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\section{Introduction}
Spallation reactions, i.e. proton-induced reactions on heavy targets at a few hundred MeV, have been the subject of many studies since 1950. They are known to be a valuable tool for the study of the de-excitation of hot nuclei because, contrarily to reactions between heavy ions, they lead to the formation of hot prefragments with only a limited excitation of the collective degrees of freedom such as rotation or compression. Their study has also been motivated by astrophysics, as cosmic rays undergo spallation reactions with the hydrogen and helium nuclei of the interstellar medium.
Recently, progresses in high-power accelerator technologies have made possible the realisation of intense neutron sources based on spallation reactions. Such sources are needed for Accelerator-Driven Systems~\cite{Bowmann,Rubbia}, and also find applications in nuclear physics~\cite{NToF} and for material physics and biology~\cite{ESS}. Furthermore, spallation reactions can also be used to produce exotic nuclei and, hence, secondary beams~\cite{ISOLDE,EURISOL}. Those new applications have motivated a large number of experimental works and created a strong demand for high-precision calculation codes.
In order to bring new data and therefore new constraints for the codes, measurements of reaction residues have been undertaken at GSI by an international collaboration. These experiments are based on the inverse-kinematics method. Fragments are identified in-flight using the FRS spectrometer~\cite{FRS}, making possible the first measurements of complete nuclide distributions. In the frame of those studies, production cross sections have already been published for several systems: Au+p at 800A~MeV~\cite{Au_Fanny,Au_Pepe}, Pb+p at 1A~GeV~\cite{Wlazlo,Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, U+p at 1A~GeV~\cite{Taieb,Monique} and U+d at 1A~GeV\cite{U_Enrique,U_Jorge}.
These results have helped to partially discriminate between the respective influence of the two main steps of the spallation process, the intranuclear cascade and the fission/evaporation process. The behavior of several codes (the ISABEL~\cite{ISABEL}, INCL4~\cite{INCL4} or BRIC~\cite{BRIC} intranuclear cascades, the ABLA fission/evaporation code~\cite{ABLA}) has proved to be now overall satisfactory for proton energies around 1 GeV. On the other hand, important failures in the description of the emission of charged particles in the Dresner evaporation code~\cite{Dresner} have been put in evidence~\cite{Au_Fanny}. In the 1 GeV energy region, the only serious, remaining deficiency is the underestimation of the lightest evaporation products, which are related to the most violent collisions. Despite large differences in the description of the spallation process, all the codes mentioned above present this weakness. This indicates that some phenomena have not been taken into account. In recent experiments conducted at GSI on lighter nuclei ($^{56}$Fe, $^{136}$Xe), our collaboration explored a range of nuclear temperatures higher than in the systems mentioned in the previous paragraph. Indications were found that fast break-up decay may play an important role in high-energy spallation reactions~\cite{Paolo}.
The question of understanding of the evolution of the reaction mechanisms with decreasing energy also remains open. This is an important point in the perspective of technical applications, because nuclear reactions in thick targets happen in a broad energy range: beam particles are subject to electronic slowing down, and also fast particles emitted in the first stage of the reactions are likely to produce additional nuclear reactions, giving rise to an {\it internuclear} cascade. To address the question of the dependance of the reaction on the energy of the incident particle, an experiment has been performed at GSI aiming at measuring production cross sections of residues in the spallation of lead by protons at 500A~MeV.
The present paper deals with the experimental results on the fragmentation-evaporation residues obtained in this experiment. It completes the results already published obtained during the same experiment for the fission products~\cite{NPA_Bea}. Detailed confrontations between the results from codes dedicated to the description of the spallation process and these data as well as other data on evaporation residues obtained by our group and other related measurements on spallation reactions like light particle production are postponed to a forthcoming paper.
The energy chosen for this experiment, which is low in comparison to the FRS standards for experiments involving nuclei as heavy as lead~\cite{achromat}, was a source of difficulties for the identification of the fragments in the spectrometer. The modified setting and the analysis methods developed for this experiment have been presented in a dedicated paper~\cite{NIM_loa}. We will briefly recall their main features in section~\ref{chap:setup}. We will then discuss in section~\ref{chap:secondary} the influence of multiple reactions taking place in the liquid hydrogen target and modifying the observed fragment production, and the method which has been employed to remove their contribution. In section~\ref{chap:kinematics} we will present the results on the reaction kinematics. Finally, in section~\ref{chap:results} we will present the production cross sections.
\section{Experimental setup and analysis process} \label{chap:setup}
The GSI synchrotron (SIS) was used to produce a 500A~MeV $^{208}$Pb pulsed beam with a pulse duration of 4 seconds and a repetition time of 8 seconds. The beam was sent onto a 87.3~mg/cm$^2$~liquid hydrogen target~\cite{target} located at the entrance of the FRagment Separator (FRS). The target window consisted of two 9~mg/cm$^2$~Ti foils on each side. The beam intensity was monitored during all the experiment by a beam-intensity monitor~\cite{SEETRAM}. In order to maximise the proportion of fully stripped fragments in the spectrometer, a 60~mg/cm$^2$~Nb foil was placed after the target.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{frs_2.eps}
\caption{Schematic view of the FRagment Separator. Each magnetic section between focal planes (the target location, $S_2$ and $S_4$) consists of two dipoles plus several quadrupoles and sextupoles (the latter are not represented here as they were not used during this experiment).}
\label{fig:FRS}
\end{center}
\end{figure}
Fragments were identified in-flight using the FRS spectrometer (see figure~\ref{fig:FRS}). The rigidity of the fragments in each of the two magnetic sections is given by:
\begin{equation}
B \rho = \frac{m_0 c}{e} \frac{A}{q} ( \beta \gamma )
\label{eqn:brho}
\end{equation}
where $B$ is the magnetic field, $\rho$ the curvature radius of the fragment trajectory, $A$ the mass number, $q$ the ionic charge of the fragment, and $\beta$ and $\gamma$ are the Lorentz relativistic coefficients. The presence of $q$ in equation~\ref{eqn:brho} is a critical point, as a large part of the fragments produced at the energy of 500A~MeV chosen for this experiment were not fully stripped.
In order to measure the nuclear charge ($Z$) of the fragment, 4 MUlti-Sampling Ionisation Chambers (MUSIC; see~\cite{MUSIC} for complete description) were placed at the exit of the spectrometer, each one filled with 2 bar of P10 gas mixture (90\% Ar, 10\% CH$_4$). The total gas thickness was 800~mg/cm$^2$, with the measurement of the energy loss effectively performed only in some 2/3 of the length of each chamber. Using such a large gas thickness was necessary in order to maximise the charge exchanges (electron pick-up and stripping) of the fragment in the gas, thus washing out the influence of the incoming ionic charge state of the fragment on the energy loss, and ensuring that the latter represents the nuclear charge of the fragment with sufficient resolution~\cite{NIM_loa}. The obtained resolution, $\Delta$Z{\it (FWHM)}/Z, ranged between 0.6\% for the lightest fragments and 0.9\% for the heaviest.
The horizontal position of the fragments at the intermediate and final focal planes (respectively $S_2$~and $S_4$; see figure~\ref{fig:FRS}) was measured using 3~mm thick plastic scintillators. The signals of these detectors were also used to measure the time of flight of the fragments in the second part of the spectrometer. The $A/q$~ratio of the fragment was deduced from the combination of these measurements, according to equation~\ref{eqn:brho}.
The thick aluminium degrader (1700~mg/cm$^2$) located at the intermediate focal plane ($S_2$) was used as a passive energy-loss measurement device. As the energy loss can be related to the variation of the magnetic rigidity and the ionic charge states in the two magnetic sections, the latter may thus be deduced from the energy loss as obtained from the nuclear-charge identification and the velocity measurement~\cite{NIM_loa}.
The resolution obtained in this measurement was not sufficient to discriminate the ionic charge states on an event-by-event basis. The only information obtained was the charge-state changing between the first and the second part of the spectrometer, an integer value that we will note $\Delta q$. Besides, the evaluation of the variation of magnetic rigidity was also used to reject fragments that underwent a nuclear reaction at $S_2$.
The mass of each fragment was determined assuming that its number of electrons in the FRS was the minimum required by its $\Delta q$ value~\cite{NIM_loa}. The production rate for each nuclide was then calculated by constructing its full velocity distribution in the first part of the FRS, using formula~\ref{eqn:brho}. For many nuclides, the momentum width was larger than the momentum acceptance of the FRS ($\pm1.5\%$); in this case several settings of the magnets were used in order to cover the full momentum distribution of the fragment. Due to the hypothesis made on the ionic charge state, some fragments were misidentified; the corresponding correction factor for the production rates was deduced from ionic charge-state probabilities calculated using the code GLOBAL~\cite{GLOBAL}.
The above procedure was performed separately for each group of fragments characterised by a given $\Delta q$ value. The probability of each $\Delta q$ value was then deduced from the scaling factor necessary for all isotopic distributions obtained for a given element to match with each other. The obtained values were found to be in good agreement with the GLOBAL calculations (discrepancies lower than 10\% for the most abundant ionic charge-state combinations, and less than 20\% for other combinations)~\cite{NIM_loa}.
The production rates were corrected for the losses in the different layers of matter located in the path of the fragments after the target area (degrader, plastic scintillators, MUSIC chambers). The total reaction cross sections were calculated using the Karol optical-model-based code~\cite{Karol}. Losses were found to be of the order of 30\%, including reactions in the MUSIC chambers (the latter being characterised by signals of first and last chambers being improperly correlated). The reaction rates in the different layers of matter are presented in table~\ref{table:reac_prob}. The dead time of the acquisition and the detector efficiencies were also taken into account. Fragment losses due to limited angular acceptance of the FRS were found to be negligible. All the measurements and the analysis procedure above were repeated with an empty target, and the resulting production rates were subtracted from the total production rates. Finally, the production cross sections were obtained by normalising the production rates to the number of atoms in the liquid hydrogen target, which had been measured in a previous experiment, and to the beam intensity.
\renewcommand{\arraystretch}{0.4}
\begin{table}[ht]
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& & & & & \\
Layer & Target windows & Stripper & Scintillator & Degrader & MUSICs \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Focal plane & \multicolumn{2}{c|}{$S_0$}&\multicolumn{2}{c|}{$S_2$}& $S_4$ \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& & & & & \\
Reaction & 2.1\% & 2.1\% & 8.4\% & 16.6\% & 4.9\% \\
probability & & & & & \\
& & & & & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
Method of & \multicolumn{2}{c|}{{\it none}} & \multicolumn{2}{c|}{$B\rho$ change} & $\Delta E$ change\\
rejection & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
& \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \\
\hline
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
Method of & \multicolumn{2}{c|}{Dedicated measurement} & \multicolumn{3}{c|}{Calculation} \\
correction & \multicolumn{2}{c|}{(empty target)} & \multicolumn{3}{c|}{(Karol model)} \\
& \multicolumn{2}{c|}{} & \multicolumn{3}{c|}{} \\
\hline
\end{tabular}
}
\caption{Reaction probabilities of the beam ($^{208}Pb$ at $500A~MeV$ at the entrance of the FRS) in the various layers of matter of the FRS beam line. Rejection method of the formed nuclei and correction method of the production rates are also mentioned. See text for details.}
\label{table:reac_prob}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
\section{Secondary reactions in the target} \label{chap:secondary}
Any fragment formed in a collision with a proton of the target may undergo additional nuclear collisions, in the target as well as in surrounding material (target window, stripper foil). These secondary reactions are expected to play an important role in the production of nuclides far from the projectile: at relativistic energies, proton-induced reactions mainly produce nuclei lighter than the heavy partner of the reaction, therefore in most cases multiple reactions will remove more nucleons than a single reaction.
There is no way to identify such multiple reactions during the analysis process. Therefore, one has to unfold their contribution by using calculated reaction cross sections or by performing a self-consistent calculation. This section is dedicated to the presentation of a new method developed for this experiment aiming at estimating the contribution of the multiple reactions with a high precision while minimising the input from codes.
\subsection{Unfolding method}
The production cross section of a nuclide $f$ from projectile (indexed as $0$ in the following) is written as:
\begin{equation}
\sigma_{0\rightarrow f} = \frac{e^{\frac{\sigma_0+\sigma_f}{2}x}}{x} \;
\left(T_f(x) - \frac{x^2}{2} \sum_{A_0 < A_i < A_f}^{} \;
\sigma_{0\rightarrow i} \; \sigma_{i\rightarrow f} \;
e^{-\frac{\sigma_0+\sigma_i+\sigma_f}{3}x} \right)
\label{eqn:secondary}
\end{equation}
where $\sigma_i$ is the total reaction cross sections of a nuclide $i$ on a nuclide of the target, $\sigma(i,j)$ is the production cross section of a nuclide $(A_j,Z_j)$ from a nuclide $(A_i,Z_i)$, $T$ is the observed production rate, and $x$ is the thickness of the target. A derivation of this equation is presented in appendix~\ref{chap:sec_calc}. This corresponds to a first-order approximation (i.e. only double reactions in the hydrogen are taken into account), but it is easily extended to higher orders. In our calculations, we actually accounted for the second order reactions: triple reactions in hydrogen, and reactions involving one reaction in a target window and one in the hydrogen (or the reverse). We found that those second order terms actually accounted for less than 20\% of the multiple reactions.
Solving this system of equations (one equation for each observed nuclide) requires the calculation of all the $\sigma$ terms. For the total cross sections, several reliable codes exist; we used the optical-model-based code of Karol~\cite{Karol}, the same one we used for the probability of nuclear reactions at the intermediate focal plane of the FRS. For the partial cross sections involving heavy target nuclei (target windows and stripper foil), the EPAX parametrisation~\cite{EPAX} offers reliable results with minimal calculation time. In the case of proton-induced reactions, the Monte-Carlo cascade-evaporation codes would seem an obvious choice, but they could hardly be used here for two reasons. First, the calculation required to evaluate the hundreds of possible reactions would have been very time consuming. Second, as one of the goals of this experiment was to produce data to constrain these codes in this poorly-known energy region, the use of those codes might have introduced an artificial consistency between the data and the codes.
In order to calculate the isotopic cross sections, we can decompose each cross section of proton-induced reactions in a product of 3 factors:
\begin{equation}
\sigma(x,y) = \sigma_x \; P_A((A_x,Z_x) \rightarrow A_y )
\; P_Z((A_x,Z_x,A_y)\rightarrow Z_y)
\end{equation}
Here, the first term is the total reaction cross section of a nuclide $x$ (we have already stated that it could be calculated using the Karol formula~\cite{Karol}), the second term is the probability to form a nuclide of mass $A_y$ from a nuclide of mass $A_x$, and the third term is the probability that the nuclide formed with a mass $A_y$ has an atomic number $Z_y$.
In order to estimate the second term, we used a property of proton-induced spallation reactions: nearly all nuclides $b$ formed from a nuclide $a$ have a mass strictly smaller than the one from $a$. For example, in the 500A~MeV experiment on $^{208}$Pb we observed no formation of any nuclide of mass 209, and nuclides of mass 208 ($^{208}$Bi) are formed in less than 0.1\% of the reactions. Furthermore, we assumed that, as far as only the probability of mass loss is concerned, the influence of the isospin of the target nuclide is weak enough to be neglected. Using these assumptions, one can solve the system of equations~\ref{eqn:secondary} isobar by isobar, in the decreasing masses order, because the term $P_A(A_x \rightarrow A_y )$ required by each equation is immediately obtained from the previously corrected data as $P_A(A_0-(A_x-A_y))$ (where $A_0$ is the projectile mass).
For the third term, this kind of simple scaling law cannot be applied because, although $P_Z$ depends only on $A_y$ for large values of $A_x-A_y$. Indeed, this well-known property defines the so-called residue corridor~\cite{Dufour}: the statistical nature of the evaporation process favors its ending close to nuclei for which the probability to evaporate a neutron and a proton (in other words, the neutron and proton separation energies), are the closest, a property which is completely independent of the entrance channel. But, in the case of short evaporation chains, the influence of the entrance channel is not suppressed; in other words, for low values of $A_x-A_y$, $P_Z$ depends not only on $A_y$ but also on $Z_x$ . This memory effect is fully taken into account in the EPAX parametrisation~\cite{EPAX}. We will describe hereafter how the EPAX formula can be used even though it is out of its energy domain applicability. Please note that, as EPAX does not take into account the fission process, this method would not be appropriate for reactions involving highly fissile nuclei such as uranium.
\subsection{Calculation of isobaric distributions using EPAX}
Some characteristics of the EPAX parametrisation~\cite{EPAX} correspond to the requirements for a multiple reactions calculation: it needs very little computing time, and it proved to be reliable, not only for reactions involving nuclides close to the stability valley, but also for proton-rich nuclides~\cite{112Sn}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{epax_a.eps}
\caption{Comparison of production rates obtained in the Pb+p at 500A~MeV experiment (dots) with calculations performed with both the standard (discontinuous lines) and a version we modified (continuous lines) of the EPAX parametrisation~\cite{EPAX}. For each isobaric spectrum, calculations have been renormalized to the data.}
\label{fig:EPAX}
\end{center}
\end{figure}
EPAX has been written in order to reproduce residues from reactions in the limiting-fragmentation regime~\cite{limit_frag}, which is reached in spallation only for projectile energies of several GeV. The mass distribution of residues formed in 1A~GeV proton-induced spallation reactions exhibits a very different shape from the one formed in the limiting-fragmentation regime~\cite{Pb_p}. Therefore, the residues from the same reaction with half the incident energy may certainly not be reproduced by the mass-loss formula of EPAX.
On the other hand, the shape of the isobaric spectra is mainly a consequence of the sequential evaporation mechanism. Therefore, there is no reason why its validity should be limited to high incident energies. To check this assumption we extracted the isobaric component of the EPAX formula and compared it to our data after proper renormalization for each isobaric spectrum. Only minor adjustments were necessary to obtain a very satisfactory reproduction of the measured production rates, as it can be seen in figure~\ref{fig:EPAX}. Such a comparison makes sense as the secondary reactions are not expected to play an important role in the production of nuclides with a mass loss of less than, roughly, 30 mass units with respect to the projectile. Unexpectedly, we observed that the charge-pickup reactions were also reasonably well reproduced by the parametrisation, despite the fact that the EPAX authors didn't take this phenomena into account during the development process.
\subsection{Contribution of the multiple reactions in the target}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{secondary72.eps}
\includegraphics[width=0.45\textwidth]{secondary78.eps}
\caption{Production cross sections in hydrogen before and after the subtraction of the multiple reactions (empty and full dots, respectively), and contribution of the multiple reactions in the target (continuous lines), for Hf (left) and Pt (right) isotopes.}
\label{fig:secondary}
\end{center}
\end{figure}
The results of multiple-reaction calculations are presented in figure~\ref{fig:secondary} for 2 isotopic distributions, each one corresponding to an extreme situation regarding the contribution of the multiple reactions. For $Z$ around 78, the multiple reactions are an important contributor for very proton-rich nuclides only. Their contribution increases and spreads towards neutron-rich nuclides with decreasing $Z$. The very proton-rich part of the isotopic distributions of the light elements such as Hf is reproduced by our calculation with differences being less than 20\% in most cases. This demonstrates the validity of our approach. As the uncertainty on this calculation could not be estimated in a systematic way, we quoted the value of 20\% mentioned above.
We chose to consider as results of the experiment only the cross sections deduced from production rates for which multiple reactions contributed for less than 50\%. This discards nearly all nuclides with $Z<70$, which represent only a very small fraction of the fragmentation residues.
\section{Kinematics of the reaction} \label{chap:kinematics}
Once nuclei are identified, their velocity in the first part of the FRS can be calculated using the equation~\ref{eqn:brho}:
\begin{equation}
( \beta \gamma )_1 = \frac{(B\rho)_1}{A/q_1}
\end{equation}
Here the index 1 stands for the first part of the FRS (before $S_2$). Using this technique, the resolution is expected to be of the same order as the one obtained for the magnetic rigidity, roughly $5.10^{-4}$. At the energy used in this experiment, this is by a factor of 3 better than what can be achieved by a time-of-flight measurement in the second part of the FRS. This high resolution makes the FRS a remarkable tool to study the kinematics of nuclear reactions.
We have already pointed out that, because of the limited momentum acceptance of the FRS, the reconstruction of the full velocity spectra of each nuclide is a necessary step in order to evaluate the production rates (section~\ref{chap:setup}). The measured velocity spectra are Lorentz transformed into the reference frame of the beam, and corrected for the contribution of the beam width and for the velocity scattering due to the passage through the target and the surrounding materials. The resulting spectra give direct access to the longitudinal momentum transfer and to the longitudinal momentum spread caused by the nuclear reactions.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{p_parrallel_mean.eps}
\includegraphics[width=0.48\textwidth]{sigma_mean.eps}
\caption{Momentum transfer (left) and momentum width (right) measured in the reactions Pb+p at 500A~MeV (triangles), Pb+p at 1A~GeV (squares) and Au+p at 800A~MeV (circles). Data are compared to Morrissey systematics~\cite{Morrissey} (continuous lines) and Goldhaber formula~\cite{Goldhaber}, the later being computed with a Fermi momentum of 118~MeV.c$^{-1}$ (dashed line) and 95~MeV.c$^{-1}$ (dashed-dotted line). All data have been normalised following the Morrissey prescription.}
\label{fig:kin}
\end{center}
\end{figure}
In figure~\ref{fig:kin} these quantities are compared to results of previous spallation experiments as well as to the well-known Morrissey systematics~\cite{Morrissey} and to the Goldhaber formula~\cite{Goldhaber}. The data were averaged over each isobaric distribution using the production cross sections as weighting factor.
A very similar tendency is obtained for all the experimental data regarding the momentum transfer. The simple linear dependence proposed by the Morrissey systematics is not fulfilled by the experiment. The longitudinal momentum transfer is underestimated for fragments corresponding to a mass loss of 10 to 45 units with respect to the projectile. This underestimation vanishes with increasing mass losses.
The momentum width exhibit a linear dependance to the square root of the mass loss. The Morrissey systematics offers a fair reproduction of the data. Using the result of a direct measurement of the Fermi momentum (118~MeV/c)~\cite{fermi_mes}, the Goldhaber formula overestimates the momentum width. This is not unexpected as this formula only takes into account the nucleons removed during the cascade stage, which lead to larger momentum fluctuations with respect to the nucleons emitted in the evaporation phase~\cite{Hanelt}. Nevertheless, a better agreement with data can be obtained by using an arbitrary Fermi momentum value of 95~MeV/c, as often done in heavy-ion calculations. The dispersion between the different data sets is probably related to the delicate corrections applied to the data, namely the beam width in momentum and position, which are difficult to estimate.
\section{Production cross sections} \label{chap:results}
\subsection{Isotopic cross sections}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{isotopic.eps}
\caption{Isotopic production cross-section distributions of residues in the reaction Pb+p at 500A~MeV.}
\label{fig:cs}
\end{center}
\end{figure}
Figure~\ref{fig:cs} shows the measured distributions of isotopic production cross-sections for elements between erbium and bismuth (see appendix~\ref{chap:annexe_xs} for the full list of cross sections). Some 250 spallation-evaporation cross sections have been measured. One observes that the cross sections of isotopic chains vary smoothly. As no even-odd fluctuations are expected~\cite{Valentina}, and considering the statistical nature of the evaporation process, this corroborates that the measured production rates do not hide any other source of fluctuations. The upper limit of 50\% production of multiple reactions in the target, that we have decided to set, removed from the distributions a growing part of the lightest isotopes when $Z$ is decreasing. The most neutron-rich Pt and Ir isotopes have not been measured because of missing settings of the magnetic fields during the experiment.
\subsection{Total cross sections}
\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c}
\hline
Reaction & $^{208}$Pb+p & $^{208}$Pb+p & $^{208}$Pb+d \\
& (500A MeV) & (1A GeV) & (1A GeV) \\
\hline
\hline
Spallation-evaporation (measured) & 1.44 (0.21) & 1.68 (0.22) & 1.91 (0.24)\\
Total (measured) & 1.67 (0.23) & 1.84 (0.23) & 2.08 (0.24) \\
Total (calculated) & 1.70 & 1.80 & 2.32 \\
\hline
\end{tabular}
\caption{Spallation cross sections (in barns) for reactions $^{208}$Pb+p at 500A~MeV and 1A~GeV, and $^{208}$Pb+d at 1A~GeV. The measured spallation-evaporation and total cross sections (adding fission) are compared to a Glauber calculation performed using updated density distributions. Values in parenthesis are the total uncertainty of the measurements.}
\label{tab:total_cs}
\end{center}
\end{table}
We have estimated the total production cross-section of evaporation residues by summing all the measured residue cross sections, obtaining a value of (1.44$\pm$0.21)~b. Our measurement does not strictly cover all the range of the possible residues. However, the nuclides for which we have no measurement are mostly the lightest fragmentation products. Considering the steepness of the mass curve in this region (see figure~\ref{fig:cs_mass}), we can assume that the contribution of these nuclides is small, and probably much smaller than the error bars.
Adding the fission cross-section for this reaction~\cite{NPA_Bea} we estimated the total cross section to (1.67 $\pm$ 0.23) b.
This value is very close to the 1.70~b found by a Glauber-type calculation performed with updated density distributions. On table~\ref{tab:total_cs} we compare those values with the ones obtained in the reactions Pb+p and Pb+d at 1A~GeV. Although the slight decrease of the total cross section with respect to 1A~GeV measurements is in agreement with the expected trend, the 500A~MeV fission cross section is higher than previous measurements conducted at this energy, and also higher than the systematics of Prokofiev~\cite{Prokofiev}. This question has been discussed in detail in the corresponding paper~\cite{NPA_Bea}. Let us only underline that the agreement of a well-established model with our experimental total cross section comes in support of our measurement.
