ b_0= \frac{17}{24}, \qquad b_1=\frac{9}{4}.$$
The numerator of the Donaldson-Futaki invariant is
$$\DF_{num}=-\frac{7}{48},$$
which is negative. The Loewy length is $\ll(k) = k$. The norm is $\|\mathcal{F}\|_2=\frac{97}{1120}$. \end{subsection}

\end{section}

redpajama_set_name: RedPajamaArXiv


text:
Photographs (13) + -
Ephemera (1) + -
MacArthur, Nellie (b. 4/4/1867?) (71) + -
Sibley's (Rochester, N.Y.) (16) + -
Cro' Nest (Rochester, N.Y.) (14) + -
Cutler, Joseph Warren (7/13/1857 - 1934) (14) + -
Baker, Marie (b. 1845) (13) + -
Dickins, Mary Belle (9/9/1876?) (13) + -
Bragdon, Charlotte (8/26/1872 - 12/15/1907) (12) + -
Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980) (12) + -
Hutchinson, Bessie J. (1877 - 1954) (12) + -
Cutler, James Goold (4/24/1848 - 4/21/1927) (11) + -
Gilmore, Joseph H. (b. 9/9/1863) (10) + -
Search: mods_subject_name_displayForm_mt:"MacArthur,\ Nellie\ \(b.\ 4\/4\/1867\?\)"
May Bragdon Diary, August 14, 1897, p. 142
Bragdon, Claude (8/1/1866 - 9/17/1946), Bragdon, May (5/16/1865 - 3/21/1947), Bellamy, Charella (1865? - 2/16/1920), Baker, Lura Ophelia (8/13/1870 - 1922), MacArthur, Nellie (b. 4/4/1867?), Hillman, John Constantine (1865 - 1932), Orchard, Helen Elizabeth (7/21/1872 - 2/6/1899), Joiner, Edith M. (7/3/1868 - 7/4/1935), Steinhausen, Meta (1879 - 1969), Bellamy, Edward (b. 11/30/1884), Harned's (Rochester, N.Y.), Sea Breeze (Rochester, N.Y.), Columbia
May Bragdon Diary, September 25, 1898 – September 28, 1898, p. 285
Hillman, John Constantine (1865 - 1932), Davis, Frank Allen (3/21/1873 - 6/1/1953), Davis, Helen Alling (4/19/1871 - 4/12/1950), Davis, Lillian May (1871 - 1910), Siemer, Martha A. (1876 - 1960), Goodyear, Katherine M. (b. 1861?), Bragdon, Claude (8/1/1866 - 9/17/1946), Cutler, James Goold (4/24/1848 - 4/21/1927), Cutler, Joseph Warren (7/13/1857 - 1934), MacArthur, Nellie (b. 4/4/1867?), Highland Park (Rochester, N.Y.)
Davis', Davis, Helen Alling (4/19/1871 - 4/12/1950), Davis, Frances Freeman (1838? - 1915), Davis, Lillian May (1871 - 1910), Davis, Charlotte Gleason (4/12/1861 - 1938), Millard, Ernest B. (1870 - 1941), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Hillman, Edith V. (1879? - 1923), Hillman, John Constantine (1865 - 1932), Baker, Lura Ophelia (8/13/1870 - 1922), Baker, Marie (b. 1845), Davis, Frank Allen (3/21/1873 - 6/1/1953), MacArthur, Nellie (b. 4/4/1867?), South Park (Rochester, N.Y.), Lyceum (Rochester, N.Y.)
May Bragdon Diary, May 24, 1904 – May 26, 1904, p. 28
Bragdon, George Chandler (4/29/1832 - 8/7/1910), MacArthur, Nellie (b. 4/4/1867?), Penfield, Edward (6/2/1866 - 2/8/1925), Bragdon, Claude (8/1/1866 - 9/17/1946), Baker, Lura Ophelia (8/13/1870 - 1922), Baker, Marie (b. 1845), Hutchison, Laura B. (b. 1870), Penfield, Florence (1853? - 1935), MacArthur, Emily Jane (1836? - 1923), Davis, Charlotte Gleason (4/12/1861 - 1938), St. Helena (Ontario, Canada), Sibley's (Rochester, N.Y.), Flinch
May Bragdon Diary, October 4, 1906 – October 8, 1906, p. 195
Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980), MacArthur, Nellie (b. 4/4/1867?), Steinhausen, Meta (1879 - 1969), Steinhausen, Theodore D. (1874 - 1944), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Frey, Dorothy Frances (b. 10/6/1904), Davis, Frances Graham (9/9/1906 - 10/6/1983), Joiner, Edith M. (7/3/1868 - 7/4/1935), Cutler, Joseph Warren (7/13/1857 - 1934), Cro' Nest (Rochester, N.Y.)
May Bragdon Diary, December 12, 1906 – December 18, 1906, p. 221
Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Bragdon, Claude (8/1/1866 - 9/17/1946), Shipherd, Jacob Rudd (6/12/1836 - 5/7/1905), Bragdon, Charlotte (8/26/1872 - 12/15/1907), MacArthur, Nellie (b. 4/4/1867?), Hillman, Edith V. (1879? - 1923), Hillman, John Constantine (1865 - 1932), Bellamy, Edward (b. 11/30/1884), Bellamy, Francis R. (1886 - 1972), Sibley's (Rochester, N.Y.), Cro' Nest (Rochester, N.Y.), Zoroaster (d. 1909), Zendavesta
