# Assessing the complexity of orbital parameters after asymmetric kick in
binary pulsars

Ali Taani1, Juan C. Vallejo2,3 and Mohammed Abu-Saleem4
1Physics Department, Faculty of Science, Al-Balqa Applied University, 19117
Salt, Jordan
2Joint Center for Ultraviolet Astronomy, AEGORA Research Group, Universidad
Complutense de Madrid, Avda Puerta de Hierro, Madrid, Spain
3Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física,
Universidad Rey Juan Carlos, Tulipán, Móstoles, Spain
4Mathematics Department, Faculty of Science, Al-Balqa Applied University,
19117 Salt, Jordan
E-mail<EMAIL_ADDRESS>
(Received date ; accepted date)

###### Abstract

The dynamical characterization of the Millisecond Pulsar (MSP) parameters is a
key issue in understanding these systems. We present an analytical analysis of
the orbital parameters of binary MSPs with long periods (P$\rm{}_{orb}>2$ d)
and circular ($e\leq$ 0.1) orbits, produced by an asymmetric kick model
imparted during the Accretion Induced Collapse (AIC) of white dwarfs process.
It turns out that the distribution of orbits peaks up to P${}_{orb,f}\leq$ 90
d with strong circularization. Considering the different assumptions about the
distribution of companion He stars $3M_{\odot}\leq M_{com}\leq 5M_{\odot}$,
the binary will affect the setups of the balance condition of minimum energy.

Our analytical approach is just a first approach to the more complete models
required for describing all binary parameters after an asymmetric kick.
Therefore, we have also run some numerical simulations in order to compare
their results with the analytical studies. We aim to initiate a first
exploration of the full complexity of the problem, when combining a variable
kick time and a variable kick vector direction. Indeed, the numerical
simulations show patterns resembling the complex behavior found in chaotic
scattering problems. Although we deal with a deterministic problem and bounded
orbits, the regular characteristic orbits are found in more realistic phases
during the AIC process. In addition, the overall process can show complex
behaviors strongly associated with the internal kick mechanisms. This would
lead us to identify the nature of regular orbits and their orbital morphology.

###### keywords:

Pulsars: evolution, complex behavior, asymmetric kick, regular and chaotic
motion, orbital dynamics.

##  1 Introduction

The discovery of Millisecond Pulsars (MSPs) has led to fresh insights into the
astrophysics of Neutron Stars (NSs) (see e.g. Bhattacharya & van den Heuvel
1991; Tauris & van den Heuvel 2009; Lorimer 2009; Taani et al. 2012a,b; Tauris
2015; Taani & Khasawaneh 2017). Interestingly, the Accretion Induced Collapse
(AIC) of massive white dwarfs (Nomoto 1987; Isern & Hernanz 1994; Ferrario &
Wickramasinghe 2007; Wickramasinghe et al. 2009; Hurley et al. 2010; Zhang et
al. 2011; Chen et al. 2011; Taani et al. 2012b, Tauris et al. 2013; Freire &
Tauris 2014; Ablimit & Li 2015a; Tauris 2015; Kwiatkowski 2015; Tauris et al.
2017; Ablimit 2021a,b) has been invoked to explain the formation of some
binary MSPs, instead of the standard model (Alpar et al. 1982; Zhang & Kojima
2006; Taani et al. 2018; Taani et al. 2019).

It has been studied that the magnetized WDs can grow in mass with low
accretion rates (Ablimit & Maeda 2019). Ablimit (2021b) showed that the
channel of magnetized ONe or CO) WD + stripped helium star binaries can
produces MSPs, magnetars and type Ia supernovae (SNe Ia which are
thermonuclear explosions of CO WDs that have grown to the Chandrasekhar
limit).

The magnitude and direction of asymmetric kicks imparted on NSs during AIC are
one of the major uncertainties in stellar population synthesis studies. Note
that there are other possible mechanisms that may affect the orbital evolution
of the WD binaries such as tides, and post-AIC mass loss and accretion (e.g.,
Ablimit & Li 2015; Tauris et al. 2013; Abu-Saleem & Taani 2021a,b; Wang et al.
2022). The physical mechanisms that cause these kicks are still open
questions, but they are presumably the result of some asymmetry in the core
collapse or subsequent SNe explosion (see, e.g., Pfahl et al. 2002;
Podsiadlowski et al. 2004; Wei et al. 2010; Janka 2017; Tauris et al. 2017;
Taani et al. 2019; Taani et al. 2020).