\subsection{Comparison to radiochemical measurements}
In recent years, a large number of measurements of spallation residues have been performed by the team of R. Michel. Of special relevance to our work is the measurement of residues of spallation of natural lead by protons at 550~MeV published by Gloris \etal~\cite{Gloris}. Cross sections with independent yields ({\it i.e.} nuclides that are not produced by $\beta$ decay) can be compared directly, while cross sections corresponding to accumulation of $\beta$-decaying nuclei require a summation of our data along the decay chain.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.9\textwidth]{gloris.eps}
\caption{Ratio between production cross-sections of residues measured in the reaction $^{208}$Pb+p at 500A~MeV (this work) and in the reaction p+$^{nat}$Pb at 550~MeV as a function of the mass of the residue for the isotopes measured in~\cite{Gloris}. Filled and empty circles represent nuclides with independent yields and cumulated yields, respectively. Calculations of the ratio between production cross sections at 550 and 500 MeV have been performed in two systems: with the same $^{208}$Pb target nuclei (continuous line) and with a different nucleus, $^{207}$Pb (dashed line), in order to study the effect of the use of natural lead in the Gloris experiment (see text).}
\label{fig:chemistry}
\end{center}
\end{figure}
The ratio between our data and those of Gloris \etal\ are presented in figure~\ref{fig:chemistry}. In the case of the cumulated yields, cross sections measured at GSI have been summed along the decay chain in order to be comparable with the radiochemical measurements. The agreement between the two data sets is overall fair for heavy residues, although a systematic shift of roughly 10\% may be guessed. Considering the error bars, all the measurements seem to be compatible, with the exception of $^{203}$Pb and $^{202}$Tl. As we have already underlined, our data points are very consistent with respect to one another. This makes such a large error in our measurement for these two nuclides rather unlikely, since it should have been clearly visible on our isotopic distributions.
For lighter nuclei the ratio decreases rapidly with increasing mass loss with respect to the projectile. This effect is the direct consequence of the differences between the measured systems: the 10\% higher energy strongly favors the production of lighter residues, up to a factor of 3 for mass losses around 35 nucleons.
This statement can be checked by using Monte-Carlo calculations. For this purpose we used the ISABEL~\cite{ISABEL} intranuclear cascade and the ABLA~\cite{ABLA} evaporation code. Although one of the purposes of these measurements is precisely to check the validity of those codes in the few hundreds of MeV region, we can assume that they are reliable enough if one only wants to calculate variations in a very limited energy and mass range, as it is the case here.
Results of the calculation of the ratio between isobaric cross sections for the two systems are also represented in figure~\ref{fig:chemistry}. A calculation that only takes into account the different incident energies offers a satisfactory reproduction of the decrease of the ratio for the light fragments. Replacing $^{208}$Pb by $^{207}$Pb (in order to mock the isotopic mixing of natural lead of which the targets of Gloris experiment were made) leads to a slight reduction of the calculated ratio for light nuclides, which improves the agreement with the data in this mass range. For heavy nuclides the calculations indicate that results at 500A~MeV should be larger than at 550A~MeV, which is not what we observed for most of the points. However, only 3 points are not compatible with the calculations when one considers the error bars. This leads us to conclude that, taking into account the differences between the systems measured in the Gloris experiment and our experiment, the agreement between these data sets is satisfactory.
\subsection{Mass spectra and comparison to previous GSI experiments}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.95\textwidth]{mass_loss.eps}
\caption{Production cross-sections of the residues of the reaction Pb+p at 500A~MeV as a function of the mass loss with respect to the projectile (full circles). Data obtained in previously mentioned experiments are also represented: Au+p at 800A~MeV (triangles), Pb+p (squares) and Pb+d (diamonds) at 1A~GeV. The isolated points at $\Delta A=0$ correspond to a single nuclide, $^{208}$Bi.}
\label{fig:cs_mass}
\end{center}
\end{figure}
Figure~\ref{fig:cs_mass} presents the production cross sections, summed for all isobars, as a function of the mass loss with respect to the projectile. The data obtained from several experiments performed at the FRS are presented here: Pb+p at 500A~MeV, Pb+p at 1A~GeV~\cite{Pb_p}, Pb+d at 1A~GeV~\cite{Pb_d}, and Au+p at 800A~MeV~\cite{Au_Fanny}.
For small mass losses, each spectrum has a nearly constant value. In this mass range, the lower the incident energy, the higher the cross sections. With increasing mass losses, the cross sections start to decrease. Here, the lower the energy, the earlier and the steeper the fall. This is easily understood as the direct consequence of the exploration by the prefragment of all the possible range of excitation energy available in each system. In this respect, the measurement with deuterons gives insights about what would be obtained in a measurement conducted with protons at twice the energy. The shape of the mass-loss curve obtained from the measurement on gold is fully compatible with the tendencies observed for lead.
In the 500A~MeV experiment, a clear separation exists between the group of the evaporation products (which does not extend beyond mass losses of 40 mass units) and the group of the fission product (which starts around mass losses of 70~\cite{NPA_Bea}). This absence of mixing could also be demonstrated by studying the velocity spectra of the light fragments, which all exhibit a quasi-perfect Gaussian shape, while the presence of fission products would have introduced a characteristic double-bumped shape due to the forward-backward selection of the fission fragments by the FRS~\cite{Pb_p}.
\subsection{Charge pick-up}
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=0.45\textwidth]{charge_pickup.eps}
\includegraphics*[width=0.45\textwidth]{iso_pickup.eps}
\caption{Left figure: charge-pickup cross sections measured on lead (full symbols) and gold (empty symbols) as a function of the incident energy. The sum of the partial cross sections measured at the FRS (this work, full dots; Keli\'c \etal~\cite{Kelic}, full squares; Rejmund \etal~\cite{Au_Fanny}, upward triangles) is compared to elemental cross sections from Waddington \etal~\cite{Waddington} (downward triangles) and Binns \etal~\cite{Binns} (diamonds), which were both extracted from CH$_2$ and C measurements. Right figure: isotopic charge-pickup cross sections at 3 energies as a function of the mass loss with respect to the heavy partner of the reaction.}
\label{fig:pickup}
\end{center}
\end{figure}
An especially interesting result of this experiment is the measurement of the production cross section of 15 isotopes of Bi (see figure~\ref{fig:pickup}). Those nuclides are formed by charge-pickup reactions. In the energy range considered here, the capture of the incident proton is not initially possible, as the incident proton energy is well above the Fermi energy of the target nuclide. Therefore the formation of $^{209}$Bi is improbable, and the formation of $^{208}$Bi is only possible via, either the formation of a resonant state ($\Delta$ and pions), or a quasi-elastic collision between the incident proton and a neutron from the target nuclide, in which the neutrons leaves with an energy very close to the initial energy of the proton. The cross section for the charge-pickup is one of the few data that bring direct constraints for the intranuclear-cascade codes.
In the left part of figure~\ref{fig:pickup} we compare our measurement of the total charge-pickup cross section to previous measurements performed by our collaboration~\cite{Au_Fanny,Kelic}, Waddington \etal~\cite{Waddington} and Binns \etal~\cite{Binns}. For a qualitative discussion we do not need to discriminate between gold and lead as those nuclides are close to one another, both in atomic number and mass. Our measurement confirms the trend of a strong increase of the total cross section of the charge-pickup with decreasing energy.
This increase of the Bi production in Pb+p experiments concerns all Bi isotopes, as it can be seen in the right part of figure~\ref{fig:pickup}. The shapes of the 500A~MeV and 1A~GeV distributions are overall similar, but the overproduction at 500A~MeV increases slowly from a factor of 2 for the heaviest isotopes to a factor of 4 for the lightest. Problems in the separation of the ionic charge states~\cite{NIM_loa} prevented us to use the kinematic spectra to distinguish between the respective contribution of the $\Delta$ resonance and the quasi-elastic reactions in the formation of the heaviest Bi isotopes, as it was done by Keli\'c \etal~\cite{Kelic}.
The shape of the Hg spectrum (obtained in the Au+p at 800A~MeV measurement) is slightly different from the lead spectra. On one hand, for mass losses up to 7 mass units, the shape of the isotopic distribution is nearly identical to the Pb spectra, with values in-between the two Pb experiments, which is consistent with a smooth evolution as a function of the projectile energy. On the other hand the production of the lightest isotopes decreases faster than in the Pb+p experiments. This difference in shape can be explained by the lower Coulomb barrier and the shorter distance from the residue corridor~\cite{Dufour} for Hg nuclides with respect to Bi nuclides, which favor the emission of protons by the excited prefragments~\cite{Summerer_pickup}.
\subsection{Isobaric cross sections}
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.95\textwidth]{isobaric3.eps}
\caption{Isobaric spectra of production cross-sections (in mb) in the reactions Pb+p and Pb+Ti at 500A~MeV (full circles and crosses, respectively). The data obtained in the experiments Au+p at 800A~MeV \cite{Au_Fanny}, Pb+p and Pb+d at 1A~GeV \cite{Pb_p,Pb_d} are also plotted (triangles, squares and diamonds, respectively). }
\label{fig:cs_isobaric}
\end{center}
\end{figure}
In figure~\ref{fig:cs_isobaric} the data from the same experiments as in previous sections are plotted as isobaric spectra.
For heavy fragments, all distributions issued from reactions of Pb with p or d are very similar, both in shape and in magnitude. Low-energy reactions slightly dominate the cross sections for masses down to 185. With decreasing masses, the isobaric spectra behave in accordance with the mass distributions: the spectra for the highest-energy reaction scale down very slowly, whereas this scaling is steeper and steeper when one considers reactions at decreasing incident energy. However, for each isobar, the shapes of the different spectra remains extremely similar, as does its centroid (this can be seen in the left part of figure~\ref{fig:cs_means}).
Data from the reactions of Pb on the dummy target (which consists mainly of Ti in the target itself and Nb for the stripper foil placed after the target) at 500A~MeV have been added to the figure~\ref{fig:cs_isobaric} in order to illustrate the so-called limiting-fragmentation regime~\cite{limit_frag}. We observe no difference of shape or centroid between the spectra issued from the reaction on heavy ions and from the reaction on hydrogen isotopes.
The gold data offer an interesting point of comparison with the lead data. The gold fragments with mass close to 197 are associated with rather cold reactions and have therefore kept a $A/Z$ ratio very close to the initial system, while Pb fragments close to the same mass have lost roughly 10 nucleons, mostly neutrons because of the hindrance of charged-particle emission due to the Coulomb barrier. Therefore the gold and lead residue spectra are strongly shifted with respect to one another. This shift slowly vanishes with the increasing mass loss, which is easily understood as the slow move of the gold fragment distributions towards the residue corridor~\cite{Dufour}. This corridor is clearly visible on the left part of figure~\ref{fig:cs_means}: the barycenter of the isobaric distributions of the residues of all reactions converge on the same line.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=0.45\textwidth]{mean_z.eps}
\includegraphics*[width=0.45\textwidth]{mean_a.eps}
\caption{Average atomic number as a function of the mass of the residue (left figure) and average mass of the residue as a function of the atomic number (right figure) in the reactions Pb+p at 500A~MeV (full circles, this work), Au+p at 800A~MeV (triangles, \cite{Au_Fanny}) and Pb+p at 1A~GeV (squares, \cite{Pb_p}). In the calculation of the average values, points have been weighted according to their cross section.}
\label{fig:cs_means}
\end{center}
\end{figure}
This universal behavior, well known for reactions between heavy ions, is here demonstrated to be valid in a very broad energy range, even for fragments which are at the very end of the mass distribution. In other words, the isobaric distributions are independent of the incident energy in the system studied if properly renormalized. This is an experimental proof that the factorisation hypothesis is valid at energies as low as a few hundreds of MeV. This further strengthens the discussion regarding the agreement between EPAX and the data obtained from proton-induced reactions in systems in which fission does not play a major role (section~\ref{chap:secondary}).
Conversely, various projectile energies lead to variations of the center of the residues isotopic distributions, as it can be seen on the right part of the figure~\ref{fig:cs_means} as a deviation of the average mass value of the residues produced at 500A MeV. If this effect is washed out by the slow variations of the mass curve for higher energy reactions, it becomes noticeable at lower energies when the fall of the mass distribution becomes so steep that the production of the most neutron-deficient isotopes is strongly inhibited. Therefore, the reproduction of the isotopic and elemental cross-sections using a scaling factor between different systems is not appropriate at energies below the fragmentation limit.
\section{Conclusion}
The production cross sections and the momentum distributions have been measured for about 250 nuclei formed in the reaction of $^{208}$Pb on protons at 500A~MeV, covering most of the nuclides created down to a mass loss of 40 units with respect to the projectile, and with cross sections as low as 5~$\mu$b. The reaction products were identified in-flight in atomic number and mass-over-ionic-charge using the FRS spectrometer. The large proportion of non-fully stripped ions in the spectrometer was accounted for in detail, thus allowing to calculate the production cross section for each nuclide. The contribution of multiple reactions in the target to the residue production was carefully subtracted.
The production cross sections are in good agreement with previous radiochemical measurements. The isobaric distributions of the production cross sections are found to be very close to the ones measured at higher energies, thus extending the validity range of the factorisation hypothesis to energies of a few hundreds of MeV. The large variations observed on the isotopic cross sections can be nearly fully ascribed to the variations of the residue distributions with mass-loss at decreasing energy. Kinematical data are consistent with previous measurements.
The data obtained in this experiment, combined with previous measurements performed with the same technique (especially in the same system at 1A~GeV), constitute a set of information that is highly relevant for the development of reliable nuclear-reaction codes and, thus, the design of ADS.
|
{
"timestamp": "2005-12-09T12:08:54",
"yymm": "0503",
"arxiv_id": "nucl-ex/0503021",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0503021"
}
|
\section{Introduction}
The purpose of this paper is to study the structure of the bounded derived
category $\Dbcoh(\boldsymbol{E})$ of coherent sheaves on a singular irreducible
projective curve $\boldsymbol{E}$ of arithmetic genus one.
In the smooth case, such structure results are easily obtained from Atiyah's
description \cite{Atiyah} of indecomposable vector bundles over elliptic
curves. However, if $\boldsymbol{E}$ has a node or a cusp, some crucial
properties fail to hold. This is illustrated by the following table.
\begin{center}
\begin{tabular}[t]{p{6cm}|c|c}
&smooth&singular\\ \hline
homological dimension of $\Coh_{\boldsymbol{E}}$
&$1$&$\infty$\\ \hline
Serre duality holds&in general&\multicolumn{1}{p{3cm}}{
with one object being perfect}\\ \hline
torsion free implies locally free&yes&no\\ \hline
indecomposable coherent sheaves are semi-stable&yes&no\\ \hline
any indecomposable complex is isomorphic to a shift of a sheaf&yes&no\\
\hline
\end{tabular}
\end{center}
Despite these difficulties, the main goal of this article is to find the common
features between the smooth and the singular case. A list of such can be
found in Remark \ref{rem:common}.
In Section \ref{sec:background}, we review the smooth case and highlight where
the properties mentioned above are used. Our approach was inspired by
\cite{LenzingMeltzer}.
Atiyah's algorithm to construct indecomposable vector bundles of any slope can
be understood as an application of a sequence of twist functors with spherical
objects. From this point of view, Atiyah shows that any indecomposable object
of $\Dbcoh(\boldsymbol{E})$ is the image of an indecomposable torsion sheaf
under an exact auto-equivalence of $\Dbcoh(\boldsymbol{E})$.
In the case of a singular Weierstra{\ss} curve $\boldsymbol{E}$, as our main
technical tool we use Harder-Narasimhan filtrations in
$\Dbcoh(\boldsymbol{E})$, which were introduced by Bridgeland
\cite{Stability}. Their general properties are studied in Section
\ref{sec:HNF}.
The key result of Section \ref{sec:dercat} is the preservation of stability
under Seidel-Thomas twists \cite{SeidelThomas} with spherical objects. This
allows us to show that, like in the smooth case, any category of semi-stable
sheaves with fixed slope is equivalent to the category of coherent torsion
sheaves on $\boldsymbol{E}$.
In the case of slope zero, this was shown in our previous work
\cite{BurbanKreussler}. For the nodal case, an explicit description of
semi-stable sheaves of degree zero via \'etale coverings was given there as
well.
A combinatorial description of semi-stable sheaves of arbitrary slope over
a nodal cubic curve was found by Mozgovoy \cite{Mozgovoy}.
On the other hand, a classification of all indecomposable objects of
$\Dbcoh(\boldsymbol{E})$ was presented in \cite{BurbanDrozd}. A description of
the Harder-Narasimhan filtrations in terms of this classification is a task for
future work.
However, if the singular point of $\boldsymbol{E}$ is a cusp, the description
of all indecomposable coherent torsion sheaves is a wild problem in the sense
of representation theory, see for example \cite{Drozd72}. Nevertheless,
stable vector bundles on a cuspidal cubic have been classified by Bodnarchuk
and Drozd \cite{Lesya}.
It turns out that semi-stable sheaves of infinite homological dimension are
particularly important, because only such sheaves appear as Harder-Narasimhan
factors of indecomposable objects in $\Dbcoh(\boldsymbol{E})$ which are not
semi-stable (Proposition \ref{prop:extreme}).
The main result (Proposition \ref{prop:spherical}) of Section \ref{sec:dercat}
is the answer to a question of Polishchuk, who asked in \cite{YangBaxter},
Section 1.4, for a description of all spherical objects on $\boldsymbol{E}$.
We also prove that the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of spherical objects.
In Section \ref{sec:tstruc} we study $t$-structures on $\Dbcoh(\boldsymbol{E})$
and stability conditions in the sense of \cite{Stability}.
We completely classify all $t$-structures on this category (Theorem
\ref{thm:tstruc}). This allows us to
deduce a description of the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ (Corollary \ref{cor:auto}).
As a second application, we calculate Bridgeland's moduli space of
stability conditions on $\boldsymbol{E}$ (Proposition \ref{prop:stabmod}).
The hearts $\mathsf{D}(\theta,\theta+1)$ of the $t$-structures constructed in
Section \ref{sec:tstruc} are finite-dimensional non-Noetherian Abelian
categories of infinite global dimension.
In the case of a smooth elliptic curve, this category is equivalent to the
category of holomorphic vector bundles on a non-commutative torus in the
sense of Polishchuk and Schwarz \cite{PolSchw}, Proposition 3.9.
It is an interesting problem to find such a differential-geometric
interpretation of these Abelian categories in the case of singular
Weierstra{\ss} curves.
Using the technique of Harder-Narasimhan filtrations, we gain new insight into
the classification of indecomposable complexes, which was obtained in
\cite{BurbanDrozd}.
It seems plausible that similar methods can be applied to study the derived
category of representations of certain derived tame associative algebras, such
as gentle algebras, skew-gentle algebras or degenerated tubular algebras, see
for example \cite{BuDro}.
The study of Harder-Narasimhan filtrations in conjunction with the action of
the group of exact auto-equivalences of the derived category should provide new
insight into the combinatorics of indecomposable objects in these derived
categories.
\textbf{Notation.} We fix an algebraically closed field $\boldsymbol{k}$ of
characteristic zero. By $\boldsymbol{E}$ we always denote a Weierstra{\ss}
curve. This is a reduced irreducible curve of arithmetic genus one, isomorphic
to a cubic curve in the projective plane. If not smooth, it has precisely one
singular point $s\in\boldsymbol{E}$, which can be a node or a cusp.
If $x\in\boldsymbol{E}$ is arbitrary, we denote by $\boldsymbol{k}(x)$ the
residue field of $x$ and consider it as a sky-scraper sheaf supported at $x$.
By $\Dbcoh(\boldsymbol{E})$ we denote the derived category of complexes of
$\mathcal{O}_{\boldsymbol{E}}$-modules whose cohomology sheaves are coherent
and which are non-zero only in finitely many degrees.
\textbf{Acknowledgement.} The first-named author would like to thank
Max-Planck-Institut f\"ur Mathematik in Bonn for financial support.
Both authors would like to thank Yuriy Drozd, Daniel Huybrechts, Bernhard
Keller, Rapha\"el Rouquier and Olivier Schiffmann for helpful discussions,
and the referee for his or her constructive comments.
\section{Background: the smooth case}\label{sec:background}
The purpose of this section is to recall well-known results about the structure
of the bounded derived category of coherent sheaves over a smooth elliptic
curve. Proofs of most of these results can be found in \cite{Atiyah},
\cite{Oda}, \cite{LenzingMeltzer} and \cite{Tu}.
The focus of our presentation is on the features and techniques
which are essential in the singular case as well.
At the end of this section we highlight the main differences between the smooth
and the singular case. It becomes clear that the failure of Serre duality is
the main reason why the proofs and even the formulation of some of the main
results do not carry over to the singular case.
The aim of the subsequent sections will then be to overcome these difficulties,
to find correct formulations which generalise to the singular case and to
highlight the common features of the bounded derived category in the smooth and
singular case.
With the exception of subsection \ref{subsec:diff}, throughout this section
$\boldsymbol{E}$ denotes a smooth elliptic curve over $\boldsymbol{k}$.
\subsection{Homological dimension}
For any two coherent sheaves $\mathcal{F}, \mathcal{G}$ on $\boldsymbol{E}$,
Serre duality provides an isomorphism $$\Ext^{\nu}(\mathcal{F},\mathcal{G})
\cong \Ext^{1-\nu}(\mathcal{G},\mathcal{F})^{\ast}.$$
This follows from the usual formulation of Serre duality and the fact that any
coherent sheaf has a finite locally free resolution. As a consequence,
$\Ext^{\nu}(\mathcal{F},\mathcal{G})=0$ for any $\nu \ge 2$, which means that
$\Coh_{\boldsymbol{E}}$ has homological dimension one. This implies that any
object $X\in\Dbcoh(\boldsymbol{E})$ splits into the direct sum of appropriate
shifts of its cohomology sheaves. To see this, start with a complex
$X=(\mathcal{F}^{-1} \stackrel{f}{\longrightarrow} \mathcal{F}^{0})$
and consider the distinguished triangle in $\Dbcoh(\boldsymbol{E})$
$$\ker(f)[1] \rightarrow X \rightarrow \coker(f) \stackrel{\xi}{\rightarrow}
\ker(f)[2].$$
Because $\xi\in\Hom(\coker(f),\ker(f)[2]) = \Ext^{2}(\coker(f), \ker(f)) =0$,
we obtain $X\cong \ker(f)[1] \oplus \coker(f)$. Using the same idea we can
proceed by induction to get the claim.
\subsection{Indecomposable sheaves are semi-stable}
It is well-known that any coherent sheaf $\mathcal{F}\in\Coh_{\boldsymbol{E}}$
has a Harder-Narasimhan filtration
$$0\subset \mathcal{F}_{n} \subset \ldots \subset \mathcal{F}_{1} \subset
\mathcal{F}_{0} = \mathcal{F}$$
whose factors $\mathcal{A}_{\nu} := \mathcal{F}_{\nu}/\mathcal{F}_{\nu+1}$ are
semi-stable with decreasing slopes $\mu(\mathcal{A}_{n})>
\mu(\mathcal{A}_{n-1}) > \ldots > \mu(\mathcal{A}_{0})$.
Using the definition of semi-stability, this implies
$\Hom(\mathcal{A}_{\nu+i}, \mathcal{A}_{\nu})
= 0$ for all $\nu\ge 0$ and $i>0$. Therefore,
$\Ext^{1}(\mathcal{A}_{0},\mathcal{F}_{1}) \cong
\Hom(\mathcal{F}_{1}, \mathcal{A}_{0})^{\ast} =0$, and the exact sequence
$0\rightarrow \mathcal{F}_{1} \rightarrow \mathcal{F} \rightarrow
\mathcal{A}_{0} \rightarrow 0$ must split.
In particular, if $\mathcal{F}$ is indecomposable, we have $\mathcal{F}_{1}=0$
and $\mathcal{F}\cong \mathcal{A}_{0}$ and $\mathcal{F}$ is semi-stable.
\subsection{Jordan-H\"older factors}
The full sub-category of $\Coh_{\boldsymbol{E}}$ whose objects are the
semi-stable sheaves of a fixed slope is an Abelian category in which any object
has a Jordan-H\"older filtration with stable factors.
If $\mathcal{F}$ and $\mathcal{G}$ are non-isomorphic stable sheaves which
have the same slope, we have $\Hom(\mathcal{F},\mathcal{G})=0$. Based on this
fact, in the same way as before, we can deduce that an indecomposable
semi-stable sheaf has all its Jordan-H\"older factors isomorphic to each
other.
\subsection{Simple is stable}
It is well-known that any stable sheaf $\mathcal{F}$ is simple,
i.e. $\Hom(\mathcal{F},\mathcal{F}) \cong \boldsymbol{k}$. On a smooth
elliptic curve, the converse is true as well, which equips us with a useful
homological characterisation of stability.
To see that simple implies stable, we suppose for a contradiction that
$\mathcal{F}$ is simple but not stable. This implies the existence of an
epimorphism $\mathcal{F}\rightarrow \mathcal{G}$ with $\mathcal{G}$ stable and
$\mu(\mathcal{F})\ge \mu(\mathcal{G})$. Serre duality implies
$\dim \Ext^{1}(\mathcal{G},\mathcal{F}) = \dim \Hom(\mathcal{F},\mathcal{G}) >
0$, hence, $\chi(\mathcal{G},\mathcal{F}) := \dim
\Hom(\mathcal{G},\mathcal{F}) - \dim \Ext^{1}(\mathcal{G},\mathcal{F}) < \dim
\Hom(\mathcal{G},\mathcal{F})$. Riemann-Roch gives
$\chi(\mathcal{G},\mathcal{F}) = (\mu(\mathcal{F}) -
\mu(\mathcal{G}))/\rk(\mathcal{F})\rk(\mathcal{G}) > 0$, hence
$\Hom(\mathcal{G},\mathcal{F})\ne 0$. But this produces a
non-zero composition $\mathcal{F}\rightarrow \mathcal{G} \rightarrow
\mathcal{F}$ which is not an isomorphism, in contradiction to the assumption
that $\mathcal{F}$ was simple.