May Bragdon Diary, April 22, 1906 – April 27, 1906, p. 116
Bragdon, Charlotte (8/26/1872 - 12/15/1907), Wilkinson, Katharine May (10/24/1876 - 6/14/1959), Bragdon, Claude (8/1/1866 - 9/17/1946), Davis', MacArthur, Nellie (b. 4/4/1867?), Joiner, Edith M. (7/3/1868 - 7/4/1935), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Andrews, Jane M. (1859 - 1948), Cro' Nest (Rochester, N.Y.), South Park (Rochester, N.Y.), Carl, Diana, Flinch
May Bragdon Diary, March 31, 1904 – April 5, 1904, p. 11
Blackall, Florence M. (b. 1875), Bragdon, George Chandler (4/29/1832 - 8/7/1910), MacArthur, Nellie (b. 4/4/1867?), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Cutler, Joseph Warren (7/13/1857 - 1934), Cutler, Anna Katherine (1845? - 1924), Davis', Davis, Charlotte Gleason (4/12/1861 - 1938), Davis, Helen Alling (4/19/1871 - 4/12/1950), Hillman, Edith V. (1879? - 1923), Hillman, John Constantine (1865 - 1932), Steinhausen, Meta (1879 - 1969), Hardenbrook, Lulu (b. 1878?), Eggleston and Archers (Rochester, N.Y.), Sibley's (Rochester, N.Y.), Flinch
May Bragdon Diary, July 25, 1908 – July 26, 1908, p. 84
Davis, Charlotte Gleason (4/12/1861 - 1938), Rogers, Mabel S. (b. 11/17/1870), MacArthur, Nellie (b. 4/4/1867?), Joiner, Edith M. (7/3/1868 - 7/4/1935), Crichton, Catharine M. (b. 1866?), Willard, Frank Almarin (8/20/1851 - 1919), Willard, Charles Bunnell (1890 - 1917), Willard, Mary F. (3/12/1853 - 1935), Bunnell, Damaris B. (1833 - 1916), Bunnell, Asael O. (1836 - 1923), Bragdon, Chandler (12/15/1907 - 1/3/1969), Wayback (N.Y.)
Bragdon, Charlotte (8/26/1872 - 12/15/1907), Bragdon, Claude (8/1/1866 - 9/17/1946), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Cutler, Joseph Warren (7/13/1857 - 1934), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Davis, Charlotte Gleason (4/12/1861 - 1938), Baker, Lura Ophelia (8/13/1870 - 1922), Crichton, Catharine M. (b. 1866?), MacArthur, Nellie (b. 4/4/1867?), Hutchinson, Bessie J. (1877 - 1954), Dickins, Mary Belle (9/9/1876?), Summerville (Rochester, N.Y.), Cro' Nest (Rochester, N.Y.)
May Bragdon Diary, October 21, 1908 – October 28, 1908, p. 136
Garfield, Helen C. (1835 - 7/18/1911), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Crawford, Letitia (b. 1863?), Scranton, Edgar M. (1860 - 1910), MacArthur, Nellie (b. 4/4/1867?), Cutler, James Goold (4/24/1848 - 4/21/1927), Flack, Reginald Gamble (1/24/1881 - 1975), Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980), Davis', Davis, Charlotte Gleason (4/12/1861 - 1938), Caswell, Herbert Marshall (b. 1860?), Williams, Herma Louise (1885 - 1957), Steinhausen, Theodore D. (1874 - 1944), Sibley's (Rochester, N.Y.), Seneca Park (Rochester, N.Y.)
Hillman, John Constantine (1865 - 1932), Bragdon, Claude (8/1/1866 - 9/17/1946), Blondin, Charles (1824 - 1897), Joiner, Edith M. (7/3/1868 - 7/4/1935), MacArthur, Nellie (b. 4/4/1867?), Orchard, Helen Elizabeth (7/21/1872 - 2/6/1899), Cutler, James Goold (4/24/1848 - 4/21/1927), Cutler, Joseph Warren (7/13/1857 - 1934), Humphrey, George P. (1864 - 1952), Bellamy, Charella (1865? - 2/16/1920), Wilder, Clara (1861? - 1944), Seneca Park (Rochester, N.Y.), Vagabonds
May Bragdon Diary, July 18, 1903 – July 23, 1903, p. 157
Stull, Mary G. (b. 1869?), Joiner, Edith M. (7/3/1868 - 7/4/1935), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Baker, Lura Ophelia (8/13/1870 - 1922), Hawley, Mary Cornelia (8/27/1868 - 1/12/1939), Baker, Marie (b. 1845), Rogers, Mabel S. (b. 11/17/1870), MacArthur, Nellie (b. 4/4/1867?), Hillman, John Constantine (1865 - 1932), Lyceum (Rochester, N.Y.), Charlotte (Charlotte, N.Y.)