On the observational side, many authors presented strong observational
evidences like the optical infrared light curves and spectra of the nickel-
rich outflows through the transient surveys (see i.e. Melatos 2007; Darbha et
al. 2010; Piro & Kulkarni 2012; Metzger & Piro 2014; Piro & Thompson 2014;
Taani 2015; Metzger et al. 2009; Melatos et al. 2021; Gautam et al. 2021),
many calculations of stellar models and their parametrization in population
synthesis codes can be found in Tutukov & Yungelson (1993), Ablimit et al.
(2019), Abdusalam et al.(2020), Ablimit (2021a) and many other similar binary
population synthesis works.

We aim to provide some insight into the possible presence of complexity when
asymmetric kicks processes are imparted on NSs during AIC. We will focus on
searching for the presence of dynamical chaos, meaning strong dependency to
the initial conditions of the system. We will follow some of the ideas applied
to chaotic scattering studies. Indeed, in a broad sense, a scattering problem
can be seen as the problem to obtain the relationship between some input
variables, that characterize an initial condition for some dynamic system, and
some output variables, that characterize the final state of the system.

Chaos can be present even in very simple two-degrees of freedom deterministic
dynamical systems, if one adds a time-dependant process to the system. In
chaotic scattering problems, complex patterns can be observed when the
incoming particle can find the target system in a state that depends on the
arrival time. That is, this state depends on the precise time at which the
incoming particle interacts with the target system.

Likewise these scattering processes, asymmetric kicks processes can add a
similar time-dependency to exploding binaries. The kick results will depend on
the precise state of the orbital parameters of the binary system when the kick
occurs. Hence, the overall binary pulsars problem can show strong sensitivity
to initial control parameters. These parameters can be, for instance, the
masses defining the binary system, or the precise time when the explosion
(kick) takes place.

Therefore, we have modified the regular two-body problem that describes the
WD-companion binary system, and we have included a kicker process that will
add the necessary time-dependency. This kicker process will modify the orbital
parameters at a given instant, and it will allow the possibility of having a
complex, may be chaotic, motion.

The structure of the paper is as follows. The Section 2 present the main
binary characteristics and introduces a simple analytical approach for the AIC
process. The Section 3 numerically explores the complex behavior of these
binaries when a variable kick process is introduced in a otherwise regular
two-body system. The Section 4 discusses those behaviors and compares the
analytical and numerical approaches. Finally, the last section summarizes the
results and makes some concluding remarks.

##  2 A simple analytical approach for the AIC process

In the case of AIC, low kick velocities are expected, of the order of 50 km/s
(Boyles et al. 2011). However, van den Heuvel (2007) argued that (6/8) double
NSs are known to have low eccentricities, suggesting that their second-born
NSs received very small kick velocities at birth ($\leq 150kms^{-1}$). In his
work, he argued that these low-mass and low-kick NSs are most likely formed by
the AIC. The change in binary parameters not only occurs from mass loss and
kick but also increases in momentum from their formation mechanism
(Arzoumanian et al. 2002; Kiel & Hurley 2009). For an instant, changing in
mass in AIC comes from converting baryonic mass to gravitational mass during
the formation of the NS (see Zeldovich & Novikov 1971; Bagchi 2011; Mardini et
al. 2019a,b; Mardini et al. 2020; Almusleh et al. 2021). This physics is
described, among others, in Fryxell & Arnett 1981 and Kalogera 1996.

As a very first approach to this problem, we present here an analytical model
to estimate the changes in orbital periods and semi-major axes because of the
kick velocity received by the WD when the AIC process takes place. In our
model, most of the mass that is lost in a binary system that undergoes AIC
comes from the WD converting baryonic mass into binding energy during the
collapse and the formation of the newborn NS, and the mass change is of the
order of 0.18 M⊙ (i.e. Ablimit & Li 2015).