\subsection{Classification}
Atiyah \cite{Atiyah} gave a description of all stable sheaves with a fixed
slope in the form $\mathcal{E}(r,d)\otimes \mathcal{L}$, where $\mathcal{L}$
is a line bundle of degree zero and $\mathcal{E}(r,d)$ is a particular stable
bundle of the fixed slope. The bundle $\mathcal{E}(r,d)$ depends on the choice
of a base point $p_{0}\in\boldsymbol{E}$ and
its construction reflects the Euclidean algorithm on the pair $(\rk,\deg)$.
We look at this description from a slightly different perspective. We use the
twist functors $T_{\mathcal{O}}$ and $T_{\boldsymbol{k}(p_{0})}$, which were
constructed by Seidel and Thomas \cite{SeidelThomas} (see also \cite{Meltzer}).
They act as equivalences on $\Dbcoh(\boldsymbol{E})$ and, hence, preserve
stability.
A stable sheaf of rank $r$ and degree $d$ is sent by $T_{\mathcal{O}}$ to one
with $(\rk,\deg)$ equal to $(r-d,d)$. If $r<d$ this is a shift of a stable
sheaf. The functor $T_{\boldsymbol{k}(p_{0})}$ sends the pair $(r,d)$ to
$(r,r+d)$ and its inverse sends it to $(r,d-r)$. Therefore, if we follow the
Euclidean algorithm, we find a composition of such functors which provides an
equivalence between the category of stable sheaves with slope $d/r$ and the
category of simple torsion sheaves. Such sheaves are precisely the structure
sheaves of closed points $\boldsymbol{k}(x)$, $x\in\boldsymbol{E}$. They are
considered to be stable with slope $\infty$.
More generally, this procedure provides an equivalence between the category of
semi-stable sheaves of rank $r$ and degree $d$ with the category of torsion
sheaves of length equal to $\gcd(r,d)$. This shows, in particular, that the
Abelian category of semi-stable sheaves with fixed slope is equivalent to the
category of coherent torsion sheaves.
\subsection{Auto-equivalences}
By $\Aut(\Dbcoh(\boldsymbol{E}))$ we denote the group of all exact
auto-equivalences of the triangulated category $\Dbcoh(\boldsymbol{E})$. This
group acts on the Grothendieck group $\mathsf{K}(\boldsymbol{E}) \cong
\mathsf{K}(\Dbcoh(\boldsymbol{E}))$.
As the kernel of the Chern character is the radical of the Euler-form
$\langle X,Y \rangle = \dim(\Hom(X,Y)) - \dim(\Hom(X,Y[1])$
which is invariant under this action, it induces an action on the even
cohomology $H^{2\ast}(\boldsymbol{E}, \mathbb{Z}) \cong \mathbb{Z}^{2}$.
Because $\dim(\Hom(\mathcal{F},\mathcal{G}))>0$ if and only if $\langle
\mathcal{F},\mathcal{G} \rangle >0$, provided $\mathcal{F}\not\cong
\mathcal{G}$ are stable sheaves, the induced action on $\mathbb{Z}^{2}$ is
orientation preserving. So, we obtain a homomorphism of groups $\varphi:
\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z})$, which is
surjective because $T_{\mathcal{O}}$ and $T_{\boldsymbol{k}(p_{0})}$ are
mapped to a pair of generators of $\SL(2,\mathbb{Z})$. Explicitly, if
$\mathbb{G}$ is an auto-equivalence, $\varphi(\mathbb{G})$ describes its
action on the pair $(\rk,\deg)$.
To understand $\ker(\varphi)$, we observe that $\varphi(\mathbb{G}) =
\boldsymbol{1}$ implies that $\mathbb{G}$ sends a simple torsion sheaf
$\boldsymbol{k}(x)$ to some $\boldsymbol{k}(y)[2k]$, because indecomposability
is retained. By the same reason, $\mathbb{G}(\mathcal{O})$ is a shifted line
bundle of degree zero.
However, $\Hom(\mathcal{L},\boldsymbol{k}(y)[l]) = 0$, if $\mathcal{L}$ is a
line bundle and $l\ne 0$. Hence, after composing $\mathbb{G}$ with a shift, it
sends all simple torsion sheaves to simple torsion sheaves, without a shift.
Because $\boldsymbol{E}$ is smooth, we can apply a result of Orlov \cite{Orlov}
which says that any auto-equivalence $\mathbb{G}$ is a Fourier-Mukai transform
\cite{Mukai}.
However, any such functor, which sends the sheaves $\boldsymbol{k}(x)$ to
torsion sheaves of length one is of the form
$\mathbb{G}(X)=f^{\ast}(\mathcal{L}\otimes X)$,
where $f:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is an automorphism and
$\mathcal{L}\in\Pic(\boldsymbol{E})$ a line bundle. Hence, $\ker(\varphi)$ is
generated by $\Aut(\boldsymbol{E}), \Pic^{0}(\boldsymbol{E})$ and even
shifts. This gives a complete description of the group
$\Aut(\Dbcoh(\boldsymbol{E}))$. A similar approach was used by Lenzing and
Meltzer to describe the group of exact auto-equivalences of tubular weighted
projective lines
\cite{LenzingMeltzerAuto}.
\subsection{Difficulties in the singular case}\label{subsec:diff}
Let now $\boldsymbol{E}$ be an irreducible but singular curve of arithmetic
genus one. The technical cornerstones of the theory as described in this
section fail to be true in this case. More precisely:
\begin{itemize}
\item the category of coherent sheaves $\Coh_{\boldsymbol{E}}$ has infinite
homological dimension;
\item there exist indecomposable complexes in $\Dbcoh(\boldsymbol{E})$ which
are not just shifted sheaves, see \cite{BurbanDrozd}, section 3;
\item Serre duality fails to be true in general;
\item not all indecomposable vector bundles are semi-stable;
\item there exist indecomposable coherent sheaves which are neither torsion
sheaves nor torsion free sheaves, see \cite{BurbanDrozd}.
\end{itemize}
Most of the trouble is caused by the failure of Serre duality.
The basic example is the following. Suppose, $s\in\boldsymbol{E}$ is a node,
then
$$\Hom(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong \boldsymbol{k}\quad
\text{ and }\quad
\Ext^{1}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong \boldsymbol{k}^{2}.$$
Serre duality is available, only if at least one of the two sheaves involved
has finite homological dimension. This might suggest that replacing
$\Dbcoh(\boldsymbol{E})$ by the sub-category of perfect complexes would solve
most of the problems. But see Remark \ref{rem:notperfect}.
In the subsequent sections we overcome these difficulties and point out the
similarities between the smooth and the singular case.
\section{Harder-Narasimhan filtrations}\label{sec:HNF}
Throughout this section, $\boldsymbol{E}$ denotes an irreducible reduced
projective curve over $\boldsymbol{k}$ of arithmetic genus one.
The notion of stability of coherent torsion free sheaves on an irreducible
curve is usually defined with the aid of the slope function
$\mu(\,\cdot\,)=\deg(\,\cdot\,)/\rk(\,\cdot\,)$. To use the phase function instead is
equivalent, but better adapted for the generalisation to derived categories
described below.
By definition, the \emph{phase} $\varphi(\mathcal{F})$ of a non-zero
coherent sheaf $\mathcal{F}$ is the unique number which satisfies $0 <
\varphi(\mathcal{F})\le 1$ and $m(\mathcal{F}) \exp(\pi
i\varphi(\mathcal{F})) = -\deg(\mathcal{F}) + i \rk(\mathcal{F})$, where
$m(\mathcal{F})$ is a positive real number, called the \emph{mass} of the
sheaf $\mathcal{F}$.
In particular, $\varphi(\mathcal{O}) = 1/2$ and all non-zero torsion sheaves
have phase one.
A torsion free coherent sheaf $\mathcal{F}$ is called semi-stable if for any
exact sequence of torsion free coherent sheaves
$$0 \rightarrow \mathcal{E} \rightarrow \mathcal{F}
\rightarrow \mathcal{G} \rightarrow 0$$
the inequality $\varphi(\mathcal{E}) \le \varphi(\mathcal{F})$, or
equivalently, $\varphi(\mathcal{F}) \le \varphi(\mathcal{G})$, holds.
It is well-known \cite{Rudakov} that any torsion free coherent sheaf
$\mathcal{F}$ on a projective variety has a Harder-Narasimhan filtration
$$0 \subset \mathcal{F}_{n} \subset \mathcal{F}_{n-1} \cdots \subset
\mathcal{F}_{1} \subset \mathcal{F}_{0} = \mathcal{F},$$
which is uniquely characterised by the property that all factors
$\mathcal{A}_{i} = \mathcal{F}_{i}/\mathcal{F}_{i+1}$ are semi-stable and
satisfy
$$\varphi(\mathcal{A}_{n}) > \varphi(\mathcal{A}_{n-1}) > \cdots >
\varphi(\mathcal{A}_{0}).$$
Originally, this concept of stability was introduced in the 1960s in order to
construct moduli spaces using geometric invariant theory. It could also be
seen as a method to understand the structure of the category of coherent
sheaves on a projective variety.
By Simpson, the notion of stability was extended to coherent sheaves of
pure dimension.
A very general approach was taken by Rudakov \cite{Rudakov}, who introduced
the notion of stability on Abelian categories. Under some finiteness
assumptions on the category, he shows the existence and uniqueness of a
Harder-Narasimhan filtration for any object of the category in question. As an
application of his work, the usual slope stability extends to the whole
category $\Coh_{\boldsymbol{E}}$ of coherent sheaves on $\boldsymbol{E}$. In
particular, any non-zero coherent sheaf has a Harder-Narasimhan filtration and
any non-zero coherent torsion sheaf on the curve $\boldsymbol{E}$ is
semi-stable.
Inspired by work of Douglas on $\Pi$-Stability for D-branes, see for example
\cite{Douglas}, it was shown by Bridgeland \cite{Stability} how to extend the
concept of stability and Harder-Narasimhan filtration to the derived category
of coherent sheaves, or more generally, to a triangulated category. These new
ideas were merged with the ideas from \cite{Rudakov} in the paper
\cite{GRK}.
We shall follow here the approach of Bridgeland \cite{Stability}. In Section
\ref{sec:tstruc} we give a description of Bridgeland's moduli space of
stability conditions on the derived category of irreducible singular curves
of arithmetic genus one. However, throughout the present chapter we stick to
the classical notion of stability on the category of coherent sheaves and the
stability structure it induces on the triangulated category.
In order to generalise the concept of a Harder-Narasimhan filtration to the
category $\Dbcoh(\boldsymbol{E})$, Bridgeland \cite{Stability} extends the
definition of the phase of a sheaf to shifts of coherent sheaves by:
$$\varphi(\mathcal{F}[n]) := \varphi(\mathcal{F})+n,$$
where $\mathcal{F}\ne 0$ is a coherent sheaf on $\boldsymbol{E}$ and
$n\in\mathbb{Z}$. A complex which is non-zero at position $m$ only has,
according to this definition, phase in the interval $(-m,-m+1]$. If
$\mathcal{F}$ and $\mathcal{F}'$ are non-zero coherent sheaves and $a,b$
integers, we have the implication:
$$\varphi(\mathcal{F}[-a]) > \varphi(\mathcal{F}'[-b])
\quad\Rightarrow\quad a\le b.$$
For any $\varphi\in\mathbb{R}$ we denote by $\mathsf{P}(\varphi)$ the Abelian
category of shifted semi-stable sheaves with phase $\varphi$. Of course,
$0\in\mathsf{P}(\varphi)$ for all $\varphi$.
If $\varphi\in(0,1]$, this is a full Abelian subcategory of
$\Coh_{\boldsymbol{E}}$. For any $\varphi\in\mathbb{R}$ we have
$\mathsf{P}(\varphi+n) = \mathsf{P}(\varphi)[n]$. A non-zero object of
$\Dbcoh(\boldsymbol{E})$ will be called \emph{semi-stable}, if it is an
element of one of the categories $\mathsf{P}(\varphi)$, $\varphi\in\mathbb{R}$.
Bridgeland's stability conditions \cite{Stability} involve so-called
central charges.
In order to define the central charge of the standard stability condition, we
need a definition of degree and rank for arbitrary objects in
$\Dbcoh(\boldsymbol{E})$.
Let $K =\mathcal{O}_{\boldsymbol{E},\eta}$ be the field of rational
functions on the irreducible curve $\boldsymbol{E}$ with generic point
$\eta\in\boldsymbol{E}$. The base change $\eta:\Spec(K)\rightarrow
\boldsymbol{E}$ is flat, so that $\eta^{\ast}(F)$,
taken in the non-derived sense, is correctly defined for any
$F\in\Dbcoh(\boldsymbol{E})$.
We define $\rk(F):=\chi(\eta^{\ast}(F))$, which is the
alternating sum of the dimensions of the cohomology spaces of the complex
$\eta^{\ast}(F)$ which are vector spaces over $K$.
In order to define the degree, we use the functor
$$\boldsymbol{R}\Hom(\mathcal{O}_{\boldsymbol{E}},\,\cdot\,): \Dbcoh(\boldsymbol{E})
\rightarrow \Dbcoh(\boldsymbol{k}),$$
and set $\deg(F):= \chi(\boldsymbol{R}\Hom(\mathcal{O}_{\boldsymbol{E}},F))$.
Here, we denoted by $\Dbcoh(\boldsymbol{k})$ the bounded derived category of
finite dimensional vector spaces over $\boldsymbol{k}$.
For coherent sheaves, these definitions coincide with the usual
definitions of rank and degree. In particular, a torsion sheaf of length $m$
which is supported at a single point of $\boldsymbol{E}$ has rank $0$ and
degree $m$.
These definitions imply that rank and degree are additive on distinguished
triangles in $\Dbcoh(\boldsymbol{E})$. Hence, they induce homomorphisms on the
Grothendieck group $\mathsf{K}(\Dbcoh(\boldsymbol{E}))$ of the triangulated
category $\Dbcoh(\boldsymbol{E})$, which is by definition the quotient of the
free Abelian group generated by the objects of $\Dbcoh(\boldsymbol{E})$ modulo
expressions coming from distinguished triangles.
Recall that $\mathsf{K}_{0}(\Coh(\boldsymbol{E})) \cong
\mathsf{K}(\Dbcoh(\boldsymbol{E}))$,
see \cite{Groth}. We denote this group by $\mathsf{K}(\boldsymbol{E})$
\begin{lemma}\label{lem:GrothGrp}
If $\boldsymbol{E}$ is an irreducible singular curve of arithmetic genus
one, we have $\mathsf{K}(\boldsymbol{E}) \cong \mathbb{Z}^{2}$ with
generators $[\boldsymbol{k}(x)]$ and $[\mathcal{O}_{\boldsymbol{E}}]$.
\end{lemma}
\begin{proof}
Recall that the Grothendieck-Riemann-Roch Theorem, see \cite{BFM} or
\cite{Fulton}, provides a homomorphism
$$\tau_{\boldsymbol{E}}:\mathsf{K}(\boldsymbol{E}) \rightarrow
A_{\ast}(\boldsymbol{E})\otimes \mathbb{Q},$$
which depends functorially on $\boldsymbol{E}$ with respect to proper direct
images.
Moreover,
$(\tau_{\boldsymbol{E}})_{\mathbb{Q}}:\mathsf{K}(\boldsymbol{E})\otimes
\mathbb{Q} \rightarrow A_{\ast}(\boldsymbol{E})\otimes \mathbb{Q}$
is an isomorphism, see \cite{Fulton}, Cor.\/ 18.3.2.
If $\boldsymbol{E}$ is an irreducible singular projective curve of
arithmetic genus one, we easily see that the Chow group
$A_{\ast}(\boldsymbol{E})$ is isomorphic to $\mathbb{Z}^{2}$.
The two generators are $[x]\in A_{0}(\boldsymbol{E})$ with
$x\in\boldsymbol{E}$ and $[\boldsymbol{E}]\in A_{1}(\boldsymbol{E})$.
Note that $[x]=[y]\in A_{0}(\boldsymbol{E})$ for any two closed points
$x,y\in\boldsymbol{E}$, because the normalisation of $\boldsymbol{E}$ is
$\mathbb{P}^{1}$.
Using \cite{Fulton}, Thm.\/ 18.3 (5), we obtain
$\tau_{\boldsymbol{E}}(\boldsymbol{k}(x))= [x] \in A_{0}(\boldsymbol{E})$
for any $x\in\boldsymbol{E}$. On the other hand, from \cite{Fulton}, Expl.\/
18.3.4 (a), we obtain $\tau_{\boldsymbol{E}}(\mathcal{O}_{\boldsymbol{E}}) =
[\boldsymbol{E}] \in A_{1}(\boldsymbol{E})$.
Therefore, the classes of $\boldsymbol{k}(x)$ and
$\mathcal{O}_{\boldsymbol{E}}$ define a basis of
$\mathsf{K}(\boldsymbol{E})\otimes \mathbb{Q}$.
However, these two classes generate the group $\mathsf{K}(\boldsymbol{E})$,
so that it must be a free Abelian group.
\end{proof}
The \emph{central charge} of the standard stability structure on
$\Dbcoh(\boldsymbol{E})$ is the homomorphism of Abelian groups
$$
Z: \mathsf{K}(\boldsymbol{E}) \rightarrow \mathbb{Z}\oplus i\mathbb{Z}
\subset \mathbb{C},
$$
which is given by
$$
Z(F) := -\deg(F) + i \rk(F).
$$
If $F$ is a non-zero coherent sheaf, $Z(F)$ is a point on the ray from the
origin through $\exp(\pi i \varphi(F))$ in $\mathbb{C}$. Its distance from the
origin was called the mass of $F$.
Although the phase $\varphi(F)$ is defined for sheaves and their shifts only,
we are able to define the slope $\mu(F)$ for any object in
$\Dbcoh(\boldsymbol{E})$ which is not equal to zero in the Grothendieck
group. Namely, the usual definition $\mu(F):=\deg(F)/\rk(F)$ gives us now a
mapping $$\mu:\mathsf{K}(\boldsymbol{E})\setminus\{0\} \rightarrow
\mathbb{Q} \cup \{\infty\},$$ which extends the usual definition of the slope
of a sheaf.
Because $Z(\mathcal{O}_{\boldsymbol{E}})=i$ and $Z(\boldsymbol{k}(x))=-1$,
Lemma \ref{lem:GrothGrp} implies that $Z$ is injective. Therefore, $\mu$ is
defined for any non-zero element of the Grothendieck group.
For arbitrary objects $X\in\Dbcoh(\boldsymbol{E})$ we have $Z(X[1]) = -Z(X)$,
hence $\mu(X[1]) = \mu(X)$ when defined.
In case of shifted sheaves, in contrast to the slope $\mu$, the
phase $\varphi$ keeps track of the position of this sheaf in the complex.
As an illustration, we include an example of an indecomposable object in
$\Dbcoh(\boldsymbol{E})$ which has a zero image in the Grothendieck group.
\begin{example}
Let $s\in\boldsymbol{E}$ be the singular point and denote, as usual, by
$\boldsymbol{k}(s)$ the torsion sheaf of length one which is supported at
$s$. This sheaf does not have finite homological dimension. To see this, we
observe first that $\Ext^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong
H^{0}(\mathcal{E}xt^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s)))$. Moreover,
as an
$\mathcal{O}_{\boldsymbol{E},s}$-module,
$\boldsymbol{k}(s)$ has an infinite periodic locally free resolution of the
form
$$
\cdots \stackrel{A}{\longrightarrow} \mathcal{O}_{\boldsymbol{E},s}^{2}
\stackrel{B}{\longrightarrow} \mathcal{O}_{\boldsymbol{E},s}^{2}
\stackrel{A}{\longrightarrow}\mathcal{O}_{\boldsymbol{E},s}^{2}
\longrightarrow \mathcal{O}_{\boldsymbol{E},s}
\longrightarrow \boldsymbol{k}(s) \longrightarrow 0
$$
where $AB=BA=f\cdot I_{2}$ is a reduced matrix factorisation of an equation
$f$ of $\boldsymbol{E} \subset \mathbb{P}^{2}$. For example, if $s$ is a
node, so that $\boldsymbol{E}$ is locally given by the polynomial $f = y^{2}
- x^{3} -x^{2}\in\boldsymbol{k}[x,y]$, we can choose
$A=\bigl(\begin{smallmatrix}
y&x^{2}+x\\x&y
\end{smallmatrix}\bigr)$ and
$B=\bigl(\begin{smallmatrix}
y&-x^{2}-x\\-x&y
\end{smallmatrix}\bigr)$ considered modulo $f$. More generally, any
singular Weierstra{\ss} cubic $f$ can be written as $y\cdot y - R\cdot S$
with $y, R,S$ all vanishing at the singular point. The off-diagonal elements
of $A$ and $B$ are then formed by $\pm R,\pm S$. Therefore, all entries of
the matrices $A$ and $B$ are elements of the maximal ideal of the local ring
$\mathcal{O}_{\boldsymbol{E},s}$. Hence, the application of $\Hom(\,\cdot\,,
\boldsymbol{k}(s))$ produces a complex with zero differential, which implies
that $\Ext^{k}(\boldsymbol{k}(s), \boldsymbol{k}(s))$ is
two-dimensional for all $k\ge 1$.
In particular, $\Ext^{2}(\boldsymbol{k}(s), \boldsymbol{k}(s)) \cong
\boldsymbol{k}^{2}$, and we can pick a non-zero element
$w\in\Hom(\boldsymbol{k}(s), \boldsymbol{k}(s)[2])$. There exists a complex
$X\in\Dbcoh(\boldsymbol{E})$ which sits in a distinguished triangle
$$X\rightarrow \boldsymbol{k}(s) \stackrel{w}{\longrightarrow}
\boldsymbol{k}(s)[2] \stackrel{+}{\longrightarrow}.$$
Because the shift by one corresponds to multiplication by $-1$ in the
Grothendieck group, this object $X$ is equal to zero in
$\mathsf{K}(\boldsymbol{E})$. On the other hand, $X$ is
indecomposable. Indeed, if $X$ would split, it must be $X\cong
\boldsymbol{k}(s) \oplus \boldsymbol{k}(s)[1]$, because the only non-zero
cohomology of $X$ is $H^{-1}(X) \cong \boldsymbol{k}(s)$ and $H^{0}(X) \cong
\boldsymbol{k}(s)$. But, because $\Hom(\boldsymbol{k}(s)[1],
\boldsymbol{k}(s)) = 0$, Lemma \ref{lem:PengXiao}, applied to the
distinguished triangle
\[\begin{CD}
\boldsymbol{k}(s)[1] @>>> X @>>> \boldsymbol{k}(s) @>{+}>{w}>
\end{CD}\]
with $X\cong \boldsymbol{k}(s) \oplus \boldsymbol{k}(s)[1]$, implies $w=0$.
\end{example}
\begin{definition}[\cite{Stability}]
A Harder-Narasimhan filtration (HNF) of an object $X \in
\Dbcoh(\boldsymbol{E})$ is a finite collection of distinguished triangles
\[
\xymatrix@C=.5em
{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
\\
}
\]
with $A_j\in\mathsf{P}(\varphi_j)$ and $A_{j}\ne 0$ for all $j$, such that
$\varphi_{n} > \varphi_{n-1} > \cdots > \varphi_{0}.$
\end{definition}
If all ingredients of a HNF are shifted by one, we obtain a HNF of $X[1]$.
The shifted sheaves $A_{j}$ are called \emph{the semi-stable HN-factors} of
$X$ and we define $\varphi_{+}(X):=\varphi_{n}$ and
$\varphi_{-}(X):=\varphi_{0}$. Later, Theorem \ref{thm:uniqueHNF}, we show that
the HNF of an object $X$ is unique up to isomorphism. This justifies this
notation. For the moment, we keep in mind that $\varphi_{+}(X)$ and
$\varphi_{-}(X)$ might depend on the HNF and not only on the object $X$.
Before we proceed, we include a few remarks about the notation we use.
Distinguished triangles in a triangulated category are either displayed in the
form
$X\rightarrow Y\rightarrow Z \stackrel{+}{\longrightarrow}
\quad\text{ or as }\quad
\xymatrix@C=.5em{
X \ar[rr] && Y, \ar[dl]\\ & Z \ar[ul]^{+}
}
$
where the arrow which is marked with $+$ is in fact a morphism $Z\rightarrow
X[1]$.
We shall use the octahedron axiom, the axiom (TR4) in Verdier's list, in the
following convenient form: if two morphisms $X\stackrel{f}{\longrightarrow} Y
\stackrel{g}{\longrightarrow} Z$ are given, for any three distinguished
triangles with bases $f, g$ and $g\circ f$ there exists a fourth distinguished
triangle which is indicated below by dashed arrows, such that we obtain the
following commutative diagram:
\begin{center}
\mbox{\begin{xy} 0;<10mm,0mm>:0,
(0,3) *+{Z'} ="top" ,
(-3,0) *+{X} ="left" ,
(3,0) *+{X'} ="right" ,
(-1.5,1.5) *+{Y} ="midleft" ,
(1.5,1.5) *+{Y'} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{Z}="center",
{"left" \ar@{->}^{f} "midleft" \ar@{->}_{g\circ f} "center"},
{"center" \ar@{->} "right" \ar@{->} "midright"},
{"midleft" \ar@{->}^{g} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
The remainder of this section is devoted to the proofs of the crucial
properties of Harder-Narasimhan filtrations in triangulated categories.