May Bragdon Diary, April 30, 1897, p. 33
Crosby, Murray Wasson (1871 - 1928), DeGraff, William (b. 1864?), Hillman, John Constantine (1865 - 1932), Ewers, William Vaux (b. 1871), Cutler, Amy Cecil (1873? - 1958?), Gilmore, Joseph H. (b. 9/9/1863), Davis, Frances Freeman (1838? - 1915), Baker, Marie (b. 1845), Backus, Harry E. (b. 1860?), Davis, Frank Allen (3/21/1873 - 6/1/1953), MacArthur, Nellie (b. 4/4/1867?), Orchard, William Henry (1863?-1926), Rogers, Mabel S. (b. 11/17/1870), Davis, Helen Alling (4/19/1871 - 4/12/1950), Perfect Little Ladies
Stull, Mary G. (b. 1869?), Stull, John M. (9/9/1870 - 8/2/1939), DeGraff, William (b. 1864?), MacArthur, Nellie (b. 4/4/1867?), MacArthur, Ida L. (b. 1860?), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Penfield, Edward (6/2/1866 - 2/8/1925), Bragdon, Claude (8/1/1866 - 9/17/1946), Davis, Helen Alling (4/19/1871 - 4/12/1950), Davis, Hamilton Clark (10/14/1864 - 2/10/1947), Joiner, Edith M. (7/3/1868 - 7/4/1935), Flinch
May Bragdon Diary, January 28, 1897 – January 29, 1897, p. 275
Gilmore, Joseph H. (b. 9/9/1863), Baker, Lura Ophelia (8/13/1870 - 1922), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Pond, Theodore Hanford (1872 - 1933), Bragdon, Claude (8/1/1866 - 9/17/1946), Orchard, Helen Elizabeth (7/21/1872 - 2/6/1899), Powers, Rose Edith (1872 - 1943), Bellamy, Charella (1865? - 2/16/1920), Behns, Joiner, Edith M. (7/3/1868 - 7/4/1935), Rogers, Mabel S. (b. 11/17/1870), MacArthur, Nellie (b. 4/4/1867?), Sibley's (Rochester, N.Y.), Gorton and McCabe (Rochester, N.Y.), Lyceum (Rochester, N.Y.), Perfect Little Ladies
Cutler, Joseph Warren (7/13/1857 - 1934), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980), Bragdon, Charlotte (8/26/1872 - 12/15/1907), Bragdon, Claude (8/1/1866 - 9/17/1946), Baker, Lura Ophelia (8/13/1870 - 1922), MacArthur, Nellie (b. 4/4/1867?), Hutchinson, Bessie J. (1877 - 1954), Dickins, Mary Belle (9/9/1876?), MacArthur, Wilkinson, Charlotte Coffin (4/24/1833 - 8/3/1909), Cutler, James Goold (4/24/1848 - 4/21/1927), Joiner, Edith M. (7/3/1868 - 7/4/1935), Scranton, Edgar M. (1860 - 1910), Sibley's (Rochester, N.Y.), Cro' Nest (Rochester, N.Y.)
May Bragdon Diary, July 6, 1907 – July 12, 1907, p. 277
Gridley, Mary R. (1876? - 1956?), Gilmore, Joseph H. (b. 9/9/1863), Bragdon, Claude (8/1/1866 - 9/17/1946), Bragdon, Charlotte (8/26/1872 - 12/15/1907), Steinhausen, Theodore D. (1874 - 1944), Hutchinson, Bessie J. (1877 - 1954), Dickins, Mary Belle (9/9/1876?), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), MacArthur, Nellie (b. 4/4/1867?), Rogers, Mabel S. (b. 11/17/1870), Flack, Ethel L. (b. 1877), Davis, Charlotte Gleason (4/12/1861 - 1938), Steinhausen, Meta (1879 - 1969), Bragdon, George Chandler (4/29/1832 - 8/7/1910), Connolly, Nan (b. 1880), Baker, Lura Ophelia (8/13/1870 - 1922), Sibley's (Rochester, N.Y.), Zoroaster (d. 1909), Autumn Chapter
May Bragdon Diary, 1910 – 1914, p. 268
Frey, Amelie D. (1882 - 1966), Rowe, Marguerite H. (1886 - 1967), Dickins, Mary Belle (9/9/1876?), Boyce, John L., Rogers, Mary J. (1865? - 1911), Bradshaw, Alice (b. 3/14/1889), Davis, Helen Alling (4/19/1871 - 4/12/1950), Baker, Lura Ophelia (8/13/1870 - 1922), MacArthur, Nellie (b. 4/4/1867?), Bragdon, Henry Wilkinson (9/6/1906 - 3/15/1980), Crawford, Letitia (b. 1863?), Arnold, James Burns (1881 - 1957), Stern, Herbert Morland (1882 - 1977), Bragdon, Katherine Elmina (12/30/1837 - 9/6/1920), Hutchinson, Bessie J. (1877 - 1954), Weller, Frieda (1889 - 1926?), Williams, Herma Louise (1885 - 1957), Davis', Davis, Charlotte Gleason (4/12/1861 - 1938), Davis, Frances Freeman (1838? - 1915), Sibley's (Rochester, N.Y.), South Park (Rochester, N.Y.), four corners (Rochester, N.Y.)