The quantitative relationship between the orbital eccentricity and the amount
of mass ejected in the AIC is (see i.e., Eqs. 19-28 in Tauris & Takens 1998),

$e_{f}-e_{i}=\frac{\Delta M_{AIC}}{M_{final}}$ (1)

where $\Delta M_{AIC}$ is the mass ejected during the AIC. Therefore, $\Delta
M_{AIC}$ = Mej = MWD \- MNS. By other hand, $M_{final}$ = $M_{comp}$ \+
$M_{NS}$, the sum of the masses of the companion and the NS. In our
calculations, we assume that the binary orbits prior to the AIC process are
circular, ei = 0. To some extent, this is justified by the fact that close
binaries are very likely to circularize as a result of mass transfer from the
evolved star before the explosion (e.g. Tauris & Takens 1998, Cai et al.
2012). The companion star may lose material from its surface because the SN
ejecta may either directly strip material from the companion by direct
transfer of momentum or evaporate the envelope through the conversion of the
blast kinetic energy into internal heat (Li 2008). Say Mrec as the amount of
mass received by the companion ($\sim 0.1-0.2M_{\odot}$ see Zhang et al.
2011). Then Mcomp,f = Mcomp,i \+ Mrec, where Mcomp,i and Mcomp,f are the
initial and final masses of the companion, respectively.

We can cancel the change in binding energy here as this is not a NS. Let us
assume that M${}_{rec}\ll M_{ej}$. As a first approach to the problem we will
analyse a planar $2D$ problem as seen in the Figure 1. In this case, we can
deduce the relation of orbital period and new parameter $\eta$ (which is
related with kick velocity) based on the previous assumptions.

Figure 1:  A simple schema of the geometry of the planar binary system under
the effect of a supernova kick, the angular momentum vector pointing along the
positive Z-axis. The WD explodes, which implies it receives a kick velocity
vector that forms a time-dependant angle $\theta$ with the X and Y axes. A
complex change of the initial orbital parameters can occur when one varies the
$t_{kick}$ time, or time when the explosion takes place. However, it can also
occur when the initial masses of the binary are considered control parameters
and they are varied whilst $t_{kick}$ remains fixed. Note that, once the kick
take place, the WD losses a given amount of mass $M_{ejec}$, while the
companion receives a different, smaller, amount of mass $M_{rec}$.

Before the AIC event the orbital energy can be expressed as (see, e.g. Hills
1983 and references therein).

$E_{orb,i}=\frac{-GM_{WD}M_{comp,i}}{2a_{i}}$ (2)

where $G$ is the gravitational constant, ai is the initial separation between
the two stars in the binary.

Hence, the orbital energy after the SNe,

$E_{orb,f}=-\frac{GM_{NS}M_{comp,f}}{2a_{f}},$ (3)

with af being the final separation.

By using the ratio of initial to final velocities (the velocity of WD relative
to it’s companion before and after the AIC event (vi/vf)), and the energy
conservation law in the equations above, we can get,

$\frac{a_{f}}{a_{i}}=[\frac{1-\frac{\Delta M}{M_{b}}}{1-(2\frac{\Delta
M}{M_{b}})-\frac{v^{2}_{f}-v^{2}_{i}}{v^{2}_{orb}}}]$ (4)

where $\Delta M$= $\Delta M_{AIC}-\Delta M_{rec}$, $M_{b}=M_{WD}+M_{comp,i}$,
and the orbital velocity $v_{orb}=\sqrt{\frac{GM_{b}}{r}}$.

By some calculations we get,

$\frac{v^{2}_{f}-v^{2}_{i}}{v^{2}_{orb}}=(\frac{v_{kick}}{v_{orb}})^{2}+2(\frac{v_{i}}{v_{orb}})(\frac{v_{kick}}{v_{orb}})cos\theta$
(5)

If we adopt $cos\theta=1$, i.e. $v_{kick}$ (kick velocity) is co-aligned with
the direction of the orbital angular momentum of the pre-explosion binary,

$\frac{a_{f}}{a_{i}}=[\frac{1-\frac{\Delta M}{M_{b}}}{1-2(\frac{\Delta
M}{M_{b}})-(\frac{v_{kick}}{v_{orb}})^{2}-2(\frac{v_{i}}{v_{orb}})(\frac{v_{kick}}{v_{orb}})}]$
(6)

The orbital speed is defined by

$v_{orb}=\sqrt{\frac{G(M_{WD}+M_{comp,i})}{r}}\sim\sqrt{\frac{GM_{WD}}{r}}$
(7)

Figure 2: The $\rm a_{f}/a_{i}$ as a function of $\eta$. It is notable that
when $\eta$ approaches 0.40, the ratios increase sharply according to Eq. 8,
(see the text). The various assumptions of $\eta$ lead to different distances
after the AIC process with different lines. Figure 3:  The same meaning as
Fig. 2, but $\rm P_{orb,f}/P_{orb,i}$ as a function of $\eta$.