These properties can be found in \cite{Stability, GRK}, where most of
them appear to be either implicit or without a detailed proof.
\begin{lemma}\label{lem:connect}
Let
$\xymatrix@C=.4em{U \ar[rr]^{f} && X \ar[dl]\\ & V \ar[ul]^{+}}$
and
$A \longrightarrow V \longrightarrow V' \stackrel{+}{\longrightarrow}$
be distinguished triangles. Then there exists a factorisation
$U\longrightarrow W \stackrel{f'}{\longrightarrow} X$
of $f$ and two distinguished triangles
$$\xymatrix@C=.5em{U \ar[rr] && W \ar[dl]\ar[rr]^{f'} && X\ar[dl]\\
& A \ar[ul]^{+} && V'.\ar[ul]^{+}}$$
\end{lemma}
\begin{proof}
If we apply the octahedron axiom to the composition $A\rightarrow V
\rightarrow U[1]$ we obtain the following commutative diagram, which gives
the claim.
\begin{center}
\mbox{\begin{xy} 0;<10mm,0mm>:0,
(0,3) *+{V'} ="top" ,
(-3,0) *+{A} ="left" ,
(3,0) *+{X[1]} ="right" ,
(-1.5,1.5) *+{V} ="midleft" ,
(1.5,1.5) *+{W[1]} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{U[1]}="center",
{"left" \ar@{->} "midleft" \ar@{->} "center"},
{"center" \ar@{->}_{f[1]} "right" \ar@{->} "midright"},
{"midleft" \ar@{->} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->}^{f'[1]} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
\end{proof}
\begin{lemma}\label{lem:split}
Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}V \ar[rr] \ar[dl]_{\cong}&& F_{n-1}V \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}V \ar[rr] && F_{0}V \ar@{=}[r]\ar[dl] & V
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $V\in\Dbcoh(\boldsymbol{E})$ and $F_{k}V \longrightarrow V
\longrightarrow V' \stackrel{+}{\longrightarrow}$ a distinguished triangle
with $1\le k \le n$. Then, $F_{k}V$ has a HNF with HN-factors $A_{n},
A_{n-1}, \ldots, A_{k}$ and $V'$ one with HN-factors $A_{k-1},
A_{k-2}, \ldots, A_{0}$.
\end{lemma}
\begin{proof}
The first statement is clear, because we can cut off the HNF of $V$ at
$F_{k}V$ to obtain a HNF of $F_{k}V$. Let us define objects $F_{i}V'$ by
exact triangles $F_{k}V \longrightarrow F_{i}V \longrightarrow F_{i}V'
\stackrel{+}{\longrightarrow}$, where the first arrow is the composition of
the morphisms in the HNF of $V$. Using the octahedron axiom, we obtain for
any $i\le k$ a commutative diagram
\begin{center}
\mbox{\begin{xy} 0;<12mm,0mm>:0,
(0,3) *+{F_{i}V'} ="top" ,
(-3,0) *+{F_{k}V} ="left" ,
(3,0) *+{A_{i-1}} ="right" ,
(-1.5,1.5) *+{F_{i}V} ="midleft" ,
(1.5,1.5) *+{F_{i-1}V'} ="midright" ,
{"left";"midright":"right";"midleft",x} *+{F_{i-1}V}="center",
{"left" \ar@{->} "midleft" \ar@{->} "center"},
{"center" \ar@{->} "right" \ar@{->} "midright"},
{"midleft" \ar@{->} "center" \ar@{->} "top"},
{"top" \ar@{-->} "midright"},
{"midright" \ar@{-->} "right"},
{"right" \ar@{-->}_{+} +(.9,-.9)},
{"midright" \ar@{->}^{+} +"midright"-"center"},
{"right" \ar@{->}^{+} +"center"-"midleft"},
{"top" \ar@{->}^{+} +(.8,.8)}
\end{xy}}
\end{center}
which implies the second claim.
\end{proof}
\begin{remark}\label{rem:split}
The statement of Lemma \ref{lem:split} is true with identical proof if we
relax the assumption of being a HNF by allowing $\varphi(A_{k}) =
\varphi(A_{k-1})$ for the chosen value of $k$.
\end{remark}
\begin{lemma}\label{lem:bounds}
If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
is a HNF of $X\in\Dbcoh(\boldsymbol{E})$ such that $A_{0}[k]$ is a sheaf,
then $H^{k}(X)\ne 0$. In particular, the following implication is true:
$$X\in \mathsf{D}^{\le m} \quad\Longrightarrow\quad
\forall i\ge0: A_{i}\in \mathsf{D}^{\le m}.$$
\end{lemma}
\begin{proof}
The assumption $A_{0}[k]\in \Coh_{\boldsymbol{E}}$ means
$H^{k}(A_{0})=A_{0}[k]\ne 0$ and $\varphi(A_{0}) \in (-k,-k+1]$. Because for
all $i>0$ we have $\varphi(A_{i}) > \varphi(A_{0})$, we obtain
$\varphi(A_{i})>-k$. This implies $H^{k+1}(A_{i})=0$ for all $i\ge 0$. The
cohomology sequences of the distinguished triangles
$F_{i+1}\longrightarrow F_{i}\longrightarrow A_{i}
\stackrel{+}{\longrightarrow}$ imply $H^{k+1}(F_{i}X)=0$ for all $i>0$ and
an exact sequence $H^{k}(X) \rightarrow H^{k}(A_{0}) \rightarrow
H^{k+1}(F_{1}X)$, hence $H^{k}(X)\ne 0$. The statement about the other
HN-factors $A_{i}$ follows now from $\varphi(A_{i})\ge \varphi(A_{0})$.
\end{proof}
\begin{proposition}
Any non-zero object $X\in\Dbcoh(\boldsymbol{E})$ has a HNF.
\end{proposition}
\begin{proof}
The existence of a HNF for objects of $\Coh_{\boldsymbol{E}}$ is
classically known, see \cite{HarderNarasimhan, Rudakov}. Therefore, we can
proceed by induction on the number of non-zero cohomology sheaves of
$X\in\Dbcoh(\boldsymbol{E})$. If $n$ is the largest integer with
$H^{n}(X)\ne 0$, we have a distinguished triangle
\begin{equation}
\tau^{\le n-1} X \longrightarrow X \longrightarrow H^{n}(X)[-n]
\stackrel{+}{\longrightarrow}
\end{equation}
By inductive hypothesis, there exists a HNF of $\tau^{\le n-1} X$. From
Lemma \ref{lem:bounds} we conclude that all HN-factors of $\tau^{\le n-1} X$
are in $\mathsf{D}^{\le n-1}$ and so $\varphi_{-}(\tau^{\le n-1} X)>-n+1$.
Because $H^{n}(X)$ is a sheaf, we have $\varphi_{+}(H^{n}(X)[-n])
\in (-n,-n+1]$, hence $\varphi_{-}(\tau^{\le n-1} X) >
\varphi_{+}(H^{n}(X)[-n])$.
We prove now for any distinguished triangle
\begin{equation}
\label{eq:induction}
U \longrightarrow X \longrightarrow V \stackrel{+}{\longrightarrow}
\end{equation}
in which $V[n]$ is a coherent sheaf that the existence of a HNF for $U$ with
$\varphi_{-}(U)> \varphi_{+}(V)$ implies the existence of a HNF of $X$.
Because $V[n]$ is a sheaf, $V$ has a HNF and we proceed by induction on the
number of HN-factors of $V$. Let $A$ be the leftmost object in a HNF of $V$,
i.e.\/ $A\in\mathsf{P}(\varphi_{+}(V))$. By Lemma \ref{lem:connect} applied
to the distinguished triangles (\ref{eq:induction}) and $A \longrightarrow V
\longrightarrow V' \stackrel{+}{\longrightarrow}$, there exist two
distinguished triangles in which $V'[n]$ is a coherent sheaf with a smaller
number of HN-factors as $V$:
$$\xymatrix@C=.5em{U \ar[rr] && W \ar[dl]\ar[rr] && X.\ar[dl]\\
& A \ar[ul]^{+} && V'\ar[ul]^{+}}$$
Because $\varphi_{-}(U)\ge \varphi(A) =\varphi_{+}(V)$, the left triangle can
be concatenated to the given HNF of $U$ in order to provide a HNF for
$W$. The start of the induction is covered as well: it is the case $V'=0$.
\end{proof}
\begin{lemma}\label{wesPT:ii}
If $X,Y\in \Dbcoh(\boldsymbol{E})$ with $\varphi_{-}(X) > \varphi_{+}(Y)$,
then $$\Hom(X,Y)=0.$$
\end{lemma}
\begin{proof}
If $X,Y$ are semi-stable sheaves, this is well-known and follows easily from
the definition of semi-stability. Because $\Hom(X,Y[k])=0$, if $X,Y$ are
sheaves and $k<0$, the claim follows if $X\in \mathsf{P}(\varphi)$ and $Y\in
\mathsf{P}(\psi)$ with $\varphi>\psi$. Let now $X\in\mathsf{P}(\varphi)$ and
$Y\in \Dbcoh(\boldsymbol{E})$ with $\varphi > \varphi_{+}(Y)$. Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{m}Y \ar[rr] \ar[dl]_{\cong}&& F_{m-1}Y \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}Y \ar[rr] && F_{0}Y \ar@{=}[r]\ar[dl] & Y
\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
be a HNF of $Y$. We have $\varphi(B_{j})\le \varphi(B_{m}) =
\varphi_{+}(Y)$, hence $\varphi(X)>\varphi(B_{j})$ and $\Hom(X,B_{j})=0$ for
all $j$. If we apply the functor $\Hom(X,\,\cdot\,)$ to the distinguished
triangles $F_{j+1}Y \longrightarrow F_{j}Y \longrightarrow B_{j}
\stackrel{+}{\longrightarrow}$, we obtain surjections $\Hom(X,F_{j+1}Y)
\twoheadrightarrow \Hom(X,F_{j}Y)$. From $\Hom(X,F_{m}Y)=
\Hom(X,B_{m})=0$, we obtain $\Hom(X,Y)=\Hom(X,F_{0}Y)=0$.
Let now $X,Y$ be arbitrary non-zero objects of $\Dbcoh(\boldsymbol{E})$
which satisfy $\varphi_{-}(X) > \varphi_{+}(Y)$. If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] & X
\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
is a HNF of $X$, we have $\varphi(A_{i})\ge \varphi(A_{0})=\varphi_{-}(X) >
\varphi_{+}(Y)$. We know already $\Hom(A_{i},Y)=0$ for all $i\ge 0$. If we
apply the functor $\Hom(\,\cdot\,,Y)$ to the distinguished triangles
$F_{i+1}X \longrightarrow F_{i}X \longrightarrow A_{i}
\stackrel{+}{\longrightarrow}$, we obtain injections $\Hom(F_{i}X,Y)
\hookrightarrow \Hom(F_{i+1}X,Y)$. Again, this implies $\Hom(X,Y)=0$.
\end{proof}
\begin{theorem}[\cite{Stability,GRK}]\label{thm:uniqueHNF}
The HNF of any non-zero object $X\in\Dbcoh(\boldsymbol{E})$ is unique up to
unique isomorphism.
\end{theorem}
\begin{proof}
If
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
and
\[\xymatrix@C=.5em{
0\; \ar[rr] && G_{m}X \ar[rr] \ar[dl]_{\cong}&& G_{m-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&G_{1}X \ar[rr] && G_{0}X \ar@{=}[r]\ar[dl] &
X\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
are HNFs of $X$, we have to show that there exist unique isomorphisms of
distinguished triangles for any $k\ge 0$
\[\begin{CD}
F_{k+1}X @>>> F_{k}X @>>> A_{k} @>{+}>>\\
@VV{f_{k+1}}V @VV{f_{k}}V @VV{g_{k}}V \\
G_{k+1}X @>>> G_{k}X @>>> B_{k} @>{+}>>
\end{CD}\]
with $f_{0}=\mathsf{Id}_{X}$. This is obtained by induction on $k\ge0$ from
the following claim: if an isomorphism $f:F\rightarrow G$ and two
distinguished triangles
$F' \longrightarrow F \longrightarrow A \stackrel{+}{\longrightarrow }$ and
$G' \longrightarrow G \longrightarrow B \stackrel{+}{\longrightarrow }$ are
given such that $A\in\mathsf{P}(\varphi), B\in\mathsf{P}(\psi)$ and $F',G'$
have HNFs with $\varphi_{-}(F')>\varphi$ and $\varphi_{-}(G')>\psi$, then
there exist unique isomorphisms $f':F'\rightarrow G'$ and $g:A\rightarrow B$
such that $(f',f,g)$ is a morphism of triangles. In particular,
$\varphi=\psi$.
Without loss of generality, we may assume $\varphi\ge \psi$. This implies
$\varphi_{-}(F'[1]) > \varphi_{-}(F') > \psi$. Lemma \ref{wesPT:ii} implies
therefore $\Hom(F',B) = \Hom(F'[1],B) = 0$. From \cite{Asterisque100},
Proposition 1.1.9, we obtain the existence and uniqueness of the morphisms
$f',g$. It remains to show that they are isomorphisms. If $g$ were zero, the
second morphism in the triangle $G' \longrightarrow G
\stackrel{0}{\longrightarrow} B \stackrel{+}{\longrightarrow }$ would be
zero. Hence, $B$ were a direct summand of $G'[1]$ which implies
$\Hom(G'[1],B)\ne 0$. This contradicts Lemma \ref{wesPT:ii}, because
$\varphi_{-}(G'[1]) > \varphi(G') > \psi=\varphi(B)$. Hence, $g\ne 0$ and
Lemma \ref{wesPT:ii} implies $\varphi(A)\le \varphi(B)$, i.e.\/
$\varphi=\psi$. So, the same reasoning as before gives a unique morphism of
distinguished triangles in the other direction. The composition of both are
the respective identities of $F' \longrightarrow F \longrightarrow A
\stackrel{+}{\longrightarrow }$ and $G' \longrightarrow G \longrightarrow B
\stackrel{+}{\longrightarrow }$ respectively, which follows again from the
uniqueness part of \cite{Asterisque100}, Proposition 1.1.9. This proves the
claim.
\end{proof}
We need the following useful lemma.
\begin{lemma}(\cite{PengXiao}, Lemma 2.5)\label{lem:PengXiao}
Let $\mathsf{D}$ be a triangulated category and
\[\begin{CD}
F @>>> G @>>> H_1 \oplus H_2 @>{+}>{(0,w)}>
\end{CD}\]
be a distinguished triangle in $\mathsf{D}$. Then $G\cong H_{1}\oplus G'$
splits and the given triangle is isomorphic to
\[\begin{CD}
F
@>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
H_1 \oplus G'
@>{\bigl(\begin{smallmatrix}1&0\\0&f' \end{smallmatrix}\bigr)}>>
H_1 \oplus H_2
@>{+}>{(0,w)}>
\end{CD}\]
Dually, if
\[\begin{CD}
F @>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
G_{1}\oplus G_{2} @>>> H @>{+}>>
\end{CD}\]
is a distinguished triangle then $H\cong G_{1}\oplus H'$ and the given
triangle is isomorphic to
\[\begin{CD}
F
@>{\bigl(\begin{smallmatrix} 0\\g \end{smallmatrix}\bigr)}>>
G_1 \oplus G_{2}
@>{\bigl(\begin{smallmatrix}1&0\\0&f' \end{smallmatrix}\bigr)}>>
G_1 \oplus H'
@>{+}>{(0,w)}>
\end{CD}\]
\end{lemma}
The results in this section are true for more general triangulated
categories than $\Dbcoh(\boldsymbol{E})$. Without changes, the proofs apply if
we replace $\Dbcoh(\boldsymbol{E})$ by the bounded derived category of an
Abelian category which is equipped with the notion of stability in the sense
of \cite{Rudakov}. In particular, these results hold for polynomial stability
on the triangulated categories $\Dbcoh(X)$ where $X$ is a projective
variety over $\boldsymbol{k}$.
\section{The structure of the bounded derived category of coherent sheaves on
a singular Weiersta{\ss} curve}\label{sec:dercat}
In this section, we prove the main results on which our understanding of
$\Dbcoh(\boldsymbol{E})$ is based. Again, $\boldsymbol{E}$ denotes a
Weierstra{\ss} curve. Our main focus is on the singular case, however all the
results remain true in the smooth case as well.
A speciality of this category is the
non-vanishing result Proposition \ref{wesPT}. Unlike the smooth case, there
exist indecomposable objects in $\Dbcoh(\boldsymbol{E})$, which are not
semi-stable. Their Harder-Narasimhan factors are characterised in Proposition
\ref{prop:extreme}. We propose to visualise indecomposable objects by their
``shadows''. As an application of our results, we give a complete
characterisation of all spherical objects in $\Dbcoh(\boldsymbol{E})$. As a
consequence, we show that the group of exact auto-equivalences acts
transitively on the set of spherical objects. This answers a question which was
posed by Polishchuk \cite{YangBaxter}.
Let us set up some notation.
For any $\varphi\in(0,1]$ we denote by $\mathsf{P}(\varphi)^{s} \subset
\mathsf{P}(\varphi)$ the full subcategory of stable sheaves with phase
$\varphi$. We extend this definition to all $\varphi\in\mathbb{R}$ by
requiring $\mathsf{P}(\varphi +n)^{s} = \mathsf{P}(\varphi)^{s}[n]$ for all
$n\in\mathbb{Z}$ and all $\varphi\in\mathbb{R}$.
We already know the structure of $\mathsf{P}(1)^{s}$. Because $\mathsf{P}(1)$
is the category of coherent torsion sheaves on $\boldsymbol{E}$, the objects
of $\mathsf{P}(1)^{s}$ are precisely the structure sheaves $\boldsymbol{k}(x)$
of closed points $x\in\boldsymbol{E}$. In order to understand the structure of
all the other categories $\mathsf{P}(\varphi)^{s}$, we use Fourier-Mukai
transforms. Our main technical tool will be the transform $\mathbb{F}$ which
was studied in \cite{BurbanKreussler}. It depends on the choice of a regular
point $p_{0}\in\boldsymbol{E}$.
Let us briefly recall its definition and main properties. It was defined
with the aid of Seidel-Thomas twists \cite{SeidelThomas}, which are functors
$T_{E}: \Dbcoh(\boldsymbol{E}) \rightarrow \Dbcoh(\boldsymbol{E})$
depending on a spherical object $E\in\Dbcoh(\boldsymbol{E})$. On objects, these
functors are characterised by the existence of a distinguished triangle
$$\boldsymbol{R}\Hom(E,F) \otimes E \rightarrow F \rightarrow T_{E}(F)
\stackrel{+}{\longrightarrow}.$$
If $p_{0}\in\boldsymbol{E}$ is a smooth point, the functor
$T_{\boldsymbol{k}(p_{0})}$ is isomorphic to the tensor product with the
locally free sheaf $\mathcal{O}_{\boldsymbol{E}}(p_{0})$, see
\cite{SeidelThomas}, 3.11. We defined
$$\mathbb{F} :=
T_{\boldsymbol{k}(p_{0})}T_{\mathcal{O}}T_{\boldsymbol{k}(p_{0})}.$$
In \cite{SeidelThomas} is was shown that twist functors can be described as
integral transforms and that $\mathbb{F}$ is isomorphic to the functor
$\FM^{\mathcal{P}}$, which is given by
$$\FM^{\mathcal{P}}(\,\cdot\,) :=
\boldsymbol{R}\pi_{2\ast}(\mathcal{P}\dtens \pi_{1}^{\ast}(\,\cdot\,)),$$
where $\mathcal{P}=\mathcal{I}_{\Delta}\otimes
\pi_{1}^{\ast}\mathcal{O}(p_{0}) \otimes
\pi_{2}^{\ast}\mathcal{O}(p_{0})[1]$. This is a shift of a coherent sheaf on
$\boldsymbol{E}\times \boldsymbol{E}$, on which we denote the ideal of the
diagonal by $\mathcal{I}_{\Delta} \subset
\mathcal{O}_{\boldsymbol{E}\times\boldsymbol{E}}$ and the two projections by
$\pi_{1}, \pi_{2}$.
In order to understand the effect of $\mathbb{F}$ on rank and degree, we look
at the distinguished triangle
$$\boldsymbol{R}\Hom(\mathcal{O},F) \otimes \mathcal{O} \rightarrow F
\rightarrow T_{\mathcal{O}}(F) \stackrel{+}{\longrightarrow}.$$
The additivity of rank and degree implies $\rk(T_{\mathcal{O}}(F))= \rk(F) -
\deg(F)$ and
$\deg(T_{\mathcal{O}}(F))= \deg(F)$. On the other hand, it is well-known that
$\deg(T_{\boldsymbol{k}(p_{0})}(F)) = \deg(F)+\rk(F)$ and
$\rk(T_{\boldsymbol{k}(p_{0})}(F)) = \rk(F)$.
So, if we use $[\mathcal{O}_{\boldsymbol{E}}], -[\boldsymbol{k}(p_{0})]$ as a
basis of $\mathsf{K}(\boldsymbol{E})$, which means that we use coordinates
$(\rk,-\deg)$, then the action of $T_{\mathcal{O}}, T_{\boldsymbol{k}(p_{0})}$
and of $\mathbb{F}$ on $\mathsf{K}(\boldsymbol{E})$ is given by the matrices
$$
\begin{pmatrix}
1&1\\0&1
\end{pmatrix},
\begin{pmatrix}
1&0\\-1&1
\end{pmatrix} \quad \;\text{and}\quad
\begin{pmatrix}
0&1\\-1&0
\end{pmatrix}\;\text{respectively.}
$$
In particular, for any object $F\in\Dbcoh(\boldsymbol{E})$ which has a slope,
we have $\mu(T_{\boldsymbol{k}(p_{0})}(F)) = \mu(F)+1$ and
$\mu(\mathbb{F}(F))=-\frac{1}{\mu(F)}$ using the usual conventions in dealing
with $\infty$.
If $F$ is a sheaf or a twist thereof, we defined the phase
$\varphi(F)$. In order to understand the effect of $\mathbb{F}$ on phases, it
is not sufficient to know its effect on the slope. This is because the slope
determines the phase modulo $2\mathbb{Z}$ only.
However, if $F$ is a coherent sheaf, the description of $\mathbb{F}$ as
$\FM^{\mathcal{P}}$ shows that $\mathbb{F}(F)$ can have non-vanishing
cohomology in degrees $-1$ and $0$ only. If, in addition, $\mathbb{F}(F)$ is a
shifted sheaf, this implies $\varphi(\mathbb{F}(F))\in (0,2]$.
From the formula for the slope it is now clear that $\varphi(\mathbb{F}(F)) =
\varphi(F)+\frac{1}{2}$ for any shifted coherent sheaf $F$.
The following result was first shown in \cite{Nachr}. We give an independent
proof here, which was inspired by \cite{Nachr}, Lemma 3.1.
\begin{theorem}\label{thm:mother}
$\mathbb{F}$ sends semi-stable sheaves to semi-stable sheaves.
\end{theorem}
\begin{proof}
Note that, by definition, a semi-stable sheaf of positive rank is
automatically torsion free. The only sheaf with degree and rank equal to
zero is the zero sheaf. Throughout this proof, we let $\mathcal{F}$ be a
semi-stable sheaf on $\boldsymbol{E}$.
If $\deg(\mathcal{F})=0$ this sheaf is torsion free and the claim
was shown in \cite{BurbanKreussler}, Thm.\/ 2.21, see also \cite{FMmin}.
For the sake of clarity we would like to stress here the fact that
\cite{BurbanKreussler}, Section 2, deals with nodal as well as cuspidal
Weierstra{\ss} curves.
Next, suppose $\deg(\mathcal{F})>0$. If
$\rk(\mathcal{F})=0$, $\mathcal{F}$ is a coherent torsion sheaf. Again, the
claim follows from \cite{BurbanKreussler}, Thm.\/ 2.21 and Thm.\/ 2.18,
where it was shown that $\mathbb{F}\circ\mathbb{F}= i^{\ast}[1]$, for any
Weierstra{\ss} curve. Here, $i:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is
the involution which fixes the singularity and which corresponds to taking
the inverse on the smooth part of $\boldsymbol{E}$ with its group structure
in which $p_{0}$ is the neutral element.
Therefore, we may suppose $\mathcal{F}$ is torsion free. As observed before,
the complex $\mathbb{F}(\mathcal{F})\in\Dbcoh(\boldsymbol{E})$ can have
non-vanishing cohomology in degrees $-1$ and $0$ only. We are going to show
that $\mathbb{F}(\mathcal{F})[-1]$ is a sheaf, which is equivalent to the
vanishing of the cohomology object
$\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))\in\Coh_{\boldsymbol{E}}$.
Recall from \cite {BurbanKreussler}, Lemma 2.13, that for any smooth point
$x\in\boldsymbol{E}$ the sheaf of degree zero $\mathcal{O}(x-p_{0})$
satisfies
$\mathbb{F}(\mathcal{O}(x-p_{0})) \cong T_{\mathcal{O}}(\mathcal{O}(x))
\cong \boldsymbol{k}(x)$. Moreover, if $s\in\boldsymbol{E}$ denotes the
singular point, $n:\mathbb{P}^{1}\rightarrow \boldsymbol{E}$
the normalisation and
$\widetilde{\mathcal{O}}:=n_{\ast}(\mathcal{O}_{\mathbb{P}^{1}})$, then
$\mathbb{F}(\widetilde{\mathcal{O}}(-p_{0})) \cong
T_{\mathcal{O}}(\widetilde{\mathcal{O}}) \cong \boldsymbol{k}(s)$. The sheaf
$\widetilde{\mathcal{O}}(-p_{0})$ has degree zero on $\boldsymbol{E}$.