redpajama_set_name: RedPajamaCommonCrawl


text: Q: find the moment of inertia of a quadrant of the ellipse (x/a)^2+(y/a)^2=1 of mass M about the x axis, if the density at a point is proportional to xy? find the moment of inertia of a quadrant of the ellipse (x/a)^2+(y/a)^2=1 of mass M about the x axis, if the density at a point is proportional to xy

redpajama_set_name: RedPajamaStackExchange


text:
Q: What Causes SDL_Renderer to be invalid? I'm trying to make a 2D game where the background is a hockey rink. The error handler said the Render-er didn't initialize thus making the texture not initialize. I know the BMP loads because I don't get an error for that. Don't worry about the camera stuff. The background is supposed to scroll. I just need the background to render. The code may not be the best, but corrections are always appreciated.
#include <SDL/SDL.h>
#include <stdio.h>
#include <iostream>
#include <fstream>
#include <string>
using namespace std;

const int fps = 60;

SDL_Window *window;

int main(int argc, char **argv)
{
    SDL_Init(SDL_INIT_EVERYTHING);
    int x = 0, y = 0;
    SDL_Surface  *screen;
    SDL_Surface *background = SDL_LoadBMP("hockeyrink.bmp");
    if(background == NULL)
    {
        SDL_ShowSimpleMessageBox(0, "Background init error", SDL_GetError(), window);
    }
    SDL_Renderer *renderer = SDL_CreateRenderer(window, -1, 0);
    if(renderer == NULL)
    {
        SDL_ShowSimpleMessageBox(0, "Renderer init error", SDL_GetError(), window);
    }
    SDL_Texture *texture = SDL_CreateTextureFromSurface(renderer,background);
    if(texture == NULL)
    {
        SDL_ShowSimpleMessageBox(0, "Texture init error", SDL_GetError(), window);
    }

    const int speed = 5;
    SDL_Rect camera;
    camera.x = 0;
    camera.y = 0;
    camera.w = 800;
    camera.h = 600;
    bool b[2] = {0,0};
    Uint32 start;
    window = SDL_CreateWindow("", 300, 100, 1024, 800, SDL_WINDOW_OPENGL);

    if (window == NULL)
    {
        cout << ("could not create window: %s/n", SDL_GetError());
        return 1;
    }
    bool running = true;
    while (running)
    {
        start = SDL_GetTicks();
        SDL_Event event;

        while(SDL_PollEvent(&event))
        {
            switch(event.type)
            {
            case SDL_QUIT:
                running = false;
                SDL_DestroyTexture(texture);
                SDL_FreeSurface(background);
                SDL_DestroyRenderer(renderer);
                SDL_DestroyWindow(window);
                SDL_Quit();
                break;
            case SDL_KEYDOWN:
                switch(event.key.keysym.sym)
                {
                case SDLK_UP:
                    b[0]=1;
                    break;
                case SDLK_LEFT:
                    b[1]=1;
                    break;
                }
                break;
            case SDL_KEYUP:
                switch(event.key.keysym.sym)
                {
                case SDLK_UP:
                    b[0]=0;
                    break;
                case SDLK_LEFT:
                    b[1]=0;
                    break;
                }
                break;
            }
        }

        if(b[0])
        {
            x+=speed;
            camera.y+=speed;
            if (camera.y > 3000-800)
            {
                camera.y=0;
            }

        }
        else if(b[1])
        {
            x-=speed;
            camera.y-=speed;

            if (camera.y <= 0)
            {
                camera.y = 2000-800;
            }