We assume here that the corresponding $v_{orb}\sim v_{esc}$. The combination
of kick components (i.e., the magnitude and direction) will be an effective
means for determining their orbital separation, their orbital periods, and, of
course, the fate of the stars (Tauris & van den Heuvel 2006; Taani 2016). This
gives rise of $a_{f}$ to a critical value of $a_{f}$ = 3$a_{i}$ ($1-e_{i})$ at
$e_{i}=0$. We assume that $a_{i}=1.2R_{\odot}$, which might be reasonable for
AIC binaries where the companion is on the MS of helium MS.

However, if the amount of angular momentum of the accreted material equals the
orbital angular momentum, this would lead to the possibility of the companion
being kicked away (Kiel et al. 2008). An assumption here is that a more
energetic explosion would be able to impart a greater initial angular velocity
onto the proto-NS than a less energetic explosion, and the result is typically
an eccentric orbit with an orbital period of a few hours (Hurley et al. 2002).

Now let us define the $\eta$ parameter as the ratio between two velocities

$\displaystyle\eta=\frac{v_{kick}}{v_{esc}}$ (8)

The ratio of final to initial orbital periods can be written as

$\frac{P_{f}}{P_{i}}=[\frac{1-\frac{\Delta M}{M_{b}}}{1-2(\frac{\Delta
M}{M_{b}})-\eta^{2}-2\eta}]^{\frac{3}{2}}$ (9)

Figures 2 & 3 illustrate the behavior of these equations as the $\eta$ value
evolve by fixing $\Delta M$ and the binary mass $M_{b}$. We have taken an
illustrative $\Delta M=M_{ej}-M_{rec}=0.18$, $M_{rec}$ is likely close to 0,
and $M_{ej}$ is likely closer to 0.18$M_{\odot}$, that correspond to the mass
of the WD that is converted into binding energy of the NS (see Zhang et al.
2011; Bagchi 2011). The binary mass was in turn computed using a WD mass of
$M_{WD}=1.1M_{\odot}$ and a companion mass of $3M_{\odot}\leq M_{comp}\leq
5M_{\odot}$. Our sample scenario seems to be reasonable for AIC binaries where
the companion is on the MS or helium MS (see other possible scenarios in
Ruiter et al. 2019).

##  3 Numerical simulations and regular or chaotic processes resulting from
AIC

In the previous sections, we have estimated the new orbital period imparted to
the newborn pulsar by relying on the kick velocity, considering an initial
circular orbit. However, this is a very special case. The general problem can
present a wider set of initial conditions, and this section aims to show the
complexity of the calculations when considering a more complete case. An
indicator of the presence of high complexity in a given system is the
detection of dynamical chaos, or strong sensitivity to initial conditions
(Alligood et al. 1996). The presence of this phenomenon means that even minor
differences in the values of the initial conditions can lead to very different
final states of the system.

The standard two-body problem can be seen as one reduced-mass particle. Hence,
it can only show regular motions. This is because, following the Poincaré-
Bendixon theorem, chaos cannot be present in continuous flows of second order
(Vallejo & Sanjuan 2019). However, if one introduces an additional dimension
into the system, the time in this case, complexity can arise. In our case,
this additional temporal dimension can be introduced by means of the kick
process.

For solving the numerical binary, we will use a standard $N$-body solver, the
Rebound package (Rein & Liu 2012). This numerical engine integrates the motion
of $N$-particles under the influence of gravity, allows the use of a variety
of integrators, and supports changes in orbital parameters at designated
times.

Each simulation starts with given initial conditions of an arbitrary, non-
circular, binary system. We have fixed the initial eccentricity to be
$e_{0}=0.01$, and the semi-major axis $a_{0}=1.0$au along all simulations. We
have also set $\Omega=0.0$, $\omega=0.0$ and $i=0$, as we are dealing with a
planar problem.