Because $\mathbb{F}$ is an equivalence, we obtain isomorphisms
\begin{align*}
\Hom(\mathbb{F}(\mathcal{F}),\boldsymbol{k}(x)) &\cong
\Hom(\mathcal{F}, \mathcal{O}(x-p_{0}))\\
\intertext{and}
\Hom(\mathbb{F}(\mathcal{F}),\boldsymbol{k}(s)) &\cong
\Hom(\mathcal{F}, \widetilde{\mathcal{O}}(-p_{0}))
\end{align*}
where $x\in\boldsymbol{E}$ is an arbitrary smooth point. These vector
spaces vanish as $\mathcal{F}$ was assumed to be semi-stable and of positive
degree.
Because cohomology of the complex $\mathbb{F}(\mathcal{F})$ vanishes in
positive degree, there is a canonical
morphism $\mathbb{F}(\mathcal{F})\rightarrow
\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))$ in $\Dbcoh(\boldsymbol{E})$, which
induces an injection of functors
$\Hom(\mathcal{H}^{0}(\mathbb{F}(\mathcal{F})), \,\cdot\,) \hookrightarrow
\Hom(\mathbb{F}(\mathcal{F}), \,\cdot\,)$. Therefore, the vanishing which was
obtained above, shows
$$\Hom(\mathcal{H}^{0}(\mathbb{F}(\mathcal{F})), \boldsymbol{k}(y)) = 0$$
for any point $y\in\boldsymbol{E}$. This implies the vanishing of the sheaf
$\mathcal{H}^{0}(\mathbb{F}(\mathcal{F}))$. Hence,
$\widehat{\mathcal{F}}:=\mathbb{F}(\mathcal{F})[-1]$ is a coherent sheaf and
the definition of $\mathbb{F}$ implies that there is an exact sequence of
coherent sheaves
$$0\rightarrow \widehat{\mathcal{F}}(-p_{0}) \rightarrow
H^{0}(\mathcal{F}(p_{0})) \otimes \mathcal{O}_{\boldsymbol{E}} \rightarrow
\mathcal{F}(p_{0}) \rightarrow 0.$$
This sequence implies, in particular, that $\widehat{\mathcal{F}}$ is
torsion free.
Before we proceed to show that $\widehat{\mathcal{F}}$ is semi-stable, we
apply duality to prove that $\mathbb{F}(\mathcal{F})$ is a sheaf if
$\deg(\mathcal{F})<0$. Let us denote the dualising functor by $\mathbb{D}:=
\boldsymbol{R}\mathcal{H}om(\,\cdot\,, \mathcal{O}_{\boldsymbol{E}})$. This
functor satisfies $\mathbb{D}\mathbb{D}\cong \boldsymbol{1}$. In
\cite{BurbanKreusslerRel}, Cor.\/ 3.4, we have shown that there exists an
isomorphism
$$\mathbb{D}\mathbb{F} [-1] \cong i^{\ast} \mathbb{F} \mathbb{D}.$$
Using $\mathbb{D}\circ[1] \cong [-1]\circ \mathbb{D}$, this implies
$$\mathbb{F} \cong \mathbb{D}i^{\ast}[-1]\mathbb{F}\mathbb{D}.$$
Because $\mathcal{F}$ is a torsion free sheaf on a curve, it is
Cohen-Macaulay and since $\boldsymbol{E}$ is Gorenstein, this implies
$\mathcal{E}xt^{i}(\mathcal{F},\mathcal{O}) = 0$ for any $i>0$.
Therefore, we have $\mathbb{D}(\mathcal{F})\cong \mathcal{F}^{\vee}$ and
this is a semi-stable coherent sheaf of positive degree. Thus, $[-1]\circ
\mathbb{F}$ sends $\mathcal{F}^{\vee}$ to a torsion free sheaf, on which
$\mathbb{D}$ is just the usual dual. Now, we see that
$\mathbb{F}(\mathcal{F})$ is a torsion free sheaf if $\mathcal{F}$ was
semi-stable and of negative degree.
It remains to prove that $\mathbb{F}$ preserves semi-stability. If
$\deg(\mathcal{F})=0$ or $\mathcal{F}$ is a torsion sheaf, this was shown
for any Weierstra{\ss} curve in \cite{BurbanKreussler}.
If $\deg(\mathcal{F})\ne 0$ the proof is based upon
$\mathbb{F}\mathbb{F}[-1]\cong i^{\ast}$, see \cite{BurbanKreussler}, Thm.\/
2.18. Suppose $\deg(\mathcal{F})>0$, then
$\mathbb{F}(\widehat{\mathcal{F}})\cong i^{\ast}(\mathcal{F})$ and this is a
coherent sheaf.
If $\widehat{\mathcal{F}}$ were not semi-stable, there would exist a
semi-stable sheaf $\mathcal{G}$ with $\mu(\widehat{\mathcal{F}}) >
\mu(\mathcal{G})$ and a non-zero morphism $\widehat{\mathcal{F}} \rightarrow
\mathcal{G}$. Because $\mu(\widehat{\mathcal{F}}) = -1/\mu(\mathcal{F})<0$,
$\mathbb{F}(\mathcal{G})$ is a coherent sheaf and application of
$\mathbb{F}$ produces a non-zero morphism $i^{\ast}(\mathcal{F}) \cong
\mathbb{F}(\widehat{\mathcal{F}}) \rightarrow
\mathbb{F}(\mathcal{G})$. However, $\mu(i^{\ast}(\mathcal{F})) =
\mu(\mathcal{F}) > -1/\mu(\mathcal{G}) = \mu(\mathbb{F}(\mathcal{G}))$
contradicts semi-stability of $i^{\ast}(\mathcal{F})$. Hence,
$\widehat{\mathcal{F}}$ is semi-stable. The proof in the case
$\deg(\mathcal{F})<0$ starts with a non-zero morphism
$\mathcal{U}\rightarrow \mathbb{F}(\mathcal{F})$ and proceeds similarly.
\end{proof}
It was shown in \cite{BurbanKreussler} that we obtain an action of the group
$\widetilde{\SL}(2,\mathbb{Z})$ on $\Dbcoh(\boldsymbol{E})$ by sending
generators of this group to $T_{\mathcal{O}}$, $T_{\boldsymbol{k}(p_{0})}$ and
the translation functor $[1]$ respectively.
Let us denote $$\mathsf{Q}:=\{\varphi\in\mathbb{R}\mid
\mathsf{P}(\varphi) \text{ contains a non-zero object}\}.$$
The action of a group $G$ on $\mathsf{Q}$ is called \emph{monotone}, if
$\varphi\le\psi$ implies $g\cdot\varphi\le g\cdot\psi$ for every $g\in G$ and
$\varphi,\psi\in \mathsf{Q}$.
\begin{proposition}\label{prop:transit}
The $\widetilde{\SL}(2,\mathbb{Z})$-action on $\Dbcoh(\boldsymbol{E})$
induces a monotone and transitive action on the set $\mathsf{Q}$. All
isotropy groups of this action are isomorphic to $\mathbb{Z}$.
\end{proposition}
\begin{proof}
As seen above, for any $\psi\in\mathsf{Q}$ and $0\ne
A\in\mathsf{P}(\psi)$, we have $\varphi(\mathbb{F}(A)) =
\varphi(A)+\frac{1}{2}$ and $\mu(T_{\boldsymbol{k}(p_{0})}(A)) = \mu(A)+1$.
Therefore, by Theorem \ref{thm:mother} it is clear that we obtain an induced
monotone action of $\widetilde{\SL}(2,\mathbb{Z})$ on $\mathsf{Q}$.
The group $\SL(2,\mathbb{Z})$ acts transitively on the set of all pairs of
co-prime integers which we interpret as primitive vectors of the lattice
$\mathbb{Z}\oplus i\mathbb{Z}\subset\mathbb{C}$. Hence, the action
of $\widetilde{\SL}(2,\mathbb{Z})$ on $\mathsf{Q}$ is transitive as
well. So, all isotropy groups are isomorphic. Finally, it is easy
to see that the isotropy group of $1\in\mathsf{Q}$ is generated by
$T_{\boldsymbol{k}(p_{0})}$.
\end{proof}
As an important consequence we obtain the following clear structure result
for the slices $\mathsf{P}(\varphi)$.
\begin{corollary}\label{cor:equiv}
The category $\mathsf{P}(\varphi)$ of semi-stable objects of phase
$\varphi\in\mathsf{Q}$ is equivalent to the category $\mathsf{P}(1)$ of
torsion sheaves. Any such equivalence restricts to an equivalence between
$\mathsf{P}(\varphi)^{s}$ and $\mathsf{P}(1)^{s}$. Under such an
equivalence, stable vector bundles correspond to structure sheaves of
smooth points. Moreover, if $\varphi\in(0,1)\cap \mathsf{Q}$,
$\mathsf{P}(\varphi)^{s}$ contains a unique torsion free sheaf, which is not
locally free. It correspond to the structure sheaf
$\boldsymbol{k}(s)\in\mathsf{P}(1)^{s}$ of the singular point.
\end{corollary}
Recall that an object $E\in\Dbcoh(\boldsymbol{E})$ is called \emph{perfect},
if it is isomorphic in the derived category to a bounded complex of locally
free sheaves of finite rank. Thus, a sheaf or shift thereof is called perfect,
if it is perfect as an object in $\Dbcoh(\boldsymbol{E})$.
If $\boldsymbol{E}$ is smooth, any object in $\Dbcoh(\boldsymbol{E})$ is
perfect. However, if $s\in\boldsymbol{E}$ is a singular point, the torsion
sheaf $\boldsymbol{k}(s)$ is not perfect.
If $\boldsymbol{E}$ is singular with one singularity $s\in\boldsymbol{E}$, the
category $\mathsf{P}(1)^{s}$ contains precisely one object which is not
perfect, the object $\boldsymbol{k}(s)$.
Hence, by Proposition \ref{prop:transit}, for any $\varphi\in\mathsf{Q}$ there
is precisely one element in $\mathsf{P}(\varphi)^{s}$ which is not perfect. We
shall refer to it as the \emph{extreme} stable element with phase
$\varphi$. So, the sheaf $\boldsymbol{k}(s)$ is the extreme stable element
with phase $1$. The extreme stable element is never locally free. A stable
object is either perfect or extreme.
We shall need the following version of Serre duality, which can be deduced
easily from standard versions:
If $E,F\in\Dbcoh(\boldsymbol{E})$ and at least one of them is perfect, then
there is a bi-functorial isomorphism
\begin{equation}
\label{wesPT:i}\Hom(E,F) \cong \Hom(F,E[1])^{\ast}.
\end{equation}
If neither of the objects is perfect, this is no longer true. For example,
$\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s))\cong \boldsymbol{k}$, but
$\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s)[1]) \cong
\Ext^{1}(\boldsymbol{k}(s),\boldsymbol{k}(s)) \cong \boldsymbol{k}^{2}$.
Any object $X$ in the Abelian category $\mathsf{P}(\varphi)$ has a
Jordan-H\"older filtration (JHF)
$$0\subset F_{n}X \subset \ldots \subset F_{1}X \subset F_{0}X = X$$
with stable JH-factors $J_{i}=F_{i}X/F_{i+1}X \in
\mathsf{P}(\varphi)^{s}$. The graded object $\oplus_{i=0}^{n}J_{i}$ is
determined by $X$. Observe that for any two objects $J\not\cong
J'\in\mathsf{P}(\varphi)^{s}$ we can apply Serre duality because at most one
of them is non-perfect.
\begin{corollary}\label{cor:sheaves}
\begin{enumerate}
\item\label{cor:i} If $\varphi,\psi \in \mathsf{Q}$ with $\varphi -1 < \psi
\le \varphi$ there exists $\Phi\in \widetilde{\SL}(2,\mathbb{Z})$ such that
$\Phi(\varphi)=1$ and $\Phi(\psi)\in(0,1]$.
\item\label{cor:ii} If $A,B\in\mathsf{P}(\varphi)^{s}$, then $A\cong B \iff
\Hom(A,B)\ne 0.$
\item\label{cor:iii} If $0\ne X\in\mathsf{P}(\varphi)$ and $0\ne
Y\in\mathsf{P}(\psi)$ with $\varphi < \psi < \varphi+1$, then $\Hom(X,Y)\ne
0$.
\item \label{cor:iv} If $J\in\mathsf{P}(\varphi)^{s}$ is not a JH-factor of
$X\in \mathsf{P}(\varphi)$, for all $i\in\mathbb{Z}$ we have
$\Hom(J,X[i])=0$.
\item \label{cor:v} If $X\in\mathsf{P}(\varphi)$ is indecomposable, all its
JH-factors are isomorphic to each other.
\item \label{cor:vi} If $X,Y\in\mathsf{P}(\varphi)$ are non-zero
indecomposable objects, both with the same JH-factor, then $\Hom(X,Y) \ne
0$.
\end{enumerate}
\end{corollary}
\begin{proof}
(\ref{cor:i}) This follows from Proposition \ref{prop:transit} because the
shift functor corresponds to an element in the centre of
$\widetilde{\SL}(2,\mathbb{Z})$ and therefore $\Phi(\mathsf{P}(\varphi)) =
\mathsf{P}(1)$ implies $\Phi(\mathsf{P}(\varphi-1)) = \mathsf{P}(0)$.
(\ref{cor:ii}) The statement is clear in case $\varphi=1$ and follows from
(\ref{cor:i}) in the general case.
(\ref{cor:iii}) Using (\ref{cor:i}) we can assume $\psi=1$, which means
that $Y$ is a coherent torsion sheaf. By Proposition \ref{prop:transit} this
implies $\varphi\in(0,1)$ and $X$ is a torsion free coherent sheaf. If
$Y\in\mathsf{P}(1)^{s}$ the statement is clear, because any torsion free
sheaf has a non-zero morphism to any $Y=\boldsymbol{k}(x)$,
$x\in\boldsymbol{E}$. If $Y\in\mathsf{P}(1)$ is arbitrary, there exists a
point $x\in\boldsymbol{E}$ and a non-zero morphism $\boldsymbol{k}(x)
\rightarrow Y$. The claim follows now from left-exactness of the functor
$\Hom(X,\,\cdot\,)$.
(\ref{cor:iv}) If $J'\in\mathsf{P}(\varphi)^{s}$ is a JH-factor of $X$, we
have $J\not\cong J'$. From (\ref{cor:ii}) and Serre duality together with
Lemma \ref{wesPT:ii} we obtain $\Hom(J,J'[i])=0$ for any
$i\in\mathbb{Z}$. Using the JHF of $X$, the claim now follows.
(\ref{cor:v}) It is easy to prove by induction that any
$X\in\mathsf{P}(\varphi)$ can be split as a finite direct sum $X\cong\oplus
X_{k}$, where each $X_{k}$ has all JH-factors isomorphic to a single element
$J_{k}\in\mathsf{P}(\varphi)^{s}$. This implies, the claim.
(\ref{cor:vi}) By (\ref{cor:i}) we may assume
$\varphi(X)=\varphi(Y)=1$. This means, both objects are indecomposable
torsion sheaves with support at the singular point $s\in\boldsymbol{E}$.
Such sheaves always have an epimorphism to and a monomorphism from the
extreme object $\boldsymbol{k}(s)$, hence the claim.
\end{proof}
It is interesting and important to note that an indecomposable semi-stable
object can be perfect even though all its JH-factors are extreme. This is
made explicit in \cite{BurbanKreussler}, Section 4, in the case of the
category $\mathsf{P}(1)$ of coherent torsion sheaves. If $\boldsymbol{E}$ is
nodal, there are two kinds of indecomposable torsion sheaves with support at
the node $s\in\boldsymbol{E}$: the so-called \emph{bands} and
\emph{strings}. The bands are perfect, whereas the strings are not
perfect. Using the action of $\widetilde{\SL}(2,\mathbb{Z})$ this carries over
to all other categories $\mathsf{P}(\varphi)$ with $\varphi\in\mathsf{Q}$.
An object $X\in\mathsf{P}(\varphi)$ will be called \emph{extreme} if it
does not have a direct summand which is perfect. This implies that, but is not
equivalent to the property that all its JH-factors are extreme. An example can
be found below, see Ex.~\ref{ex:extremefactors}.
From the above we deduce that any $X\in\mathsf{P}(\varphi)$ can be split as a
direct sum $X\cong X^{e}\oplus X^{p}$ with $X^{e}$ extreme and $X^{p}$
perfect. All direct summands of the extreme part have the unique
extreme stable element with phase $\varphi$ as its JH-factors. On the
other hand, all the direct summands of $X^{p}$ are perfect and they can have
any object of $\mathsf{P}(\varphi)^{s}$ as JH-factor.
\begin{corollary}
Any coherent sheaf $\mathcal{F}$ with $\End(\mathcal{F}) = \boldsymbol{k}$
is stable.
\end{corollary}
\begin{proof}
The assumption implies that $\mathcal{F}$ is indecomposable.
If $\mathcal{F}$ were not even semi-stable, it would have at least two
HN-factors. Using Corollary \ref{cor:sheaves}, we may assume that
$\varphi_{+}(\mathcal{F})=1$. Thus, $\mathcal{F}$ is a coherent sheaf which
is neither torsion nor torsion free. This implies that there is a
non-invertible endomorphism
$\mathcal{F} \rightarrow \boldsymbol{k}(s) \rightarrow \tors(\mathcal{F})
\rightarrow \mathcal{F}$, in contradiction to the assumption. Hence,
$\mathcal{F}\in\mathsf{P}(\varphi)$ is semi-stable. Let
$\mathcal{J}\in\mathsf{P}(\varphi)$ be its JH-factor.
From Corollary \ref{cor:sheaves} (\ref{cor:vi}) we obtain a non-zero
endomorphism $\mathcal{F}\rightarrow \mathcal{J}\rightarrow \mathcal{F}$,
which can only be an isomorphism, if $\mathcal{F}\cong \mathcal{J}$, so
$\mathcal{F}$ is indeed stable.
\end{proof}
The following method can be used to visualise the structure of the category
$\Dbcoh(\boldsymbol{E})$: the vertical slices in Figure \ref{fig:slices} are
thought to correspond to the categories $\mathsf{P}(t)^{s}$ of stable objects.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1,1){\line(0,1){3}}\put(1,0.8){\makebox(0,0)[t]{$2$}}
\put(4,1){\line(0,1){3}}\put(4,0.8){\makebox(0,0)[t]{$1$}}
\put(7,1){\line(0,1){3}}\put(7,0.8){\makebox(0,0)[t]{$0$}}
\put(10,1){\line(0,1){3}}\put(10,0.8){\makebox(0,0)[t]{$-1$}}
\thinlines
\put(4.9,1){\line(0,1){3}}\put(4.9,0.8){\makebox(0,0)[t]{$t$}}
\put(1.8,2.5){$\Coh_{\boldsymbol{E}}[1]$}
\put(5.3,2.5){$\Coh_{\boldsymbol{E}}$}
\put(7.6,2.5){$\Coh_{\boldsymbol{E}}[-1]$}
\end{picture}
\end{center}
\caption{slices}\label{fig:slices}
\end{figure}
They are non-empty if and only if $t\in\mathsf{Q}$, i.e.\/
$\mathbb{R}\exp(\pi it) \cap \mathbb{Z}^{2} \ne \{(0,0)\}$. A point on such a
slice represents a stable object. The extreme stable objects are those which
lie on the dashed upper horizontal line. The labelling below the picture
reflects the phases of the slices. We have chosen to let it decrease from the
left to right in order to have objects with cohomology in negative degrees on
the left and with positive degrees on the right.
By Proposition \ref{prop:transit}, the group $\widetilde{\SL}(2,\mathbb{Z})$
acts on the set of all stable objects, hence it acts on such pictures.
This action sends slices to slices and acts transitively on the set of
slices with phase $t\in\mathsf{Q}$. The dashed line of extreme stable objects
is invariant under this action.
Any indecomposable object $0\ne X\in\Dbcoh(\boldsymbol{E})$ has a
\emph{shadow} in such a picture: it is the set of all stable objects which
occur as JH-factors in the HN-factors of $X$. If this set consists of more
than one point, the shadow is obtained by connecting these points by line
segments.
The following proposition shows that the shadow of an indecomposable object
which consists of more than one point is completely contained in the
extreme line.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1,1){\line(0,1){3}}\put(1,0.8){\makebox(0,0)[t]{$2$}}
\put(4,1){\line(0,1){3}}\put(4,0.8){\makebox(0,0)[t]{$1$}}
\put(7,1){\line(0,1){3}}\put(7,0.8){\makebox(0,0)[t]{$0$}}
\put(10,1){\line(0,1){3}}\put(10,0.8){\makebox(0,0)[t]{$-1$}}
\thinlines
\put(1.8,2.5){$\Coh_{\boldsymbol{E}}[1]$}
\put(5.3,2.5){$\Coh_{\boldsymbol{E}}$}
\put(7.6,2.5){$\Coh_{\boldsymbol{E}}[-1]$}
\put(4,2){\circle*{0.2}}\put(4.2,2){\makebox(0,0)[l]{$X_{1}$}}
\put(8.2,3.4){\circle*{0.2}}\put(8.4,3.4){\makebox(0,0)[l]{$X_{2}$}}
\put(0.3,4){\circle*{0.2}}
\thicklines\put(0.3,4){\line(1,0){1.5}}
\put(1.8,4){\circle*{0.2}}
\thicklines\put(1.8,4){\line(1,0){0.8}}
\put(2.6,4){\circle*{0.2}}\put(1.3,4.2){\makebox(0,0)[b]{$X_{3}$}}
\put(4.3,4){\circle*{0.2}}
\thicklines\put(4.3,4){\line(1,0){1}}
\put(5.3,4){\circle*{0.2}}\put(4.8,4.2){\makebox(0,0)[b]{$X_{4}$}}
\put(6.3,4){\circle*{0.2}}\put(6.3,4.2){\makebox(0,0)[b]{$X_{5}$}}
\end{picture}
\end{center}
\caption{shadows}\label{fig:example}
\end{figure}
Figure \ref{fig:example} shows the shadows of five different
indecomposable objects:
\begin{itemize}
\item $X_{1}\in\Coh_{\boldsymbol{E}}$ an indecomposable torsion sheaf,
\item $X_{2}\in \Coh_{\boldsymbol{E}}[-1]$ the shift of an indecomposable
semi-stable locally free sheaf,
\item $X_{3}$ a genuine complex with three extreme HN-factors, one in
$\Coh_{\boldsymbol{E}}[2]$ and the other two in $\Coh_{\boldsymbol{E}}[1]$,
\item $X_{4}$ an indecomposable torsion free sheaf which is not semi-stable,
\item $X_{5}\in\Coh_{\boldsymbol{E}}$ an indecomposable and semi-stable
torsion free sheaf which could be perfect or not (a band or a string in the
language of representation theory).
\end{itemize}
The shadow of an indecomposable object is a single point if and only if this
object is semi-stable.
\begin{proposition}\label{prop:extreme}
Let $X \in \Dbcoh(\boldsymbol{E})$ be an indecomposable object which is not
semi-stable. Then, all HN-factors of $X$ are extreme.
\end{proposition}
\begin{proof}
Let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $X$. If the HN-factor $A_{i}$ were not
extreme, it could be split into a direct sum $A_{i} \cong A_{i}'
\oplus A_{i}''$ with $0\ne A_{i}'$ perfect and $A_{i}',
A_{i}''\in\mathsf{P}(\varphi_{i})$.
Because $\varphi_{-}(F_{i+1}X) > \varphi_{i}=\varphi(A_{i}')$, Lemma
\ref{wesPT:ii} and Serre duality imply
$$\Hom(A_{i}', F_{i+1}X[1]) \cong \Hom(F_{i+1}X, A_{i}')^{\ast} = 0.$$
Hence, we can apply Lemma \ref{lem:PengXiao} to the distinguished triangle
$$F_{i+1}X \rightarrow F_{i}X \rightarrow A_{i}
\stackrel{+}{\longrightarrow}$$
and obtain a decomposition $F_{i}X \cong F_{i}'X \oplus A_{i}'$.
We proceed by descending induction on $j\le i$ to show that there exist
decompositions $F_{j}X \cong F_{j}'X \oplus A_{i}'$. This is
obtained from Lemma \ref{lem:PengXiao} applied to the distinguished triangle
$$F_{j}'X\oplus A_{i}' \rightarrow F_{j-1}X \rightarrow A_{j-1}
\stackrel{+}{\longrightarrow}$$
and using Lemma \ref{wesPT:ii}, Serre duality and
$\varphi(A_{i}') > \varphi(A_{j-1})$ to get
$$\Hom(A_{j-1}, A_{i}'[1]) \cong \Hom(A_{i}', A_{j-1})^{\ast} = 0.$$
We obtain a decomposition $X=F_{0}X \cong F_{0}'X \oplus A_{i}'$ in which
we have $A_{i}'\ne 0$. Because $X$ was assumed to be indecomposable, we
should have $X\cong A_{i}'$, but this was excluded by assumption. This
contradiction shows that all HN-factors $A_{i}$ are necessarily
extreme.
\end{proof}
\begin{corollary}\label{cor:types}
There exist four types of indecomposable objects in the category
$\Coh_{\boldsymbol{E}}$:
\begin{enumerate}
\item \label{type:i} semi-stable with perfect JH-factor;
\item \label{type:ii} semi-stable, perfect but its JH-factor extreme;
\item \label{type:iii} semi-stable and extreme;
\item \label{type:iv} not semi-stable, with all its HN-factors extreme.