            SDL_RenderCopy(renderer, texture, NULL, NULL);
            SDL_RenderPresent(renderer);
        }
        if(1000/fps>SDL_GetTicks()-start)
        {
            SDL_Delay(1000/fps-(SDL_GetTicks() - start));
        }
    }

    return 0;
}

A: When you create renderer, the window pointer does not point to anything(i.e. it is uninitialized).
You must create the window before  creating the renderer.
So put the lines:
    window = SDL_CreateWindow("", 300, 100, 1024, 800, SDL_WINDOW_OPENGL);

    if (window == NULL)
    {
        cout << ("could not create window: %s/n", SDL_GetError());
        return 1;
    }

before the line: SDL_Renderer *renderer = SDL_CreateRenderer(window, -1, 0);


text:
Autumn Foliage Forecast 2006
famous viewing spots and color changing forecast
- Early to Mid November -
Kyoto Prefecture
Takao | Kyoto City
Jjingoji Temple Autumn Leaves
<Early November>
Takao, Makino-o and Togano-o, the three ridges on the southeast side of Mt. Takaosan are famous areas for autumn colored leaves. Particularly in the precincts of Jingoji Temple, a range of maple trees freely stretch their branches and creates a brilliant mood of the ancient capital. Along Kiyotakigawa River there are walking and hiking courses and you can easily enjoy the autumn leaves. For there are restaurants, tea houses and accommodation, you can explore the area for an entire day.
>> Kyoto Prefectural Guide
Arashiyama | Kyoto City
<Mid November>
You can get the very best view of Arashiyama when you stand back and put the Togetsukyo bridge in front of the hillside of Arashiyama full of red and yellow trees. It is only 375 meters high and is part of a mountain range formed by mountains of similar scale, but is a very famous mountain sung in many Japanese poems and songs from the ancient ages. On the second Sunday of November, the Arashiyama Momiji Festival is held where the scene of the gorgeous picture scroll is re-created in reality with a boat on Oikawa River. Arashiyama is not only the name for the mountain but also for the entire neighborhood as well. There are a number of scenic and historical spots and also temples. In spring the mountain gets embellished with astonishing cherry blossoms.
Autumn Leaves of Tenryuji Temple
Botanical Gardens, Graduate School of Science The University of Tokyo (Koishikawa Botanical Gardens) | Bunkyo-ku
The site of Koishikawa Botanical Gardens belonged to the fifth Edo Shogun, and after his death became a herb farm. With 300 years of history, it is the oldest botanical garden in Japan. Now it has about 1,400 kinds of trees and you can enjoy colored leaves in every bit of the garden. Especially recommended is the area near the glasshouse where Japanese maple and lime trees unsoaringly boast their vivid red and soft yellow leaves. The view seen from the pond is also very pretty. Although it is situated in the heart of Tokyo, the garden is uncrowded and provides a quiet atmosphere for couples to spend time.
Yoyogi Park | Shibuya-ku
Yoyogi Park, with its area of 540,000 square meters, is one of the largest parks in central Tokyo. The park is rich in trees including 1,000 zelkova trees, 200 ginko trees and 100 Japanese maple trees which all turn red and gold in autumn. The best spots to appreciate autumn leaves are the zelkova woods on the Harajuku side, the thicket near the West Gate and the Himalayan cedar trees near the cycling center. There are tons of benches and grass fields inside the park, so you can sit down and relax pretty much anywhere, or take a walk among the tall and thick trees. In spring you can also have a fabulous view cherry blossoms on 618 cherry trees.
Denen Chofu | Ota-ku
Denen Chofu, especially the western side of the station, is widely known as a rich and quiet neighborhood. 3 of the 5 main roads radiating from the station have abundant ginko trees and in autumn the roads turn into stretchs of golden tunnels. The trees go with the residences so beautifully, that Denen Chofu had been awarded the Best City Scenery in 1991. The beauty has not changed since then, and especially the view on the far-left road of the 5 is the best.
Koishikawa Korakuen | Bunkyo-ku
Designated as a special historic spot and special place of scenery, Koishikawa Korakuen is a Japanese Garden with rich Chinese flavor and astonishing view. Once you set your foot inside, you can hardly believe you're still in the heart of the city. In autumn the 570 maple trees and approx. 300 zelkowa trees unveil splendorous beauty and particularly the view from Togetsukyo Bridge is worth having. The name Korakuen means "to fall behind power and drown in pleasure."
Photo: Korakuen Autumn 2005
Kewaizaka | Kamakura City
You simply cannot miss Kewaizaka if you want to enjoy the autumn beauty of Kamakura. It's located on the way from Zeniarai Benten to Genjiyama Park, and has a 50-meter-chain of red Japanese maple leaves, but is also a wild area with rough rocks showing their surfaces. There are several historical legends telling stories about this "slope". The end of the slope is Genjiyama Park which has a fine view and is a good place to have a picnic. The slope is very steep in some places and gets slippery in rainy weather so it would be better to wear a pair of sneakers when walking through Kewaizaka.
>> Kanagawa Prefectural Guide
Kourankei Gorge | Asuke Town
Kourankei Gorge located in Asuke town, the eastern end of Aichi prefecture, has been known as a famous place for colored leaves from long ago. It is created by the erosion of the rivers Asukegawa and Tomoegawa, and the 1km-long gorge between Tomoe bridge and Kouran bridge is called Kourankei. The legend says that a Buddhist monk planting maple trees as he chanted the Buddhist sutra along the river is the beginning of this rich nature and scenery. A red tunnel covers the gorge as to match with the red bridge. There is a hot spring resort nearby which is said to have good effects on nerve pain and rheumatism.
>> Aichi Prefectural Guide
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Rad's new location will be in the new West Woodland Business Center (1128 NW 52nd St), where they've leased 26,457 square feet of the 4-story, 72,000 sq. ft. industrial building, and plan to both manufacture and sell their bikes there. The company has almost doubled their output since they launched in 2015, and have outgrown their multiple Ballard locations. They're currently leasing three locations — an office, a retail store, and a warehouse.
Rad will take over the new space in January 2019.
*An earlier version of this story referred to east Ballard as West Woodland. We've changed it to "east Ballard" to avoid confusion.
Congrats on the success Mike and the Rad Team!
I don't see any RVs in that artist rendering.
What are these things costing? And being these brave people feel they're now an automobile, and are faster in many cases, shouldn't these be regulated, taxed and licensed? After all, most here are big government folks, who seem to love big daddy picking the winners/losers. Plus the revenue stream would mean more $$ rolling into local coffers, perhaps fending off an income tax and or worse. Pay to play is the name or the game.
So are these the people we thank for unleashing a maelstrom of illegal electric bikes on our trails? I love how they took the Uber approach: openly defy the law and force the local governments to negotiate.
I have e bike. It's wonderful. I'm a 77 yr old who never goes over 13 on Burke-G. It's not the bike, it's weather rider responsible!!!
Hey, I'm in terrible shape and want to go at near-motorcycle speeds on the bike trail. Do you have any products I might be interested in? I already bought a fat bike with tires like a toddler's toy, so I might want to upgrade to an absurdly expensive e-bike.
Your new building is gorgeous! Reminds me of the old Burger King on Market and 15th.
Great news! Maybe this will lessen the amount of rad bikes cruising down Market St. on the sidewalks doing 20mph. Buy AMERICAN not Chinese! Congrats though on being successful. I wish I could afford one of these bikes actually, but I'd buy one made in the USA.