The masses of the two bodies will be $M_{WD}$ for the exploding white dwarf,
and $M_{comp}$ for the companion, with $M_{comp}>M_{WD}$. Once the explosion
takes place, at an arbitrary time $t_{kick}$, the WD star losses $M_{ejec}$
mass and suffers the necessary kick, changing its velocity vector. The
$M_{comp}$ mass corresponds to the initial mass of the companion star, that
gains some small amount of mass $M_{rec}$ after the explosion.

There is firm evidence that many newborn NSs receive this momentum kick at
birth (Podsiadlowski et al. 2004), which can raise the observed high
velocities (typically $\sim$ 400 kms1). Note that in AIC, low kick velocities
are expected to be much lower than this value (see Boyles et al. 2011).

Chaotic behavior can be generated by this kick, even for the lowest values of
the kick. This is because the system strongly depends on $t_{kick}$: a kick
produced when the stars are in different orbital states will produce different
final results. Depending on the kick strength, the resulting stars will or
will not remain bounded.

Hence, the first control parameter to take into account is the kick velocity
vector $\vec{v}_{kick}$. This vector defines the recoil that the exploded star
receives. This will modify the orbital velocity of the kicked body, that will
have a final mass $M_{WD}-M_{ejec}$ after the explosion.

This kick vector is defined by its the modulus and its direction, and the
recoil velocity components added to the kicked star will strongly depend on
the selection of this direction. The kick direction can depend on many
different factors on a real kick scenario. As a first approach to it, we have
defined the following kick process. We have implemented a rotating vector,
with an arbitrary intrinsic period $T_{kicker}$. At the precise kick time,
$t_{kick}$, the kick is modeled as a vector of modulus $v_{kick}$, pointing to
the direction of the rotating vector at that time.

Hence, depending on $t_{kick}$, the kick velocity vector will be pointing to a
different direction, defined by a variable $\theta$ in Figure 1. This kick
will finally modify the orbit of the stars, leading to different orbital
parameters after the explosion.

The period of the rotating vector has been fixed to be $T_{kicker}=0.1$yrs in
all simulations, as we want to address the complexity of the final dynamics
and not the precise values resulting from each simulation. We will compare our
numerical results with the previous section analytical calculations. We will
also analyse the resulting complexity by focusing on the values of the semi-
major axis $a$ and the orbital period $T$. In addition, we have also analysed
the changes in eccentricity in some cases.

|
---|---
Figure 4: The consequences of a kick on a typical binary system (see system
details in the text). The initial orbital period is $T_{0}=0.4$yrs. The kick
take place at $t_{kick}=1.0$yrs. Upper diagrams show how the $v_{x}$ and
$v_{y}$ components for the white dwarf are modified after the kick. The bottom
diagrams show the effect in semi-major axis and orbital period. Left panel
corresponds to a kick velocity modulus $v_{kick}=5.0$km/s, when the change is
relatively small. The right panel corresponds to $v_{kick}=50.0$km/s, when the
changes are larger. One can observe noticeable changes in the final orbital
parameters of the system in both cases.

The Figure 5 reflects a typical case of the impact of a kick on the initial
binaries. The system masses are set to be $M_{WD}=1.1$ $M_{\odot}$ and
$M_{comp}=5.0$ $M_{\odot}$, resulting on an initial period of $T_{0}=0.7$yrs.
The kick is defined to happen at $t_{kick}=1$yr after the simulations starts.
Remaining parameters are set to be $M_{ejec}=0.2M_{\odot}$,
$M_{rec}=0.01M_{\odot}$. The left panel corresponds to a kick velocity modulus
$v_{kick}=5.0$km/s, when the change is relatively small. The right panel
corresponds to $v_{kick}=50.0$km/s, when the changes are larger. One can
observe noticeable changes in the final orbital parameters of the system in
both cases. Figure 1 also shows that the final period $T$ and semi-major axis
$a$ increase, but the system remains bounded in both cases. Hence, we have
defined a fully deterministic dynamical system that can show a very complex
behavior despite the simplicity of the defined kick mechanism, and it even can
present strong dependency on initial conditions.