\end{enumerate}
\end{corollary}
A similar statement is true for $\Dbcoh(\boldsymbol{E})$. In this case, the
objects of types (\ref{type:i}), (\ref{type:ii}) and (\ref{type:iii}) are
shifts of coherent sheaves, whereas genuine complexes are possible for objects
of type (\ref{type:iv}).
Types (\ref{type:ii}), (\ref{type:iii}) and (\ref{type:iv}) were not available
in the smooth case.
Examples of type (\ref{type:i}) are simple vector bundles and structure sheaves
$\boldsymbol{k}(x)$ of smooth points $x\in\boldsymbol{E}$. All indecomposable
objects with a shadow not on the extreme line fall into type (\ref{type:i}).
Under the equivalences of Corollary \ref{cor:equiv}, indecomposable semi-stable
locally free sheaves with extreme JH-factor correspond, in the nodal case,
precisely to those torsion sheaves with support at the node $s$, which are
called bands, see \cite{BurbanKreussler}.
Examples of type (\ref{type:iii}) are the stable coherent sheaves which are not
locally free and the structure sheaf $\boldsymbol{k}(s)$ of the singular point
$s\in\boldsymbol{E}$. Moreover, in the nodal case, the torsion sheaves with
support at $s$, which are called strings in \cite{BurbanKreussler}, are of
type (\ref{type:iii}) as well. Examples of objects of type (\ref{type:iv}) are
given below.
\begin{example}
We shall construct torsion free sheaves on nodal $\boldsymbol{E}$ with an
arbitrary finite number of HN-factors. This implies that the number of points
in a shadow of an indecomposable object in $\Dbcoh(\boldsymbol{E})$ is not
bounded.
Recall from \cite{DrozdGreuel} that any indecomposable torsion free sheaf
which is not locally free, is isomorphic to a sheaf
$\mathcal{S}(\boldsymbol{d}) = p_{n\ast} \mathcal{L}(\boldsymbol{d})$. We use
here the notation of \cite{BurbanKreussler}, Section 3.5, so that $p_{n}:
\boldsymbol{I_{n}} \rightarrow \boldsymbol{E}$ denotes a certain morphism
from the chain $\boldsymbol{I_{n}}$ of $n$ smooth rational curves to the
nodal curve $\boldsymbol{E}$. If $\boldsymbol{d}=(d_{1},\ldots,d_{n})
\in\mathbb{Z}^{n}$, we denote by $\mathcal{L}(\boldsymbol{d})$ the line
bundle on $\boldsymbol{I_{n}}$ which has degree $d_{\nu}$ on the $\nu$-th
component of $\boldsymbol{I_{n}}$. We know $\rk(\mathcal{S}(\boldsymbol{d}))
= n$ and $\deg(\mathcal{S}(\boldsymbol{d})) = 1+\sum d_{\nu}$. We obtain,
in particular, that for any $\varphi\in\mathsf{Q} \cap (0,1)$ there exist
$n\in\mathbb{Z}$ and $\boldsymbol{d}(\varphi)\in\mathbb{Z}^{n}$ such that
$\mathcal{S}(\boldsymbol{d}(\varphi))$ is the unique extreme element in
$\mathsf{P}(\varphi)^{s}$. On the other hand, if $\boldsymbol{d}'\in
\mathbb{Z}^{n'}, \boldsymbol{d}''\in \mathbb{Z}^{n''}$ and
$\boldsymbol{d} = (\boldsymbol{d}_{+}', \boldsymbol{d}'')\in
\mathbb{Z}^{n'+n''}$, where $\boldsymbol{d}_{+}'$ is obtained from
$\boldsymbol{d}'$ by adding $1$ to the last component, we have an exact
sequence
$$0\rightarrow \mathcal{S}(\boldsymbol{d}') \rightarrow
\mathcal{S}(\boldsymbol{d}) \rightarrow
\mathcal{S}(\boldsymbol{d}'') \rightarrow 0$$
see for example \cite{Mozgovoy}. Hence, if we start with a sequence
$0<\varphi_{0} <\varphi_{1}< \ldots <\varphi_{m} <1$ where
$\varphi_{\nu}\in\mathsf{Q}$ and define
$$\boldsymbol{d}^{(m)} = \boldsymbol{d}(\varphi_{m})\quad\text{ and }\quad
\boldsymbol{d}^{(\nu)} =
(\boldsymbol{d}_{+}^{(\nu+1)},\boldsymbol{d}(\varphi_{\nu})) \text{ for }
m > \nu \ge 0,$$
we obtain an indecomposable torsion free sheaf
$\mathcal{S}(\boldsymbol{d}^{(0)})$ whose HN-factors are the extreme stable
sheaves $\mathcal{S}(\boldsymbol{d}(\varphi_{\nu})) \in
\mathsf{P}(\varphi_{\nu}), 0\le \nu \le m$. The HNF of this sheaf is given by
$$\mathcal{S}(\boldsymbol{d}^{(m)}) \subset
\mathcal{S}(\boldsymbol{d}^{(m-1)}) \subset \ldots \subset
\mathcal{S}(\boldsymbol{d}^{(0)}).$$
The sheaf $\mathcal{S}(\boldsymbol{d}^{(0)})$ is of type (\ref{type:iv}) and
not perfect.
\end{example}
\begin{example}\label{ex:extremefactors}
Suppose $\boldsymbol{E}$ is nodal and let
$\pi:C_{2}\rightarrow\boldsymbol{E}$ be an \'etale morphism of degree
two, where $C_{2}$ denotes a reducible curve which has two components, both
isomorphic to $\mathbb{P}^{1}$ and which intersect transversally at two
distinct points.
By $i_{\nu}:\mathbb{P}^{1}\rightarrow \boldsymbol{E},\;\nu=1,2$ we denote the
morphisms which are induced by the embeddings of the two components of
$C_{2}$.
There is a $\boldsymbol{k}^{\times}$-family of
line bundles on $C_{2}$, whose restriction to one component is
$\mathcal{O}_{\mathbb{P}^{1}}(-2)$ and to the other is
$\mathcal{O}_{\mathbb{P}^{1}}(2)$. The element in $\boldsymbol{k}^{\times}$
corresponds to a gluing parameter over one of the two singularities of
$C_{2}$. If $\mathcal{L}$ denotes one such line bundle,
$\mathcal{E}:=\pi_{\ast}\mathcal{L}$ is an
indecomposable vector bundle of rank two and degree zero on $\boldsymbol{E}$.
Let us fix notation so that $i_{1}^{\ast}\mathcal{E}
\cong \mathcal{O}_{\mathbb{P}^{1}}(-2)$ and $i_{2}^{\ast}\mathcal{E} \cong
\mathcal{O}_{\mathbb{P}^{1}}(2)$. There is an exact sequence of coherent
sheaves on $\boldsymbol{E}$
\begin{equation}\label{eq:nonssvb}
0\rightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow
\mathcal{E}
\rightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2) \rightarrow 0.
\end{equation}
Because the torsion free sheaves $i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}$ and
$i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2)$ have rank one and
$\boldsymbol{E}$ is irreducible, they are stable. Because $\varphi(i_{2\ast}
\mathcal{O}_{\mathbb{P}^{1}}) = 3/4$ and $\varphi(i_{1\ast}
\mathcal{O}_{\mathbb{P}^{1}}(-2)) = 1/4$, Theorem \ref{thm:uniqueHNF}
implies that the HNF of $\mathcal{E}$ is given by the
exact sequence (\ref{eq:nonssvb}). The HN-factors are the two torsion
free sheaves of rank one $i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}$ and
$i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}(-2)$, which are not locally
free. These are the extreme stable elements with phases $3/4$ and $1/4$
respectively. Therefore, the indecomposable vector bundle $\mathcal{E}$ is a
perfect object of type (\ref{type:iv}) which satisfies
$\varphi_{-}(\mathcal{E})=1/4$ and $\varphi_{+}(\mathcal{E})=3/4$.
\end{example}
\begin{remark}\label{rem:notperfect}
This example shows that the full sub-category of perfect complexes in the
category $\Dbcoh(\boldsymbol{E})$ is not closed under taking
Harder-Narasimhan factors. We interpret this to be an indication that the
derived category of perfect complexes is not an appropriate object for
homological mirror symmetry on singular Calabi-Yau varieties.
\end{remark}
\begin{remark}
It seems plausible that methods similar to those of this section could be
applied to study the derived category of representations of certain derived
tame associative algebras. Such may include gentle algebras, skew-gentle
algebras and degenerated tubular algebras.
The study of Harder-Narasimhan filtrations in conjunction with the action of
the group of exact auto-equivalences of the derived category may provide new
insight into the combinatorics of indecomposable objects in these derived
categories.
\end{remark}
\begin{proposition}\label{wesPT}
Suppose $X,Y\in \Dbcoh(\boldsymbol{E})$ are non-zero.
\begin{enumerate}
\item \label{wesPT:iii} If $\varphi_{-}(X) <
\varphi_{+}(Y) < \varphi_{-}(X)+1$, then $\Hom(X,Y)\ne 0$.
\item \label{wesPT:iv} If $X$ and $Y$ are indecomposable objects which are
not of type (\ref{type:i}) in Corollary \ref{cor:types} and which satisfy
$\varphi_{-}(X) = \varphi_{+}(Y)$, then $\Hom(X,Y)\ne 0$.
\end{enumerate}
\end{proposition}
\begin{proof}
If $X$ and $Y$ are semi-stable objects, the claim
(\ref{wesPT:iii}) was proved in Corollary \ref{cor:sheaves} (\ref{cor:iii}).
Similarly, (\ref{wesPT:iv}) for two semi-stable objects follows from
Corollary \ref{cor:sheaves} (\ref{cor:vi}), because there is only one
non-perfect object in $\mathsf{P}(\varphi)^{s}$.
For the rest of the proof we treat both cases, (\ref{wesPT:iii}) and
(\ref{wesPT:iv}) simultaneously.
For the proof of (\ref{wesPT:iv}) we keep in mind that Proposition
\ref{prop:extreme} implies that no HN-factor has a perfect summand, if
the object is indecomposable but not semi-stable.
If $X\in\mathsf{P(\varphi)}$ is semi-stable but $Y\in\Dbcoh(\boldsymbol{E})$
is arbitrary, we let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{m}Y \ar[rr] \ar[dl]_{\cong}&& F_{m-1}Y \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}Y \ar[rr] && F_{0}Y \ar@{=}[r]\ar[dl] &
Y\\
& B_{m} \ar[lu]^{+} && B_{m-1} \ar[lu]^{+} & & & & & & B_0 \ar[lu]^{+}&
}\]
be a HNF of $Y$. As $\varphi(B_{m})=\varphi_{+}(Y)$ we know already
$\Hom(X,B_{m})\ne 0$.
By assumption, we have $\varphi(B_{i}[-1]) = \varphi(B_{i})
-1 \le \varphi_{+}(Y)-1 < \varphi(X)$. Hence, by Lemma \ref{wesPT:ii},
$\Hom(X, B_{i}[-1]) =0$ and the cohomology sequence of the distinguished
triangle $F_{i+1}Y\rightarrow F_{i}Y\rightarrow B_{i}
\stackrel{+}{\rightarrow}$ provides an inclusion $\Hom(X, F_{i+1}Y) \subset
\Hom(X, F_{i}Y)$. This implies $0\ne \Hom(X,B_{m})\subset \Hom(X,Y)$.
Finally, in the general case, we let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be a HNF of $X$. As $\varphi(A_{0})=\varphi_{-}(X)$ we have
$\Hom(A_{0},Y)\ne 0$.
Because $\varphi_{-}(F_{1}X[1]) = \varphi_{-}(F_{1}X) +1 =
\varphi(A_{1}) +1 > \varphi_{-}(X)+1 > \varphi_{+}(Y)$, Lemma
\ref{wesPT:ii} implies $\Hom(F_{1}X[1], Y)=0$. The distinguished triangle
$F_{1}X \rightarrow X \rightarrow A_{0} \stackrel{+}{\rightarrow}$ gives us
now an inclusion $0\ne \Hom(A_{0},Y) \subset \Hom(X,Y)$ and so the claim.
\end{proof}
In \cite{YangBaxter}, Polishchuk asked for the classification of all
spherical objects in the bounded derived category of a singular projective
curve of arithmetic genus one. Below, we shall solve this problem for
irreducible curves.
Let $\boldsymbol{E}$ be an irreducible projective curve of arithmetic genus
one over our base field $\boldsymbol{k}$.
Recall that in this case an object $X\in\Dbcoh(\boldsymbol{E})$ is
\emph{spherical} if
$$X \text{ is perfect and }\quad
\Hom(X,X[i]) \cong
\begin{cases}
\boldsymbol{k} & \text{if }\; i \in \{0,1\} \\
0 & \text{if }\; i \not\in \{0,1\}
\end{cases}
$$
\begin{proposition}\label{prop:spherical}
Let $\boldsymbol{E}$ be an irreducible projective curve of arithmetic genus
one and $X\in\Dbcoh(\boldsymbol{E})$. Then the following are equivalent:
\begin{enumerate}
\item\label{spher:i} $X$ is spherical;
\item \label{spher:ii}$\Hom(X,X[i]) \cong
\begin{cases}
\boldsymbol{k} & \text{if }\; i = 0 \\
0 & \text{if }\; i = 2 \;\text{ or }\; i<0;
\end{cases}$
\item\label{spher:iii} $X$ is perfect and stable;
\item\label{spher:iv} there exists $n\in\mathbb{Z}$ such that $X[n]$ is
isomorphic to a simple vector bundle or to a torsion sheaf of length one
which is supported at a smooth point of $\boldsymbol{E}$.
\end{enumerate}
In particular, the group of exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of all spherical
objects.
\end{proposition}
\begin{proof}
The implication (\ref{spher:i})$\Rightarrow$(\ref{spher:ii}) is obvious.
Let us prove (\ref{spher:ii})$\Rightarrow$(\ref{spher:iii}).
First, we observe that $\Hom(X,X) \cong \boldsymbol{k}$ implies that
$X$ is indecomposable. Suppose, $X$ is not semi-stable. This is equivalent
to $\varphi_{+}(X)>\varphi_{-}(X)$. By Proposition \ref{prop:extreme}
we know that all HN-factors of $X$ are extreme.
Let $M\ge 0$ be the unique integer with $M\le \varphi_{+}(X) -
\varphi_{-}(X) < M+1$.
If $M< \varphi_{+}(X) - \varphi_{-}(X) <M+1$,
Proposition \ref{wesPT} (\ref{wesPT:iii}) implies $\Hom(X,X[-M])\ne
0$. Under the assumption (\ref{spher:ii}), this is possible only if $M=0$.
On the other hand, if $M=\varphi_{+}(X) - \varphi_{-}(X)$, we obtain from
Proposition \ref{wesPT} (\ref{wesPT:iv}) $\Hom(X,X[-M]) \ne 0$.
Again, this implies $M=0$.
So, we have $0< \varphi_{+}(X) - \varphi_{-}(X) <1$.
If we apply the functor $\Hom(\,\cdot\,,X)$ to
$F_{1}X\stackrel{u}{\rightarrow} X \rightarrow A_{0}
\stackrel{+}{\longrightarrow}$,
the rightmost distinguished triangle of the HNF of $X$, we obtain the exact
sequence
$$\Hom(F_{1}X[1],X) \rightarrow \Hom(A_{0},X) \rightarrow \Hom(X,X)
\rightarrow \Hom(F_{1}X,X),$$
in which the leftmost term $\Hom(F_{1}X[1],X)=0$ by Lemma \ref{wesPT:ii},
because $\varphi_{-}(F_{1}X[1]) > \varphi_{-}(X)+1>\varphi_{+}(X)$. The third
morphism in this sequence is not the zero map, as it sends $\mathsf{Id}_{X}$
to $u\ne 0$. Because $\Hom(X,X)$ is one dimensional, this is only possible
if $\Hom(A_{0},X)=0$. But Proposition \ref{wesPT} (\ref{wesPT:iii}) and
$\varphi(A_{0})<\varphi_{+}(X)< \varphi(A_{0})+1$ imply $\Hom(A_{0},X)\ne 0$.
This contradiction shows that $X$ must be semi-stable.
We observed earlier that all the
JH-factors of an indecomposable semi-stable object are isomorphic to each
other. Therefore, any indecomposable semi-stable object which is not stable
has a space of endomorphisms of dimension at least two. So, we conclude
$X\in\mathsf{P}(\varphi)^{s}$ for some $\varphi\in\mathbb{R}$.
Because $\Hom(\boldsymbol{k}(s),\boldsymbol{k}(s)[2]) \cong
\Ext^{2}(\boldsymbol{k}(s),\boldsymbol{k}(s))\ne 0$, the transitivity of the
action of $\widetilde{\SL}(2,\mathbb{Z})$ on the set $\mathsf{Q}$ implies
that none of the extreme stable objects satisfies the condition
(\ref{spher:ii}). Hence, $X$ is perfect and stable.
To prove (\ref{spher:iii})$\Rightarrow$(\ref{spher:i}), we observe that
the group of automorphisms of the curve $\boldsymbol{E}$ acts
transitively on the regular locus $\boldsymbol{E}\setminus\{s\}$. Hence, by
Proposition \ref{prop:transit}, the group of auto-equivalences of
$\Dbcoh(\boldsymbol{E})$ acts transitively on the set of all perfect stable
objects. Because, for example, the structure sheaf
$\mathcal{O}_{\boldsymbol{E}}$ is spherical, it is now clear that all
perfect stable objects are indeed spherical and that the group of
exact auto-equivalences of $\Dbcoh(\boldsymbol{E})$ acts transitively on the
set of all spherical objects.
To show the equivalence with (\ref{spher:iv}), it remains to recall that any
perfect coherent torsion free sheaf on $\boldsymbol{E}$ is locally free. This
follows easily from the Auslander-Buchsbaum formula because we are working in
dimension one.
\end{proof}
\section{Description of $t$-structures in the case of a
singular Weierstra\ss{} curve}\label{sec:tstruc}
The main result of this section is a description of all $t$-structures on the
derived category of a singular Weierstra\ss{} curve $\boldsymbol{E}$. This
generalises results of \cite{GRK} and \cite{Pol1}, where the smooth case was
studied. As an application, we obtain a description of the group
$\Aut(\Dbcoh(\boldsymbol{E}))$ of all exact auto-equivalences of
$\Dbcoh(\boldsymbol{E})$. A second application is a description of Bridgeland's
space of stability conditions on $\boldsymbol{E}$.
Recall that a $t$-structure on a triangulated category $\mathsf{D}$ is a pair
of full subcategories $(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 0})$ such that,
with the notation $\mathsf{D}^{\ge n} := \mathsf{D}^{\ge
0}[-n]$ and $\mathsf{D}^{\le n} := \mathsf{D}^{\le 0}[-n]$ for any $n\in
\mathbb{Z}$, the following holds:
\begin{enumerate}
\item $\mathsf{D}^{\le 0} \subset \mathsf{D}^{\le 1}$ and $\mathsf{D}^{\ge 1}
\subset \mathsf{D}^{\ge 0}$;
\item $\Hom(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 1}) = 0$;
\item\label{def:tiii} for any object $X \in \mathsf{D}$ there exists a
distinguished triangle
$$A \rightarrow X \rightarrow B \stackrel{+}{\longrightarrow}$$
with $A \in \mathsf{D}^{\le 0}$ and $B \in \mathsf{D}^{\ge 1}.$
\end{enumerate}
If $(\mathsf{D}^{\le 0}, \mathsf{D}^{\ge 0})$ is a $t$-structure then ${\sf
A} = \mathsf{D}^{\le 0} \cap \mathsf{D}^{\ge 0}$ has a structure of an
Abelian category. It is called the \emph{heart} of the $t$-structure.
In this way, $t$-structures on the derived category
$\Dbcoh(\boldsymbol{E})$ lead to interesting Abelian categories embedded into
it. The natural $t$-structure on $\Dbcoh(\boldsymbol{E})$ has $\mathsf{D}^{\le
n}$ equal to the full subcategory formed by all complexes with non-zero
cohomology in degree less or equal to $n$ only. Similarly, the full subcategory
$\mathsf{D}^{\ge n}$ consists of all complexes $X$ with $H^{i}(X)=0$ for all
$i<n$. The heart of the natural $t$-structure is the Abelian category
$\Coh_{\boldsymbol{E}}$.
In addition to the natural $t$-structure we also have many interesting
$t$-structures on $\Dbcoh(\boldsymbol{E})$.
In order to describe them, we introduce the following notation. We continue to
work with the notion of stability and the notation introduced in the previous
section.
If $\mathsf{P}\subset\mathsf{P}(\theta)^{s}$ is a subset, we denote by
$\mathsf{D}[\mathsf{P}, \infty)$ the full subcategory of
$\Dbcoh(\boldsymbol{E})$ which is defined as follows:
$X\in\Dbcoh(\boldsymbol{E})$ is in $\mathsf{D}[\mathsf{P},
\infty)$ if and only if $X=0$ or all its HN-factors, which have at least one
JH-factor which is not in $\mathsf{P}$, have phase $\varphi>\theta$.
Similarly, $\mathsf{D}(-\infty,\mathsf{P}]$ denotes the category which is
generated by $\mathsf{P}$ and all $\mathsf{P}(\varphi)$ with
$\varphi<\theta$.
If $\mathsf{P}=\mathsf{P}(\theta)^{s}$ we may abbreviate
$\mathsf{D}[\theta,\infty) = \mathsf{D}[\mathsf{P}, \infty)$ and
$\mathsf{D}(-\infty,\theta] = \mathsf{D}(-\infty,\mathsf{P}]$.
Similarly, if $\mathsf{P}=\emptyset$ we use
the abbreviations $\mathsf{D}(\theta,\infty)$ and $\mathsf{D}(-\infty,\theta)$.
For any open, closed or half-closed interval $I\subset\mathbb{R}$ we define
the full subcategories $\mathsf{D}I$ precisely in the same way. Thus, an
object $0\ne X\in\Dbcoh(\boldsymbol{E})$ is in $\mathsf{D}I$ if and only if
$\varphi_{-}(X)\in I$ and $\varphi_{+}(X)\in I$.
\begin{proposition}\label{prop:texpl}
Let $\theta\in\mathbb{R}$ and $\mathsf{P}(\theta)^{-} \subset
\mathsf{P}(\theta)^{s}$ be arbitrary. Denote by $\mathsf{P}(\theta)^{+} =
\mathsf{P}(\theta)^{s} \setminus \mathsf{P}(\theta)^{-}$ the complement of
$\mathsf{P}(\theta)^{-}$. Then,
$$\mathsf{D}^{\le0} := \mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)$$
defines a $t$-structure on $\Dbcoh(\boldsymbol{E})$ with
$$\mathsf{D}^{\ge1} := \mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}].$$
The heart $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ of it is
the category $\mathsf{D}[\mathsf{P}(\theta)^{-},
\mathsf{P}(\theta)^{+}[1]]$, which consists of those objects
$X\in\Dbcoh(\boldsymbol{E})$ whose HN-factors either have
phase $\varphi\in(\theta,\theta+1)$ or have all its JH-factors in
$\mathsf{P}(\theta)^{-}$ or $\mathsf{P}(\theta)^{+}[1]$.
\end{proposition}
\begin{proof}
The only non-trivial property which deserves a proof is (\ref{def:tiii}) in
the definition of $t$-structure. Given $X\in \Dbcoh(\boldsymbol{E})$, we
have to show that there exists a distinguished triangle $A\rightarrow X
\rightarrow B \stackrel{+}{\rightarrow}$ with $A\in \mathsf{D}^{\le0}$ and
$B\in \mathsf{D}^{\ge1}$. In order to construct it, let
\[\xymatrix@C=.5em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
be the HNF of $X$. Because $\varphi(A_{i+1})>\varphi(A_{i})$ for all $i$,
there exists an integer $k$, $0\le k \le n+1$ such that $\varphi(A_{k})\ge
\theta >\varphi(A_{k-1})$. If $\varphi(A_{k})>\theta$, this implies
$A_{i}\in \mathsf{D}^{\le0}$, if $i\ge k$ and $A_{i}\in \mathsf{D}^{\ge1}$,
if $i<k$. In particular, $F_{k}X\in \mathsf{D}^{\le0}$. In this case, we
define $A:=F_{k}X$ and let $A=F_{k}X \rightarrow X$ be the composition of
the morphisms in the HNF.
If, however, $\varphi(A_{k})=\theta$, there is a splitting $A_{k}\cong
A_{k}^{-} \oplus A_{k}^{+}$ such that all JH-factors of $A_{k}^{-}$ (resp.\/
$A_{k}^{+}$) are in $\mathsf{P}(\theta)^{-}$ (resp.\/
$\mathsf{P}(\theta)^{+}$).
Now, we apply Lemma \ref{lem:connect} to the distinguished triangles
$F_{k+1}X \stackrel{f}{\longrightarrow} F_{k}X \longrightarrow A_{k}
\stackrel{+}{\longrightarrow}$
and
$A_{k}^{-} \longrightarrow A_{k} \longrightarrow A_{k}^{+}
\stackrel{+}{\longrightarrow}$, given by the splitting of $A_{k}$,
to obtain a factorisation $F_{k+1}X \rightarrow A \rightarrow F_{k}X$ of
$f$ and two distinguished triangles
$$\xymatrix@C=.5em{F_{k+1}X \ar[rr] && A \ar[dl]\ar[rr] && F_{k}X.\ar[dl]\\
& A_{k}^{-} \ar[ul]^{+} && A_{k}^{+}\ar[ul]^{+}}$$
Part of the given HNF of $X$ together with the left one of these two
triangles form a HNF of $A$, whence $A\in \mathsf{D}^{\le0}$.