Bill Mandell was quoted in the April 2017 Emergency Department Legal Letter article on visitors and guns in hospital emergency departments and suggested a useful policies for hospitals to consider.
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Volo Beauty, a beauty tech startup focused on creating innovative hair technology, debuted its first product, the Volo Go Cordless hair dryer, at CES in Las Vegas.
"Debuting our product at a tech-driven show is important to us because it highlights the Volo difference: this isn't your run-of-the-mill hair dryer. This is brand new cordless technology that redefines a category and, more importantly, solves a problem by offering active women more freedom in their heat styling, as well as healthier hair," said Ryan Goldman, CEO, Volo Beauty.
The patent-pending Volo Go is the company's first fully engineered product, a hair dryer redesigned to be cordless, battery-powered and healthier due to infrared heat. The dryer features infrared light therapy, radiant heat, ionic technology said to quicken overall drying time, and lithium-ion battery power.


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Famous Baseball Quotes Filled With Timeless Wisdom
by Work-At-Home-Jobs | Nov 5, 2013 | Jobs And Work |
Baseball History May Be Ever Changing But Some Things Never Change
Baseball history, like all history, certainly is ever changing, but some aspects of the game have remained unchanged for well over a half-century. There are a few reasons for this – over time the basic rules of the game have for the most part remained unaltered; the development of essential skills continues to involve an investment of time and personnel by ball clubs; and fans have always flocked to see money players and exciting teams.
In 1976, Major League Baseball (MLB) was changed forever with the birth of free agency. Since its inception, the owners had held power over all players. They could trade anyone at anytime and control, with relative ease, what individuals would be paid. Great players, like Babe Ruth, usually commanded solid salaries but with free agency players were able to negotiate their contracts and to go to a team willing to pay their price.
Still, as it had always been, players had to have the skills a team needed to get their price. The one major difference was that players were now able to sign guaranteed contracts, which stated that they would be paid their salary no matter how they performed and even if they were injured.
Seventy-three years before free agency, professional baseball underwent a change that would influence the way in which the Majors conducted business and found players. In 1903 the National Association of Professional Base Ball Leagues, better known as the Minor Leagues, was formed in order to create some order in which Major League teams acquired players from small market clubs. In the 1930s the great Branch Rickey developed the structure for what we know today as the "Minor Leagues." Rickey's formalization of the "Minors," which became dedicated to developing players who could perform in the Majors, was jokingly called the "farm system" because small town clubs were raising young players "like corn" down on the farm.
Since the 1930s, MLB has relied upon affiliate farm teams to develop players for the big leagues to supply promising prospects for trades, or to simply provide adequate replacements when necessary. Today, the Minor League system is highly developed, bringing players up through A, AA, and AAA ball. When a team is looking to make a trade for a solid Major Leaguer, one way they can sweeten the deal is by including prospects from the minors. Additionally, one way for owners to keep costs down is to bring up players from the "farm team" when they're ready. By doing this a MLB team can save millions of dollars.
Bringing up an adequate second baseman from the minors and paying him the minimum $ 327,000 for the season can prove to have more value than paying a veteran infielder 2.5 million dollars. Using a certain number of non-veterans allows a team to spend more money on other positions, especially pitching, which is always at a premium and comes at a high price.
The Minor Leagues have always been a cost-saving venture for clubs but with today's exorbitant salaries, the strength of the players union, and most clubs carrying payrolls of under one-hundred million dollars, the strategic use of Minor League players can make the difference in both turning a profit and winning the World Series.
Using players from the farm club actually gives owners more power, since those team members who have been brought up are not eligible for salary arbitration until they have three years in the Majors and cannot become free agents until they've accumulated six or more years in the big leagues. Today, for a brief part of a player's career, National and American League owners have the power they used to possess over every player prior to 1976.
Teams looking to win a championship and attract as much revenue as possible have often invested money in key players. Throughout baseball's history, there have been owners willing to pay more than others. In 1919, some of the Chicago White Sox, which was owned by Charles Comiskey, decided to throw the World Series to their National League counterparts, the Cincinnati Reds.
The White Sox players felt they were both underpaid and under-appreciated by their boss and figured if they bet on the underdog Reds and insured that the National Leaguers would win, they'd make a tidy profit. They did just that, but they also got caught. It became known as the Black Sox Scandal of 1919, and it's a constant reminder to owners of how a man like Comiskey, who had a whole lot of money, could be penny-wise and pound-foolish.
Of course, spending a lot of money does not insure a team will win the World Series or even the League Championship. Since the New York Yankees payroll exceeded one hundred millions dollars five years ago, they've been unable to win it all. The last few years Steinbrenner and company have been handing over two hundred million in team pay. Last year's champs, the Chicago White Sox, paid out a little over seventy-five million to their players. It was ironic, because they had not won the Series since 1917, prior to the big scandal.