|
---|---
|
Figure 5: The consequences of different kicks on a typical binary system, as
the initial mass $M_{comp}$ slightly increases and remaining parameters are
kept constant. The Figure contains four panels, corresponding to four
different $v_{kick}$ velocities. Each panel has an upper diagram, showing the
resulting period $T_{kick}/T_{0}$ in blue. For reference, the initial period
$T_{0}$ is showed in green. $T_{0}$ decreases as the $M_{comp}$ increases. The
middle diagrams show the resulting semi-major axes $a_{kick}/a_{0}$ (in blue).
Finally, the bottom diagrams, in red, show the resulting values of the
eccentricity of the system after the kick. One can see complex patterns,
resulting of the strong dependency on initial conditions.

The Figure 5 analyses the consequences of a kick as we vary the initial mass
$M_{comp}$. We will keep constant the remaining parameters aiming to explore
the strong sensitivity to initial conditions. The Figure contains four panels,
corresponding to four different $v_{kick}$ velocities. Each panel has an upper
diagram, showing the resulting period $T_{kick}/T_{0}$ in blue. The initial
period $T_{0}$ in green, for reference ($T_{0}$ decreases as the $M_{1}$
increases). The middle diagrams show the resulting semi-major axes
$a_{kick}/a_{0}$ (in blue). Finally, the bottom diagrams, in red, show the
resulting values of the eccentricity of the system after the kick. One can see
complex patterns, resulting of the strong dependency on initial conditions.

As expected, the Figure 5 shows that larger periods, longer semi-major axes
and larger eccentricity values are obtained as $v_{kick}$ increases. Indeed,
beyond certain $v_{kick}$, the system might be left un-bounded in some cases.

Interestingly, the Figure 5 shows a complex pattern of spikes. These spikes
reflect a variety of changes in the final orbital parameters when the same
explosion is applied to the initial systems defined by different $M_{comp}$.
In other words, the WD explodes at the same instant in all cases. Therefore,
the kick occurs at the same time in all modeled binaries.

Figure 5 shows that even very small variations in the control parameter
$M_{comp}$ produces very different results. This is because as $M_{comp}$
changes, the $v_{kick}$ is added to systems with slightly different orbital
periods and different orbital velocities. One gets different results as
$M_{comp}$ varies, and the system shows a strong dependency on initial
conditions.

This pattern indeed resembles a fractal-like structure. The Figure 5 shows a
variety of interleaved small and large changes, and also interleaved
increments and decrements in the final values of the semi-major axes and
eccentricities. In short, different $T_{0}$ values make or not the whole
system more sensitive to the kick, even when the explosion occurs at the same
time in all cases. The stars are in different relative positions at $t_{kick}$
and one adds kick vectors pointing to different directions when the kick takes
place. Hence, one can get a complex behavior resulting from both dependencies.

##  4 Discussions

Our method based on the equations gives analytical expressions for parameters
of the binaries after asymmetric kick and probability of disruption of a
binary depending on the ratio of kick velocity and velocity of components
prior to explosion and on relative amount of the ejected matter. We assume
here the explosion to be instantaneous (infinitely short duration) and would
takes place in very close binaries, in the range of LMXBs and close double
pulsars, such as the PSR 1913+16, J0737- 3039AB and the close WD-pulsar system
PSR 0655+64 (van den Heuvel 2009). As a result, $v_{final}\approx
v_{initial}\approx 0.1c$. To analytically find the separation distance at the
moment of explosion, we considered a system in which the initial conditions
are $\Delta M=0.01M_{\odot}$, $M_{WD}=1.1M_{\odot}$ and
$M_{comp,i}=3-6M_{\odot}$.

In Figure 2, we note that several values of $\eta$ appear to correlate with
the orbital parameters. In addition, our results cover the observational
orbital parameters up to $P_{orb}\geq 90$ d. We should emphasize here that the
degree of interaction is critically dependent on the critical value of $\eta$.
At $\eta<0.40$ the kick velocity has very small effect on the orbital
distributions, as we can show that Mej is on the order of 0.1 (the mass ”lost”
in the AIC itself). On the other hand, when $\eta\geq 0.40$ this would lead to
a significant increase of their orbital distribution. This ratio also serves
to demonstrate the sensitivity of binary evolution, and has also crucial
characteristics in examining influence of physical parameters on the binary
evolution. It has recently been argued that, the majority of the observed
double NS systems can best be explained with small kicks in the second SNe
(Tauris et al. 2017).