Again, we let $A\rightarrow X$
be obtained by composition with the morphisms in the HNF of $X$.
In any case, we choose a distinguished triangle $A\rightarrow X \rightarrow B
\stackrel{+}{\rightarrow}$, where $A\rightarrow X$ is the morphism chosen
before. From Lemma \ref{lem:split} or Remark \ref{rem:split} we obtain $B\in
\mathsf{D}^{\ge1}$. This proves the proposition.
\end{proof}
We shall also need the following standard result.
\begin{lemma}\label{lem:tsummands}
Let $(\mathsf{D}^{\le 0}, \sf{D}^{\ge 0})$ be a $t$-structure on a
triangulated category. If $X \oplus Y \in \sf{D}^{\le 0}$ then
$X \in \sf{D}^{\le 0}$ and $Y \in \sf{D}^{\le 0}$.
The corresponding statement holds for $\sf{D}^{\ge 0}$.
\end{lemma}
\begin{proof}
Let $A \stackrel{f}{\longrightarrow} X \stackrel{g}{\longrightarrow} B
\stackrel{+}{\longrightarrow}$ be a distinguished triangle with
$A\in\mathsf{D}^{\le 0}$ and $B\in\mathsf{D}^{\ge 1}$, which exists due to
the definition of a $t$-structure. If $X\not\in \mathsf{D}^{\le 0}$, we
necessarily have $g\ne 0$ and $B \ne 0$.
Because $\Hom(\sf{D}^{\le 0}, \sf{D}^{\ge 1}) = 0$, the composition $X
\oplus Y \stackrel{p}{\longrightarrow} X \stackrel{g}{\longrightarrow} B$,
in which $p$ denotes the natural projection, must be zero. If
$i:X\rightarrow X\oplus Y$ denotes the canonical morphism, we
obtain $g=g\circ p\circ i=0$, a contradiction. In the same way it follows
that $Y\in\mathsf{D}^{\le 0}$.
\end{proof}
Recall that an Abelian category is called \emph{Noetherian}, if any sequence of
epimorphisms stabilises, this means that for any sequence of epimorphisms
$f_{k}:A_{k}\rightarrow A_{k+1}$ there exists an integer $k_{0}$ such that
$f_{k}$ is an isomorphism for all $k\ge k_{0}$.
\begin{lemma}\label{lem:heart}
The heart $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ of the $t$-structure,
which was described in Proposition \ref{prop:texpl}, is Noetherian if and
only if $\mathsf{P}(\theta) \ne \{0\}$ and $\mathsf{P}(\theta)^{-} =
\emptyset$. In this case, ${\sf A}(\theta, \emptyset) = \mathsf{D}(\theta,
\theta +1]$.
\end{lemma}
\begin{proof}
If $\mathsf{P}(\theta) = \{0\}$ then
$\mathsf{A}(\theta,\mathsf{P}(\theta)^{-}) =
\mathsf{D}(\theta,\theta+1)$. This category is not Noetherian.
To prove this, we follow the proof of Polishchuk in the smooth case
\cite{Pol1}, Proposition 3.1.
We are going to show for any non-zero locally free shifted sheaf $E \in
\mathsf{D}(\theta, \theta +1)$, the
existence of a locally free shifted sheaf $F$ and an epimorphism
$E\twoheadrightarrow F$ in $\mathsf{D}(\theta, \theta +1)$, which is not an
isomorphism. This will be sufficient to show that $\mathsf{D}(\theta, \theta
+1)$ is not Noetherian.
By applying an appropriate shift, we may assume $0<\theta<1$. Under this
assumption, for every stable coherent sheaf $G$ we have
\begin{align*}
G\in\mathsf{D}(\theta, \theta +1) &\iff \theta<\varphi(G)\le 1\\
G[1]\in\mathsf{D}(\theta, \theta +1) &\iff 0<\varphi(G)< \theta.
\end{align*}
For any two objects $X,Y\in\Dbcoh(\boldsymbol{E})$ we define the Euler form
to be
$$\langle X,Y \rangle = \rk(X)\deg(Y) - \deg(X)\rk(Y)$$
which is the imaginary part of $\overline{Z(X)}Z(Y)$. If $X$ and $Y$ are
coherent sheaves and one of them is perfect, we have
$$\langle X,Y \rangle = \chi(X,Y) := \dim\Hom(X,Y) - \dim \Ext^{1}(X,Y).$$
This remains true, if we apply arbitrary shifts to the sheaves $X,Y$, where
we understand $\chi(X,Y)=\sum_{\nu} (-1)^{\nu} \dim \Hom(X,Y[\nu]).$
Let $E\in\mathsf{D}(\theta, \theta +1)$ be an arbitrary non-zero locally
free shifted sheaf. We look at the strip in the plane between the lines
$L(0):= \mathbb{R}\exp(i\pi\theta)$ and $L(E):=L(0)+Z(E)$. This strip must
contain lattice points in its interior.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,6)
\put(1.5,2){\vector(1,0){9.5}}\put(11,1.9){\makebox(0,0)[t]{$-\deg$}}
\put(6,0){\vector(0,1){6}}\put(5.8,6){\makebox(0,0)[r]{$\rk$}}
\put(2,0){\line(2,1){9}}
\put(11,4.3){\makebox(0,0)[t]{$\theta$}}
\put(2.8,0){\makebox(0,0)[l]{$\theta+1$}}
\put(1,1.5){\line(2,1){9}}
\put(6,2){\vector(-1,1){1}}\put(4.8,2.9){\makebox(0,0)[t]{$F$}}
\put(6,2){\vector(2,3){2}}\put(8.2,4.9){\makebox(0,0)[t]{$E$}}
\put(5,3){\vector(3,2){3}}
\put(10.5,4.5){\makebox(0,0)[b]{$L(0)$}}
\put(10.5,6){\makebox(0,0)[t]{$L(E)$}}
\end{picture}
\end{center}
\caption{}\label{fig:strip}\end{figure}
Therefore, there exists a lattice point $Z_{F}$ in this strip which enjoys
the following properties:
\begin{enumerate}
\item\label{nopoint} the only lattice points on the closed triangle whose
vertices are $0, Z(E), Z_{F}$, are its vertices;
\item\label{phase} $\varphi_{F} > \varphi(E)$.
\end{enumerate}
By $\varphi_{F}$ we denote here the unique number which satisfies
$\theta <\varphi_{F} < \theta+1$ and $Z_{F}\in \mathbb{R}\exp(i\pi
\varphi_{F})$.
Because $\SL(2,\mathbb{Z})$ acts transitively on $\mathsf{Q}$, there exists
a stable non-zero locally free shifted sheaf $F\in\mathsf{D}(\theta, \theta
+1)$ with $Z(F)=Z_{F}$ and $\varphi(F)=\varphi_{F}$.
The assumption $\mathsf{P}(\theta)=\{0\}$ implies
$\mathbb{R}\exp(i\pi\theta)\cap\mathbb{Z}^{2} = \{0\}$, hence, $Z(E)$ is the
only lattice point on the line $L(E)$. This implies that $Z(F)$ is not on
the boundary of the stripe between $L(0)$ and $L(E)$. In particular,
$Z(E)-Z(F)$ is contained in the same half-plane of $L(0)$ as $Z(E)$ and
$Z(F)$, see Figure \ref{fig:strip}.
Condition (\ref{nopoint}) implies $\langle E,F \rangle = 1$. Because $E$ is
locally free, condition (\ref{phase}) implies
$$\Ext^{1}(E,F) = \Hom(F,E) = 0.$$
Hence, $\Hom(E,F)\cong \boldsymbol{k}$. The evaluation map gives, therefore,
a distinguished triangle
$$\Hom(E,F)\otimes E \rightarrow F \rightarrow T_{E}(F)
\stackrel{+}{\longrightarrow}$$
with $T_{E}(F)\in \Dbcoh(\boldsymbol{E})$.
If $C:=T_{E}(F)[-1]$ we obtain a distinguished triangle
\begin{equation}
\label{eq:mutation}
C\rightarrow E\rightarrow F \stackrel{+}{\longrightarrow}
\end{equation}
with $Z(C)=Z(E)-Z(F)$.
Because $E$ is a stable non-zero shifted locally free sheaf, it is spherical
by Proposition \ref{prop:spherical} and so $T_{E}$ is an equivalence. This
implies that $T_{E}(F)$ is spherical and, by Proposition
\ref{prop:spherical} again, $C$ is a stable non-zero shifted locally free
sheaf. All morphisms in the distinguished triangle (\ref{eq:mutation}) are
non-zero because $C, E, F$ are indecomposable, see Lemma \ref{lem:PengXiao}.
Using Lemma \ref{wesPT:ii}, this implies $\theta-1<\varphi(C)<\theta+1$.
However, we have seen in which half-plane $Z(C)$ is contained, so that we
must have $\theta<\varphi(C)<\theta+1$, which implies
$C\in\mathsf{D}(\theta,\theta+1)$. The distinguished triangle
(\ref{eq:mutation}) and the definition of the structure of Abelian category
on the heart $\mathsf{D}(\theta,\theta+1)$ imply now that the
morphism $E\rightarrow F$ in (\ref{eq:mutation}) is an epimorphism in
$\mathsf{D}(\theta,\theta+1)$. This gives an infinite chain of epimorphisms
which are not isomorphisms, so that the category
$\mathsf{D}(\theta,\theta+1)$ is indeed not Noetherian.
In order to show that $\mathsf{A}(\theta,\mathsf{P}(\theta)^{-})$ is not
Noetherian for $\mathsf{P}(\theta)^{-} \ne \emptyset$ we may assume $\theta
= 0$. If there exists a stable element $\boldsymbol{k}(x) \in
\mathsf{P}(0)^{-}[1]\subset \mathsf{P}(1)$, where $x\in\boldsymbol{E}$ is a
smooth point, we have exact sequences
\begin{equation}
\label{eq:sequence}
0 \rightarrow \mathcal{O}(mx)
\rightarrow \mathcal{O}((m+1)x)
\rightarrow \boldsymbol{k}(x)
\rightarrow 0
\end{equation}
in $\Coh_{\boldsymbol{E}}$ with arbitrary
$m\in\mathbb{Z}$.
Hence the cone of the morphism $\mathcal{O}(mx) \rightarrow
\mathcal{O}((m+1)x)$ is isomorphic to $\boldsymbol{k}(x)[0]$. Because
$\boldsymbol{k}(x)[0]$ is an object of $\mathsf{D}^{\le-1}$, with regard to
the $t$-structure which is defined by $\mathsf{P}({0})^{-}$, we obtain
$\tau_{\ge0}(\boldsymbol{k}(x)[0])=0$, which is the cokernel of
$\mathcal{O}(mx) \rightarrow \mathcal{O}((m+1)x)$
in the Abelian category $\mathsf{A}(0,\mathsf{P}(0)^{-})$, see
\cite{Asterisque100}, 1.3.
Hence, there is an exact sequence
$$0 \rightarrow \boldsymbol{k}(x)[-1] \rightarrow \mathcal{O}(mx)
\rightarrow \mathcal{O}((m+1)x) \rightarrow 0$$
in $\mathsf{A}(0,\mathsf{P}(0)^{-})$
and we obtain an infinite chain of epimorphisms
$$ \mathcal{O}(x) \rightarrow \mathcal{O}(2x) \rightarrow \mathcal{O}(3x)
\rightarrow \cdots$$ in the category $\mathsf{A}(0,\mathsf{P}(0)^{-})$,
which, therefore, is not Noetherian.
If $\mathsf{P}(0)^{-}[1]$ contains $\boldsymbol{k}(s)$ only, where
$s\in\boldsymbol{E}$ is the singular point, we proceed as follows. First,
recall that there exist coherent torsion modules with support at $s$
which have finite injective dimension, see for example
\cite{BurbanKreussler}, Section 4. To describe examples of them, we can
choose a line bundle $\mathcal{L}$ on $\boldsymbol{E}$ and a section
$\sigma\in H^{0}(\mathcal{L})$, such that the cokernel of
$\sigma:\mathcal{O}\rightarrow \mathcal{L}$ is a coherent torsion module
$\mathcal{B}$ of length two with support at $s$. If we embed
$\boldsymbol{E}$ into $\mathbb{P}^{2}$, such a line bundle $\mathcal{L}$ is
obtained as the tensor product of the restriction of
$\mathcal{O}_{\mathbb{P}^{2}}(1)$ with $\mathcal{O}_{\boldsymbol{E}}(-x)$,
where $x\in\boldsymbol{E}$ is a smooth point. The section $\sigma$
corresponds to the line in the plane through $x$ and $s$. By twisting with
$\mathcal{L}^{\otimes m}$ we obtain exact sequences
$$ 0 \rightarrow \mathcal{L}^{\otimes m} \rightarrow \mathcal{L}^{\otimes
(m+1)} \rightarrow \mathcal{B} \rightarrow 0$$ in $\Coh_{\boldsymbol{E}}$.
Because $\mathcal{B}$ is a semi-stable torsion sheaf with support at $s$,
all its JH-factors are isomorphic to $\boldsymbol{k}(s)$ and we conclude as
above.
\end{proof}
\begin{proposition}\label{prop:eitheror}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a $t$-structure on
$\Dbcoh(\boldsymbol{E})$ and $B$ a semi-stable indecomposable object
in $\Dbcoh(\boldsymbol{E})$. Then either $B\in \mathsf{D}^{\le0}$ or $B\in
\mathsf{D}^{\ge1}$.
\end{proposition}
\begin{proof}
Let $X\stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} Y
\stackrel{+}{\longrightarrow}$ be a distinguished triangle with $X\in
\mathsf{D}^{\le0}$ and $Y\in \mathsf{D}^{\ge1}$. Suppose $X\ne 0$ and $Y\ne
0$ in $\Dbcoh(\boldsymbol{E})$. We decompose both objects into
indecomposables $X=\bigoplus X_{i}$ and $Y=\bigoplus Y_{j}$. By Lemma
\ref{lem:tsummands} we have $X_{i}\in \mathsf{D}^{\le0}$ and $Y_{j}\in
\mathsf{D}^{\ge1}$. If one of the components of the morphisms
$Y[-1]\rightarrow X=\bigoplus X_{i}$ or $\bigoplus Y_{j}=Y\rightarrow X[1]$
were zero, by Lemma \ref{lem:PengXiao} we would obtain a direct summand
$X_{i}$ or $Y_{j}$ in $B$. Because $B$ was assumed to be indecomposable,
this implies the claim of the proposition.
For the rest of the proof we suppose that all components of these two
morphisms are non-zero. This implies that $X_{i}$ and $Y_{j}$ are
non-perfect for all $i,j$. Indeed, if $X_{i}$ were perfect, we could
apply Serre duality (\ref{wesPT:i}) to obtain $\Hom(Y,X_{i}[1]) =
\Hom(X_{i},Y)^{\ast}$, which is zero because $X_{i}\in \mathsf{D}^{\le0}$
and $Y\in \mathsf{D}^{\ge1}$. The case with perfect $Y_{j}$ can be dealt
with similarly.
Using Lemma \ref{lem:PengXiao} again, it follows that none of the components
of $f:\bigoplus X_{i} \rightarrow B$ or $g:B\rightarrow \bigoplus Y_{j}$ is
zero, because none of the $X_{i}$ could be a direct summand of $Y[-1]$ and
none of the $Y_{j}$ could be a summand of $X[1]$.
Using Lemma \ref{wesPT:ii}, this implies
$\varphi_{-}(X_{i}) \le \varphi(B) \le \varphi_{+}(Y_{j})$ for all $i,j$.
If there exist $i,j$ such that $\varphi_{-}(X_{i}) -
\varphi_{+}(Y_{j})\not\in \mathbb{Z}$, there exists an integer $k\ge 0$ such
that $\varphi_{-}(X_{i}[k]) < \varphi_{+}(Y_{j}) < \varphi_{-}(X_{i}[k])
+1$. Using Proposition \ref{wesPT} (\ref{wesPT:iii}) this implies
$\Hom(X_{i}[k], Y_{j}) \ne 0$. But, for any
integer $k\ge 0$ we have $X_{i}[k]\in \mathsf{D}^{\le0}$ and because
$Y_{j}\in \mathsf{D}^{\ge1}$, we should have $\Hom(X_{i}[k], Y_{j}) =
0$. This contradiction implies $\varphi_{-}(X_{i}) - \varphi_{+}(Y_{j}) \in
\mathbb{Z}$ for all $i,j$. But, if $k=\varphi_{+}(Y_{j}) -
\varphi_{-}(X_{i})$, we still have $\Hom(X_{i}[k], Y_{j}) \ne 0$, which
follows from Proposition \ref{wesPT} (\ref{wesPT:iv}) because $X_{i}$ and
$Y_{j}$ are not perfect. The conclusion is now that we must have $X=0$ or
$Y=0$, which implies the claim.
\end{proof}
\begin{lemma}\label{lem:inequ}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a $t$-structure on
$\Dbcoh(\boldsymbol{E})$. If $F\in \mathsf{D}^{\le0}$ and $G\in
\mathsf{D}^{\ge1}$, then $\varphi_{-}(F)\ge \varphi_{+}(G)$.
\end{lemma}
\begin{proof}
Suppose $\varphi_{-}(F)< \varphi_{+}(G)$. It is sufficient to derive a
contradiction for indecomposable objects $F$ and $G$.
Because, for any $k\ge0$, $F[k]\in \mathsf{D}^{\le0}$, we may replace $F$
by $F[k]$ and can assume $0< \varphi_{+}(G) - \varphi_{-}(F)\le 1$. Now,
there exists a stable vector bundle $\mathcal{B}$ on $\boldsymbol{E}$ and an
integer $r$ such that
$$\varphi_{-}(F) < \varphi(\mathcal{B}[r]) < \varphi_{+}(G) \le
\varphi_{-}(F) + 1.$$
By Proposition \ref{prop:eitheror},
$\mathcal{B}[r]$ is in $\mathsf{D}^{\le0}$ or in
$\mathsf{D}^{\ge1}$. But, from Proposition \ref{wesPT} (\ref{wesPT:iii}) we
deduce $\Hom(F, \mathcal{B}[r])\ne 0$ and $\Hom(\mathcal{B}[r],G)\ne 0$. If
$\mathcal{B}[r]\in \mathsf{D}^{\ge1}$, the first inequality contradicts $F\in
\mathsf{D}^{\le0}$ and if $\mathcal{B}[r]\in \mathsf{D}^{\le0}$, the second
one contradicts $G\in \mathsf{D}^{\ge1}$.
\end{proof}
\begin{theorem}\label{thm:tstruc}
Let $(\mathsf{D}^{\le0}, \mathsf{D}^{\ge0})$ be a t-structure on
$\Dbcoh(\boldsymbol{E})$. Then there exists a number $\theta\in \mathbb{R}$
and a subset
$\mathsf{P}(\theta)^{-}\subset \mathsf{P}(\theta)^{s}$, such that
$${\sf D}^{\le 0} = \mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)
\quad\text{ and }\quad
{\sf D}^{\ge 1} = \mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}].$$
\end{theorem}
\begin{proof}
From Lemma \ref{lem:inequ} we deduce the existence of $\theta \in
\mathbb{R}$ such that $\mathsf{D}(\theta,\infty)\subset\mathsf{D}^{\le 0}$
and $\mathsf{D}(-\infty, \theta)\subset\mathsf{D}^{\ge 1}.$
If we define
$\mathsf{P}(\theta)^{-}=\mathsf{P}(\theta)^{s}\cap \mathsf{D}^{\le 0}$
and
$\mathsf{P}(\theta)^{+}=\mathsf{P}(\theta)^{s}\cap \mathsf{D}^{\ge1}$,
Proposition \ref{prop:eitheror} implies
$\mathsf{P}(\theta)^{s} = \mathsf{P}(\theta)^{-}\cup\mathsf{P}(\theta)^{+}$.
Hence, $\mathsf{D}[\mathsf{P}(\theta)^{-}, \infty)\subset \mathsf{D}^{\le0}$
and $\mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}]\subset \mathsf{D}^{\ge 1}$.
From Proposition \ref{prop:texpl} we know that
$(\mathsf{D}[\mathsf{P}(\theta)^{-}, \infty),
\mathsf{D}(-\infty,\mathsf{P}(\theta)^{+}[1]])$ defines a
$t$-structure. Now, the statement of the theorem follows.
\end{proof}
\begin{remark}
In the case of a smooth elliptic curve Theorem \ref{thm:tstruc} was proved in
\cite{GRK}. If $\theta\not\in\mathsf{Q}$ the heart
$\mathsf{D}(\theta,\theta+1)$ of the corresponding $t$-structure is a
finite-dimensional non-Noetherian Abelian category of infinite global
dimension. In the smooth case, they correspond to the category of holomorphic
vector bundles on a non-commutative torus in the sense of Polishchuk and
Schwarz \cite{PolSchw}. It is an interesting problem to find a similar
interpretation of these Abelian categories in the case of a singular
Weierstra{\ss} curve $\boldsymbol{E}$.
\end{remark}
To complete this section we give two applications of Theorem
\ref{thm:tstruc}. The first is a description of the group of exact
auto-equivalences of the triangulated category $\Dbcoh(\boldsymbol{E})$. The
second application is a description of Bridgeland's space of all stability
structures on $\Dbcoh(\boldsymbol{E})$. In both cases, $\boldsymbol{E}$ is an
irreducible curve of arithmetic genus one over $\boldsymbol{k}$.
\begin{corollary}\label{cor:auto}
There exists an exact sequence of groups
$$
\boldsymbol{1} \longrightarrow \Aut^0(\Dbcoh(\boldsymbol{E}))
\longrightarrow \Aut(\Dbcoh(\boldsymbol{E}))
\longrightarrow \SL(2,\mathbb{Z}) \longrightarrow \boldsymbol{1}
$$
in which $\Aut^0(\Dbcoh(\boldsymbol{E}))$ is generated by tensor products
with line bundles of degree zero, automorphisms of the curve and the shift
by $2$.
\end{corollary}
\begin{proof}
The homomorphism $\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z})$
is defined by describing the action of an auto-equivalence on
$\mathsf{K}(\boldsymbol{E})$ in terms of the coordinate functions $(\deg,
\rk)$.
That this is indeed in $\SL(2,\mathbb{Z})$ follows, for example, because
$\Aut(\Dbcoh(\boldsymbol{E}))$ preserves stability and the Euler-form
\begin{align*}
\langle \mathcal{F},\mathcal{G}\rangle &=
\dim\Hom(\mathcal{F},\mathcal{G}) -
\dim\Hom(\mathcal{G},\mathcal{F})\\
&= \rk(\mathcal{F}) \deg(\mathcal{G}) -
\deg(\mathcal{F})\rk(\mathcal{G})
\end{align*}
for stable and perfect sheaves $\mathcal{F},\mathcal{G}$.
Clearly, tensor products with line bundles of degree
zero, automorphisms of the curve and the shift by $2$ are contained in the
kernel of this homomorphism. In order to show that the kernel coincides with
$\Aut^0(\Dbcoh(\boldsymbol{E}))$, we let $\mathbb{G}$ be an arbitrary exact
auto-equivalence of $\Dbcoh(\boldsymbol{E})$.
Then, $\mathbb{G}(\Coh_{\boldsymbol{E}})$ is still Noetherian and it is the
heart
of the $t$-structure $(\mathbb{G}(\mathsf{D}^{\le0}),
\mathbb{G}(\mathsf{D}^{\ge0}))$.
From Theorem \ref{thm:tstruc} and Lemma \ref{lem:heart} we know all
Noetherian hearts of $t$-structures. We obtain
$\mathbb{G}(\Coh_{\boldsymbol{E}}) = \mathsf{D}(\theta,\theta+1]$ with
$\mathsf{P}(\theta)\ne \{0\}$.
Now, by Corollary \ref{cor:sheaves} there exists
$\Phi\in\widetilde{\SL}(2,\mathbb{Z})$ which maps
$\mathsf{D}(\theta,\theta+1]$ to $\mathsf{D}(0, 1]=\Coh_{\boldsymbol{E}}$.
This implies that the auto-equivalence $\Phi\circ\mathbb{G}$ induces an
auto-equivalence of the category $\Coh_{\boldsymbol{E}}$.
It is well-known that such an auto-equivalence has the form $f^*(\mathcal{L}
\otimes \,\cdot\,)$, where $f:\boldsymbol{E} \rightarrow \boldsymbol{E}$ is an
isomorphism and $\mathcal{L}$ is a line bundle.
Note that $f^*(\mathcal{L} \otimes \,\cdot\,)$ is sent to the identity in
$\SL(2,\mathbb{Z})$, if and only if $\mathcal{L}$ is of degree zero.
The composition of $\Phi\circ\mathbb{G}$ with the inverse of $f^*(\mathcal{L}
\otimes \,\cdot\,)$ satisfies the assumptions of \cite{BondalOrlov}, Prop.~A.3,
hence is isomorphic to the identity.
Because the kernel of the homomorphism $\widetilde{\SL}(2,\mathbb{Z})
\rightarrow \SL(2,\mathbb{Z})$, which is induced by the action of
$\widetilde{\SL}(2,\mathbb{Z})$ on $\Dbcoh(\boldsymbol{E})$ and the above
homomorphism $\Aut(\Dbcoh(\boldsymbol{E})) \longrightarrow
\SL(2,\mathbb{Z})$, is generated by the element of
$\widetilde{\SL}(2,\mathbb{Z})$ which acts as the shift by $2$, the claim
now follows.