Most analysts agree that a MLB club needs to wisely spend about seventy million to be competitive. That's why some pundits believe that the Toronto Blue Jays, who reached that magic salary mark this season by adding quality players and about thirty million to their pay roll, are a club to watch.
The basic idea at work regarding salary prior to free agency is that a happy player – one who feels he's being treated and paid fairly – can equal a happy owner with a winning team. After free agency, the thought became that in order to compete a team had to pay a player what the market dictated and owners, to a degree, had the ability to control that market. These are, in essence, two sides of the same coin.
If we could resurrect Cy Young, Shoeless Joe Jackson, and Home Run Baker and bring them out to the ballpark, they'd certainly still understand the game, even though it has changed. Since their time, there has been the development of specialty pitchers, creation of artificial turf, and the addition of the designated hitter in the American League. The mound, strike zone, and scoreboards are all different too.
Player skills are still fundamentally the same, except more advanced. They would get the idea that a better player should command more money than a player with less skill. But the overall economics of the game would probably baffle those great players. That has added a new type of off-field strategy that owners who want to win must master. Since the turn of the 19th century, it's been the greatest change MLB has seen.
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December 3, 2021 - Media Release
2021 National Volunteers Month
The Public-Private Partnership Center joins the celebration of the National Volunteer Month this December pursuant to Presidential Proclamation No. 55, series of 1998. The month-long celebration aims to: Build nationwide public awareness and appreciation of volunteerism and the role it plays in social, economic and political development of the country. Create the environment for bolstering... continue reading
December 2, 2021 - In the News
Public, private partnership can sustain opening of economy – Concepcion
MANILA, Philippines — Presidential adviser for entrepreneurship and Go Negosyo founder Joey Concepcion yesterday stressed the importance of the partnership between private and public sectors for the sustainable reopening of the economy. "What we want to see moving forward is we continue to maintain the opening of the economy on a more sustainable basis. Yes,... continue reading
December 2, 2021 - Bid Announcement
Invitation to Bid-Supply and Delivery of 25 Units of Laptop Computers with Operating System and One (1) Unit Multipurpose Printer with ADF (Rebidding)
The Public-Private Partnership Center of the Philippines (PPP Center), through the National Expenditure Program (NEP) intends to apply the sum of One Million Five Hundred Twenty Eight Thousand (Php 1,528,000.00) being the ABC to payments under the contract for Supply and Delivery of 25 Units of Laptop Computers with Operating System (Procurement Project)/CN 2022-03. Bids... continue reading
November 29, 2021 - Media Release
PPP Center orients UP Manila on PPP concepts and processes
The PPP Center conducted an online capacity building activity (OCBA) titled PPP 101: Orientation on PPP Concepts and Processes for the University of the Philippines (UP) Manila on November 24, 2021 via Zoom. Vice Chancellor for Planning and Development Michael Tee led the participants from UP Manila composed of officials and staff from the Office... continue reading
November 28, 2021 - In the News
Private partners eyed for Kennon Road rehab
MANILA, Philippines — The government is now looking for potential partners from the private sector for the rehabilitation of the Kennon Road as part of efforts to boost tourism activities in Baguio City. The Public-Private Partnership Center said the Kennon Road project is undergoing another round of consultation with potential private sector partners through market... continue reading
CCLEX seen to drive tourism; Cordova braces for visitor influx
The once sleepy town of Cordova is preparing for the expected influx of tourists with the opening of the Cebu-Cordova Link Expressway (CCLEX) in March 2022. Guenevere Flores, the town's consultant on tourism, said the creation of a Tourism Code is now being crafted at Cordova's Sangguniang Bayan, which will also create a Tourism Division.... continue reading
Iloilo City to hold Dec. 1 investors' meet for P139-M slaughterhouse project
THE ILOILO City government is holding a conference on Dec. 1 to entice investors for its P139.35-million slaughterhouse upgrade project under a public-private partnership scheme. "The potential revenue for this project is worth your time and investment. May this meeting become a blueprint for local leaders, private partners, and participants in ensuring a dynamic and... continue reading
PPP Center, DPWH conduct market sounding activities on Kennon Road Project
The Rehabilitation/ Reconstruction / Improvement, Operation, and Maintenance of the Kennon Road Project is undergoing another round of consultation with potential private sector partners through market sounding activities from November 22-29, 2021. The results of the market sounding activities will form part of the feasibility study (FS) being conducted by the Department of Public Works... continue reading
PPP Center conducts workshop on developing PPP project concept notes
The Public-Private Partnership (PPP) Center conducted the PPP Project Concept Note (PCN) Development Workshop on October 18-29, 2021. The two-week webinar, attended by 42 participants from 12 government agencies, was conducted to guide implementing agencies in crafting PCNs, which is vital in the preparation of sound and viable PPP projects. The PPP PCN Development Workshop... continue reading
PPP Center holds 2nd batch of Introductory Course on Joint Venture Arrangements for LGUs
The Public-Private Partnership (PPP) Center held the second batch of Introductory Course on Joint Venture Arrangements (JVA) for Local Government Units (LGUs) on November 15-19, 2021 via Zoom. A total of 104 participants composed of officers and staff from 44 LGUs, one regional government agency, and two NEDA Regional Offices (NROs) attended the online capacity... continue reading