|
---|---
Figure 6: The consequences of different kicks on a binary system, with kicked
star fixed to be $M_{WD}=1.2M_{\odot}$, and three different values of
$M_{comp}$, hence, of $T_{0}$. The leftmost diagram corresponds to
$t_{kick}=0.7$ and the rightmost diagram corresponds to $t_{kick}=0.8$. In
each diagram, top panel show the evolution of $T_{kick}/T_{0}$ with $\eta$,
while bottom panel show the evolution of $a_{kick}/a_{0}$.

The Figure 6 shows the consequences of different kicks on a binary system,
with a kicked star fixed to be $M_{WD}=1.2M_{\odot}$, and three different
values of $M_{comp}$, hence, of $T_{0}$. The two diagrams correspond to
$t_{kick}=0.7$ and $0.8$. In each diagram, top panel show the evolution of
$T_{kick}/T_{0}$ with $\eta$, while bottom panel show the evolution of
$a_{kick}/a_{0}$.

The shapes of the curves are roughly similar, following the same evolution
that was described in the analytical study.

In general, these curves show that the final period and semi-major axis
increases with $\eta$ (that is, with $v_{kick}$). However, this increase is
stronger for certain values of $M_{comp}$. Notably, for each $M_{comp}$, there
is a different $\eta$ limit value which makes the system unbounded, reflected
by the vertical line. One can also see that the results are different for each
kick time, as the leftmost and rightmost diagrams show different limit $\eta$
values.

Hence, the Figure 6 is reflecting again the strong dependency to the initial
control parameter values. There is no simple evolution of the plotted curves
for each $M_{comp}$. One observes that the green curve corresponding to the
largest mass $M_{comp}=5.0$ $M_{\odot}$ flattens with the increase in
$t_{kick}$ value, reflecting smaller changes in the final orbital parameters.
Conversely, the blue curve corresponding to the smallest mass $M_{comp}=3.0$
$M_{\odot}$, rises as the $t_{kick}$ value increases. This reflects larger
changes in the final orbital parameter as the $\eta$ parameter grows in these
cases.

Because this complexity, further numerical explorations on the parametric
space are needed as a natural extension to the work done.

Further observational evidence that at least some NSs receive low kicks at
birth. The observed of pulsations with a period of 11.1 ms in the accreting
binary, J0900-3144 (Burgay et al. 2006), has confirmed this scenario.
Additionally, Champion et al. (2008) proposed that the AIC of a massive and
rapidly rotating WD ($M$ = $1.05_{\odot}\pm 0.015_{\odot}$) could produce the
observed orbital parameters of PSR J1903+0327 ($M_{NS}$= $1.667_{\odot}\pm
0.021_{\odot}$). Recently, Freire et al. (2011) and Pijloo et al. (2012) have
questioned the plausibility of the existence of such a MSP-WD-MS triple system
and have pointed out several shortcomings associated with such a proposition.
Further challenges the conventional formation scenarios not only is the
pulsars companion a main-sequence star, but the orbit is also highly eccentric
(e = 0.44) with 95-d orbital period, neither of which are predicted by the
standard formation scenarios. On the hand, Abdikamalov et al. (2010) have
presented results from an extensive set of general-relativistic AIC
simulations using a microphysical finite-temperature equation of state during
collapse; they investigated a set of 114 progenitor models in axisymmetric
rotational equilibrium, with a wide range of rotational configurations and
resulting WD masses. They contrast the gravitational-wave signals of AIC and
rotating massive star iron core collapse and find that they can be
distinguished.

##  5 Conclusions

We present a model to estimate the new orbital parameters relying on the
dynamical effects of the asymmetric mass ejection, during the accreting WD
binaries in the initially tight binary system in the AIC process. Assuming the
kick velocity axis coincident with the spin axis of the newborn NS, whose
rotation is presumably co-aligned with the orbital angular momentum of the
pre-explosion binary. As a result, the instability here is associated with
many dissipative processes such as kicks, mass loos and shock waves. These
effects imparted on the newborn pulsars, can account for the differences in
their orbital parameter distributions. Our results are valid for all orbital
periods of MSPs up to $P_{orb,f}\leq 90~{}d$ in circular orbits in which the
complete co-rotation has achieved. While the final separation distance is more
than three times as large as it’s initial value.