\end{proof}
For our second application, we recall Bridgeland's definition of stability
condition on a triangulated category \cite{Stability}.
Recall that we set $\mathsf{K}(\boldsymbol{E}) =
\mathsf{K}_{0}(\Coh(\boldsymbol{E})) \cong
\mathsf{K}_{0}(\Dbcoh(\boldsymbol{E}))$.
Following Bridgeland \cite{Stability}, we call a pair $(W,\mathsf{R})$ a
\emph{stability condition} on $\Dbcoh(\boldsymbol{E})$, if
$$W:\mathsf{K}(\boldsymbol{E})\rightarrow\mathbb{C}$$
is a group homomorphism and $\mathsf{R}$ is a compatible slicing of
$\Dbcoh(\boldsymbol{E})$. A \emph{slicing} $\mathsf{R}$ consists of a
collection of full additive subcategories $\mathsf{R}(t) \subset
\Dbcoh(\boldsymbol{E})$, $t\in\mathbb{R}$, such that
\begin{enumerate}
\item $\forall t\in\mathbb{R}\quad \mathsf{R}(t+1) = \mathsf{R}(t)[1]$;
\item If $t_{1}>t_{2}$ and $A_{\nu}\in\mathsf{R}(t_{\nu})$, then
$\Hom(A_{1},A_{2}) =0$;
\item each non-zero object $X\in\Dbcoh(\boldsymbol{E})$ has a HNF
\[\xymatrix@C=.4em{
0\; \ar[rr] && F_{n}X \ar[rr] \ar[dl]_{\cong}&& F_{n-1}X \ar[rr]
\ar[dl]&& \dots \ar[rr] &&F_{1}X \ar[rr] && F_{0}X \ar@{=}[r]\ar[dl] &
X\\
& A_{n} \ar[lu]^{+} && A_{n-1} \ar[lu]^{+} & & & & & & A_0 \ar[lu]^{+}&
}\]
in which $0\ne A_{\nu}\in\mathsf{R}(\varphi_{\nu})$ and
$\varphi_{n}>\varphi_{n-1}> \ldots > \varphi_{1}>\varphi_{0}$.
\end{enumerate}
Compatibility means for all non-zero $A\in\mathsf{R}(t)$
$$W(A)\in \mathbb{R}_{>0}\exp(i\pi t).$$
By $\varphi^{\mathsf{R}}$ we denote the phase function on
$\mathsf{R}$-semi-stable objects. Similarly, we denote by
$\varphi^{\mathsf{R}}_{+}(X)$ and
$\varphi^{\mathsf{R}}_{-}(X)$ the largest, respectively smallest, phase of an
$\mathsf{R}$-HN factor of $X$.
The standard stability condition, which was studied in the previous section,
will always be denoted by $(Z, \mathsf{P})$. This stability condition has a
slicing which is \emph{locally finite}, see \cite{Stability}, Def.\/ 5.7.
A slicing $\mathsf{R}$ is called locally finite, iff there exists $\eta>0$
such that for any $t\in\mathbb{R}$ the quasi-Abelian category
$\mathsf{D}^{\mathsf{R}}(t-\eta, t+\eta)$ is of finite length, i.e. Artinian
and Noetherian.
This category consists of those objects $X\in\Dbcoh(\boldsymbol{E})$ which
satisfy $t-\eta<\varphi^{\mathsf{R}}_{-}(X) \le \varphi^{\mathsf{R}}_{+}(X) <
t+\eta$.
In order to obtain a good moduli space of stability conditions, Bridgeland
\cite{Stability} requires the stability conditions to be
\emph{numerical}. This means that the central charge $W$ factors
through the numerical Grothendieck group. This makes sense if for any two
objects $E,F$ of the triangulated category in question, the vector spaces
$\bigoplus_{i} \Hom(E,F[i])$ are finite-dimensional. This condition is not
satisfied for $\Dbcoh(\boldsymbol{E})$, if $\boldsymbol{E}$ is
singular. However, in the case of our interest, we do not need such an extra
condition, because the Grothendieck group $\mathsf{K}(\boldsymbol{E})$ is
sufficiently small. From Lemma \ref{lem:GrothGrp} we know
$\mathsf{K}(\boldsymbol{E}) \cong \mathbb{Z}^{2}$ with generators
$[\mathcal{O}_{\boldsymbol{E}}]$ and $[\boldsymbol{k}(x)]$,
$x\in\boldsymbol{E}$ arbitrary.
Because $Z(\boldsymbol{k}(x))=-1$ and $Z(\mathcal{O}_{\boldsymbol{E}})=i$, it
is now clear that any homomorphism $W:\mathsf{K}(\boldsymbol{E}) \rightarrow
\mathbb{C}$ can be written as $W(E)=w_{1}\deg(E) + w_{2}\rk(E)$ with $w_{1},
w_{2}\in\mathbb{C}$. Equivalently, if we identify $\mathbb{C}$ with
$\mathbb{R}^{2}$, there exists a $2\times 2$-matrix $A$ such that $W=A\circ Z$.
\begin{definition}
By $\Stab{\boldsymbol{E}}$ we denote the set of all stability conditions $(W,
\mathsf{R})$ on $\Dbcoh(\boldsymbol{E})$ for which $\mathsf{R}$ is a locally
finite slicing.
\end{definition}
\begin{lemma}\label{lem:notaline}
For any $(W, \mathsf{R}) \in \Stab(\boldsymbol{E})$ there exists a matrix
$A\in\GL(2,\mathbb{R})$, such that $W=A\circ Z$.
\end{lemma}
\begin{proof}
As seen above, there exists a not necessarily invertible matrix $A$ such
that $W=A\circ Z$. If $A$ were not invertible, there would exist a number
$t_{0}\in\mathbb{R}$ such that $W(\mathsf{K}(\boldsymbol{E})) \subset
\mathbb{R}\exp(i\pi t_{0})$. This implies that there may exist a non-zero
object in $\mathsf{R}(t)$ only if $t-t_{0}\in\mathbb{Z}$. The assumption
that the slicing $\mathsf{R}$ is locally finite implies now that
$\mathsf{R}(t)$ is of finite length for any $t\in\mathbb{R}$. On the other
hand, the heart of the $t$-structure, which is defined by $(W,\mathsf{R})$
is $\mathsf{R}(t_{0})$ up to a shift. However, in Lemma \ref{lem:heart} we
determined all Noetherian hearts of $t$-structures on
$\Dbcoh(\boldsymbol{E})$ and none of them is Artinian. This contradiction
shows that $A$ is invertible.
\end{proof}
\begin{lemma}\label{lem:function}
If $(W,\mathsf{R}) \in \Stab(\boldsymbol{E})$, there exists a unique
strictly increasing function $f:\mathbb{R} \rightarrow \mathbb{R}$ with
$f(t+1) = f(t)+1$ and $\mathsf{R}(t) = \mathsf{P}(f(t))$.
\end{lemma}
\begin{proof}
By definition, $W(\mathsf{R}(t)) \subset \mathbb{R}_{>0} \exp(i\pi
t)$. By Lemma \ref{lem:notaline}, there exists a linear isomorphism $A$ such
that $W=A^{-1}\circ Z$. This implies that there is a function
$f:\mathbb{R}\rightarrow \mathbb{R}$ such that $Z(\mathsf{R}(t))
\subset \mathbb{R}_{>0} \exp(i\pi f(t))$.
On the other hand, $\mathsf{R}(t)$ is the intersection of two hearts of
$t$-structures. By Proposition \ref{prop:texpl} these hearts are of the form
$\mathsf{D}[\mathsf{P}(\theta_{1})^{-}, \mathsf{P}(\theta_{1})^{+}[1]]$ and
$\mathsf{D}[\mathsf{P}(\theta_{2})^{-}, \mathsf{P}(\theta_{2})^{+}[1]]$ with
$\theta_{1}\le \theta_{2}$. These have non-empty intersection only if
$\theta_{2} \le \theta_{1}+1$. Their intersection is contained in
$\mathsf{D}[\theta_{2},\theta_{1}+1]$, see Figure \ref{fig:intersection}.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{10mm}
\begin{picture}(11,5)
\multiput(0,4)(0.2,0){56}{\line(1,0){0.1}}
\put(0,1){\line(1,0){11.1}}
\thicklines
\put(1.5,2){\line(0,1){2}}\put(1.5,0.8){\makebox(0,0)[t]{$\theta_{2}+1$}}
\put(1.4,3){\makebox(0,0)[r]{$\mathsf{P}(\theta_{2})^{+}[1]$}}
\put(5.5,1){\line(0,1){1}}\put(5.5,0.8){\makebox(0,0)[t]{$\theta_{2}$}}
\put(5.6,1.4){\makebox(0,0)[l]{$\mathsf{P}(\theta_{2})^{-}$}}
\put(4.5,2.3){\line(0,1){1}}\put(4.5,0.8){\makebox(0,0)[t]{$\theta_{1}+1$}}
\put(4.4,3){\makebox(0,0)[r]{$\mathsf{P}(\theta_{1})^{+}[1]$}}
\put(8.5,1){\line(0,1){1.3}}\put(8.5,0.8){\makebox(0,0)[t]{$\theta_{1}$}}
\put(8.5,3.3){\line(0,1){0.7}}
\put(8.6,1.5){\makebox(0,0)[l]{$\mathsf{P}(\theta_{1})^{-}$}}
\thinlines
\multiput(1.5,1)(0,0.2){5}{\line(0,1){0.1}}
\multiput(5.5,2)(0,0.2){10}{\line(0,1){0.1}}
\multiput(4.5,1)(0,0.2){7}{\line(0,1){0.1}}
\multiput(4.5,3.3)(0,0.2){4}{\line(0,1){0.1}}
\multiput(8.5,2.3)(0,0.2){5}{\line(0,1){0.1}}
\put(1.9,4){\line(-2,-3){0.4}}
\put(2.7,4){\line(-2,-3){1.2}}
\multiput(1.5,1)(0.8,0){3}{\line(2,3){2}}
\put(3.9,1){\line(2,3){1.6}}
\put(4.7,1){\line(2,3){0.8}}
\put(8.1,4){\line(2,-3){0.4}}
\put(7.3,4){\line(2,-3){1.2}}
\multiput(8.5,1)(-0.8,0){3}{\line(-2,3){2}}
\put(6.1,1){\line(-2,3){1.6}}
\put(5.3,1){\line(-2,3){0.8}}
\end{picture}
\end{center}
\caption{}\label{fig:intersection}
\end{figure}
Moreover, if $\theta_{2}<\theta_{1}+1$, there exist $\alpha,
\beta\in\mathsf{Q}$ with $\theta_{2}< \alpha < \beta < \theta_{1}+1\le
\theta_{2}+1$. In this case we have two non-trivial subcategories
$\mathsf{P}(\alpha)\subset \mathsf{R}(t)$ and $\mathsf{P}(\beta)\subset
\mathsf{R}(t)$. However, because $0<\beta-\alpha<1$ and
$Z(\mathsf{R}(t)) \subset \mathbb{R}_{>0} \exp(i\pi f(t))$, we cannot have
$Z(\mathsf{P}(\alpha)) \subset \mathbb{R}_{>0} \exp(i\pi\alpha)$ and
$Z(\mathsf{P}(\beta)) \subset \mathbb{R}_{>0} \exp(i\pi\beta)$.
Hence, $\theta_{2}=\theta_{1}+1=f(t)$ and we obtain $\mathsf{R}(t) \subset
\mathsf{P}(f(t))$.
From $\mathsf{R}(t+m)=\mathsf{R}(t)[m]$ we easily obtain
$f(t+m)=f(t)+m$. Moreover, $f(t_{2})=f(t_{1})+m$ with $m\in\mathbb{Z}$
implies $t_{2}-t_{1}\in\mathbb{Z}$, because the image of $W$ is not
contained in a line by Lemma \ref{lem:notaline}.
Next, we show that $f$ is strictly increasing. Suppose $t_{1}<t_{2}$,
$t_{2}-t_{1}\not\in\mathbb{Z}$ and both $\mathsf{R}(t_{i})$ contain non-zero
objects $X_{i}$. For any $m\ge0$ we have $\Hom(X_{2}, X_{1}[-m]) = 0$. If
$f(t_{2}) < f(t_{1})$, we choose $m\ge0$ such that $f(t_{2}) < f(t_{1}) -m <
f(t_{2}) +1$ and obtain $X_{2}\in\mathsf{P}(f(t_{2}))$ and $X_{1}[-m] \in
\mathsf{P}(f(t_{1})-m)$. But this implies, by Corollary \ref{cor:sheaves}
(\ref{cor:iii}), $\Hom(X_{2}, X_{1}[-m]) \ne 0$, a contradiction. Hence, we
have shown that $f$ is strictly increasing with $f(t+1)=f(t)+1$ and
$\mathsf{R}(t)\subset \mathsf{P}(f(t))$. In particular, any $\mathsf{R}$-HNF
is a $\mathsf{P}$-HNF as well. Therefore, all $\mathsf{P}$-semi-stable
objects are $\mathsf{R}$-semi-stable and we obtain $\mathsf{R}(t) =
\mathsf{P}(f(t))$.
\end{proof}
It was shown in \cite{Stability} that the group
$\widetilde{\GL}^{+}(2,\mathbb{R})$ acts naturally on the moduli space of
stability conditions $\Stab(\boldsymbol{E})$.
This group is the universal cover of $\GL^{+}(2, \mathbb{R})$ and has the
following description:
$$\widetilde{\GL}^{+}(2,\mathbb{R}) =
\{(A,f) \mid A\in\GL^{+}(2,\mathbb{R}), f:\mathbb{R}\rightarrow \mathbb{R}
\text{ compatible}\},$$
where compatibility means that $f$ is strictly increasing, satisfies
$f(t+1)=f(t)+1$ and induces the same map on $S^{1}\cong\mathbb{R}/2\mathbb{Z}$
as $A$ does on $S^{1}\cong\mathbb{R}^{2}\setminus\{0\}/\mathbb{R}^{\ast}$.
The action is simply $(A,f)\cdot (W,\mathsf{Q})=(A^{-1}\circ W,\mathsf{Q}\circ
f)$. So, this action basically is a relabelling of the slices.
The following result generalises \cite{Stability}, Thm.\/ 9.1, to the singular
case.
\begin{proposition}\label{prop:stabmod}
The action of $\widetilde{\GL}^{+}(2,\mathbb{R})$ on $\Stab(\boldsymbol{E})$
is simply transitive.
\end{proposition}
\begin{proof}
If $(W,\mathsf{R})\in\Stab(\boldsymbol{E})$, the two values
$W(\mathcal{O}_{\boldsymbol{E}})$ and $W(\boldsymbol{k}(p_{0}))$ determine a
linear transformation $A^{-1}\in\GL(2, \mathbb{R})$ such that $W=A^{-1}\circ
Z$, see Lemma \ref{lem:notaline}.
By construction, the function $f:\mathbb{R}\rightarrow \mathbb{R}$ of Lemma
\ref{lem:function} induces the same mapping on
$S^{1}\cong\mathbb{R}/2\mathbb{Z}$ as $A^{-1}$ does on
$S^{1}\cong\mathbb{R}^{2}\setminus\{0\}/\mathbb{R}^{\ast}$. Therefore,
$A\in\GL^{+}(2, \mathbb{R})$ and we obtain $(A,f)\in\widetilde{\GL}^{+}(2,
\mathbb{R})$ which satisfies $(W,\mathsf{R}) = (A,f)\cdot
(Z,\mathsf{P})$. Finally, if $(A,f)\cdot (Z,\mathsf{P}) =(Z,\mathsf{P})$ for
some $(A,f)\in\widetilde{\GL}^{+}(2, \mathbb{R})$, we obtain $f(t)=t$ for
all $t\in\mathbb{R}$. This implies easily $A=\boldsymbol{1}$.
\end{proof}
The group $\Aut(\Dbcoh(\boldsymbol{E}))$ acts on $\Stab(\boldsymbol{E})$ by
the rule $$\mathbb{G} \cdot (W, \mathsf{R}) := (\overline{\mathbb{G}}\circ W,
\mathbb{G}(\mathsf{R})).$$
Here, $\overline{\mathbb{G}}\in\SL(2,\mathbb{Z})$ is the image of
$\mathbb{G}\in\Aut(\Dbcoh(\boldsymbol{E}))$ under the homomorphism of Corollary
\ref{cor:auto} and $\mathbb{G}(\mathsf{R})(t):= \mathbb{G}(\mathsf{R}(t))$.
Because automorphisms of $\boldsymbol{E}$ and twists by line
bundles act trivially on $\Stab(\boldsymbol{E})$, we obtain
$$\Stab(\boldsymbol{E})/\Aut(\Dbcoh(\boldsymbol{E})) \cong \GL^{+}(2,
\mathbb{R})/\SL(2,\mathbb{Z}),$$
which is a $\mathbb{C}^{\times}$-bundle over the coarse moduli space of
elliptic curves. This result coincides with Bridgeland's result in the smooth
case. The main reason for this coincidence seems to be the irreducibility of
the curve. Example \ref{ex:marginalst} below shows that the situation is
significantly more difficult in the case of reducible degenerations of
elliptic curves.
\begin{remark}\label{rem:common}
Our results show that singular and smooth Weierstra{\ss} curves
$\boldsymbol{E}$ share the following properties:
\begin{enumerate}
\item A coherent sheaf $\mathcal{F}$ is stable if and only if
$\End(\mathcal{F}) \cong \boldsymbol{k}$.
\item Any spherical object is a shift of a stable vector bundle or of a
structure sheaf $\boldsymbol{k}(x)$ of a smooth point $x\in\boldsymbol{E}$.
\item The category of semi-stable sheaves of a fixed slope is
equivalent to the category of coherent torsion sheaves.
Such an equivalence is induced by an auto-equivalence of
$\Dbcoh(\boldsymbol{E})$.
\item There is an exact sequence of groups\\
$\boldsymbol{1} \rightarrow \langle \Aut(\boldsymbol{E}),
\Pic^{0}(\boldsymbol{E}),[2]\rangle \rightarrow
\Aut(\Dbcoh(\boldsymbol{E})) \rightarrow \SL(2,\mathbb{Z}) \rightarrow
\boldsymbol{1}.$
\item $\widetilde{\GL}^{+}(2,\mathbb{R})$ acts transitively on
$\Stab(\boldsymbol{E})$.
\item $\Stab(\boldsymbol{E})/\Aut(\Dbcoh(\boldsymbol{E})) \cong \GL^{+}(2,
\mathbb{R})/\SL(2,\mathbb{Z}).$
\end{enumerate}
\end{remark}
\begin{example}\label{ex:marginalst}
Let $C_{2}$ denote a reducible curve which has two components, both
isomorphic to $\mathbb{P}^{1}$ and which intersect transversally at two
distinct points. This curve has arithmetic genus one and appears as a
degeneration of a smooth elliptic curve.
On this curve, there exists a line bundle $\mathcal{L}$ which
fails to be stable with respect to some stability conditions. To construct
an explicit example, denote by $\pi:\widetilde{C}_{2}\rightarrow C_{2}$ the
normalisation, so that $\widetilde{C}_{2}$ is the disjoint union of two
copies of $\mathbb{P}^{1}$. There is a $\boldsymbol{k}^{\times}$-family of
line bundles whose pull-back to $\widetilde{C}_{2}$ is
$\mathcal{O}_{\mathbb{P}^{1}}$ on one component and
$\mathcal{O}_{\mathbb{P}^{1}}(2)$ on the other. The element in
$\boldsymbol{k}^{\times}$
corresponds to a gluing parameter over one of the two singularities. Let
$\mathcal{L}$ denote one such line bundle.
If $i_{\nu}:\mathbb{P}^{1}\rightarrow C_{2},\;\nu=1,2$ denote the embeddings
of the two components, we fix notation so that $i_{1}^{\ast}\mathcal{L}
\cong \mathcal{O}_{\mathbb{P}^{1}}$ and $i_{2}^{\ast}\mathcal{L} \cong
\mathcal{O}_{\mathbb{P}^{1}}(2)$. There is an exact sequence of coherent
sheaves on $C_{2}$
\begin{equation}\label{eq:linebundle}
0\rightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow \mathcal{L}
\rightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}} \rightarrow 0.
\end{equation}
Moreover, the only non-trivial quotients of $\mathcal{L}$ are
$\mathcal{L}\twoheadrightarrow i_{1\ast} \mathcal{O}_{\mathbb{P}^{1}}$
and
$\mathcal{L}\twoheadrightarrow i_{2\ast} \mathcal{O}_{\mathbb{P}^{1}}(2)$.
For arbitrary positive real numbers $a,b$ we define a centred
slope-function $W_{a,b}$ on the category $\Coh_{C_{2}}$ by
$$W_{a,b}(F):= -\deg(F) + i(a\cdot \rk(i_{1}^{\ast}F) + b\cdot
\rk(i_{2}^{\ast} F)),$$
where $\deg(F)=h^{0}(F) - h^{1}(F)$. For example,
\begin{align*}
W_{a,b}(i_{1\ast}\mathcal{O}_{\mathbb{P}^{1}}(d)) &= -d-1+ia
\quad\text{ and }\\
W_{a,b}(i_{2\ast}\mathcal{O}_{\mathbb{P}^{1}}(d)) &= -d-1+ib.
\end{align*}
Using the exact sequence (\ref{eq:linebundle}), we obtain
$W_{a,b}(\mathcal{L}) = -2+i(a+b)$.
Using results of \cite{Rudakov}, it is easy to see that $W_{a,b}$ has the
Harder-Narasimhan property in the sense of \cite{Stability}. Hence, by
\cite{Stability}, Prop.\/ 5.3, $W_{a,b}$ defines a stability condition on
$\Dbcoh(C_{2})$. With respect to this stability condition, the line bundle
$\mathcal{L}$ is stable precisely when $2/(a+b) < 1/a$, which is
equivalent to $a<b$. It is semi-stable, but not stable, if $b=a$. If $a>b$,
$\mathcal{L}$ is not even semi-stable.
\end{example}
This example illustrates an interesting effect, which was not available on an
irreducible curve of arithmetic genus one. It is an interesting problem to
describe the subset in $\Stab(\boldsymbol{E})$ for which a given line
bundle $\mathcal{L}$ is semi-stable. This is a closed subset, see
\cite{Stability}. Physicists call the boundary of this set the line of
marginal stability, see e.g. \cite{AspinwallDouglas}. The example above
describes the intersection of this set with a two-parameter family of
stability conditions in $\Stab(\boldsymbol{E})$.
\begin{remark}
In the case of an irreducible curve of arithmetic genus one, we have shown
in Proposition \ref{prop:spherical} that $\Aut(\Dbcoh(\boldsymbol{E}))$ acts
transitively on the set of all spherical objects on
$\boldsymbol{E}$. Polishchuk \cite{YangBaxter} conjectured that this is
likewise true in the case of reducible curves with trivial dualising sheaf.
However, on the curve $C_{2}$ there exists a
spherical complex which has non-zero cohomology in two different degrees, see
\cite{BuBu}. This indicates that the reducible case is more difficult and
involves new features.
\end{remark}
|
{
"timestamp": "2006-02-14T17:10:40",
"yymm": "0503",
"arxiv_id": "math/0503496",
"language": "en",
"url": "https://arxiv.org/abs/math/0503496"
}
|
"\\section{Introduction}\n\\label{sec:intro}\nKaonic atoms and kaonic nuclei \ncarry important infor(...TRUNCATED)
| {"timestamp":"2005-08-25T03:10:23","yymm":"0503","arxiv_id":"nucl-th/0503039","language":"en","url":(...TRUNCATED)
|
"\\section{Introduction}\n\\label{intro}\nThere is the evident nigh affinity between the classical p(...TRUNCATED)
| {"timestamp":"2005-05-20T06:21:32","yymm":"0503","arxiv_id":"math/0503624","language":"en","url":"ht(...TRUNCATED)
|
"\\section{Introduction}\n\\mlabel{sec:intro}\n\n\nIt is well-known that the natural functor from th(...TRUNCATED)
| {"timestamp":"2007-05-31T19:49:29","yymm":"0503","arxiv_id":"math/0503647","language":"en","url":"ht(...TRUNCATED)
|
"\\section{Introduction}\n\\label{Introduction}\nIn this article we prove new theorems which are hig(...TRUNCATED)
| {"timestamp":"2005-03-30T10:19:28","yymm":"0503","arxiv_id":"math/0503699","language":"en","url":"ht(...TRUNCATED)
|
"\\section{Introduction}\n\nIt is common knowledge that Lifshitz formula \\cite{19}\ndescribes the v(...TRUNCATED)
| {"timestamp":"2005-03-06T19:39:25","yymm":"0503","arxiv_id":"quant-ph/0503064","language":"en","url"(...TRUNCATED)
|
"\\section{Introduction}\n\nCommunities of ecologically similar species that compete with each other(...TRUNCATED)
| {"timestamp":"2005-03-17T10:51:17","yymm":"0503","arxiv_id":"q-bio/0503026","language":"en","url":"h(...TRUNCATED)
|
"\\section{From ASDYM equations to Einstein--Weyl structures}\n\\setcounter{equation}{0}\nThe idea o(...TRUNCATED)
| {"timestamp":"2005-06-10T19:44:19","yymm":"0503","arxiv_id":"nlin/0503030","language":"en","url":"ht(...TRUNCATED)
|
"\\section{Introduction}\n\nModels of non-linear spatially extended systems exhibit a variety of spa(...TRUNCATED)
| {"timestamp":"2005-08-24T23:54:38","yymm":"0503","arxiv_id":"nlin/0503039","language":"en","url":"ht(...TRUNCATED)
|
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