Alexandria, Va., January 8, 2013–ServRight, a leading nationwide provider of customized IT service and support solutions to manufacturers and channel partners, announced today that the company has completed the acquisition of OnSite Maintenance Center of Salt Lake City.
The terms of the agreement were not disclosed. OnSite will now be known as OnSite Maintenance, a ServRight Company.
"OnSite Maintenance is a world-class finishing equipment service provider to the Quick Print Industry," said Marty Evans, Vice President and General Manager of ServRight. "Their expertise in the document finishing market and their direct field force in 29 cities in the US and Canada are a perfect complement to ServRight's existing nationwide network of 3500 technicians.
"Now, in addition to ServRight's dominance in product areas such as Scanners, Printers, Servers, POS, and Digital Signage, with OnSite's skill sets in the document finishing market we are in an enviable position to expand our customized support solutions with our combined as well as new partners," said Evans.
Greg Clark, CEO and Founder of OnSite Maintenance, LLC., will remain engaged with the service business during 2013 as the two companies complete the merger.
OnSite Maintenance Center is a national service organization, dedicated to maintaining and repairing finishing, binding and framing equipment for the quick print and framing industries. Our technicians are located in every major city in the United States, so you know that you will have service support where ever you are. OnSite can be reached for service at 1-866-ONSITE6 (1-866-667-4836) or visit us at OnSite Maintenance, a ServRight Company.