Consequently, it is seen that all orbits retain their circularization
characteristics after AIC process along the stellar evolution during the
instantaneous mass loss erasing any remnant eccentricity. The orbital period
shows a steady increase in the magnitude, before the ratio of kick to escape
($\eta$) is reached a value of 0.40. After this stage, $\eta\geq 0.40$, it is
anticipated that the distribution of the kick velocities rising to a sharp
peak and would produce longer orbital periods. This ratio serves to
demonstrate the sensitivity of binary evolution, since in all cases, the kick
to the escape is smaller than the original orbital velocity in the fate of the
binary. The magnitude and circularization of the orbits are determined by the
nature of the companion star which we considered $3M_{\odot}\leq M_{com}\leq
5M_{\odot}$ at the evolution of a binary under the instantaneous mass loss.
However, the initial separations should be distributed according to the
observed distribution of semi-major axes.

However, a small portion of mass ratio ($M_{WD}/M_{comp}$) could be affected
to the Porb,f and it exhibits considerable non-diversity in behavior. By
comparing results from our model with observed binary MSPs, we can enhance our
understanding of their formation, evolution and distributions. Therefore their
total contribution is expected to be around $20\%$ (Taani et al. 2012a) of
population, since many objects might have formed in such an environment. The
distribution functions of systemic velocities, correlation with orbital
parameters and other effects associated with dynamical characterisation of MSP
binary parameters can be studied in terms of constrain the Galactic potential
and their modeling spatial distribution in the Galaxy (Kalogera 1996), and
also in globular clusters by assuming their smallest possible systemic
velocity.

In addition, we have approached a more complete scenario and numerically
explored the case of planar, but non-circular binaries. We have implemented a
very simple rotating mechanism to show the important role played by the
direction of the kick in the resulting kicked system. Despite this simplicity,
we have seen that the combination of a variable kick time and a variable kick
vector direction leads to a complex behavior, as different kick velocity
components are combined with different velocity components of the kicked star.
We have observed that even for initial small eccentricities and semi-major
axes, the analysed problem seems to present strong sensitivity to initial
conditions. This means to observe a complex behavior, with very different
results for every set of control parameters.

This strong sensitivity lead to a complex behavior, reflected in the plots
showing the dependency of the new orbital parameters with the mass of one of
the stars. Indeed, these plots resemble fractal-like behavior. The obtained
patterns are similar to those obtained in some other scattering problems with
presence of chaos, mainly plots of differential cross sections or delay-time
functions (Seoane et al. 2013). The presence of this complex behavior in a
deterministic system is a sign of strong sensitivity to initial conditions,
but it is not a synonym of fully chaotic motion. Assessing the presence of
chaos is a necessary step for classifying the resulting orbits in regular
orbits or chaotic ones. Hence, further work is still needed to formally
characterise this complexity. This is an interesting research to extend the
results of our work.

Although, the prevalence of AIC is still a matter of heated debate both
observationally and theoretically. However, our results are promising
approach, and more stringent constraints can be applied to the kick velocity
and mass-loss and separation between the components and the new formed binary.
This will lead understanding the binary stellar evolution with the birth of
the binary and double pulsar systems. More observations are required to
explain these features and behaviors.

## Acknowledgements

A. Taani would like to thank NAOC and IHEP institutes for supporting his visit
as Visiting Scientist Researcher. Juan C. Vallejo thanks the Spanish Ministry
of Economy, Industry and Competitiveness for grants ESP2017-87813-R. We are
grateful to Alexander Tutukov, Postnov Konstantin, Manjari Bagchi, Iminhaji
Ablimit and Serena Repetto for their comments and suggestions that allowed to
improve the clarity of the original version. The authors would also like to
thank the anonymous referee for the careful reading of the manuscript and for
all suggestions and comments which allowed us to improve both the quality and
the clarity of the paper.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * [Abdikamalov & et al.(2010)] Abdikamalov E. B., Ott C. D., Rezzolla L. et al., 2010, PhRvD, 81, 4012A
  * [1] Abdusalam K.; Ablimit I.; Hashim P.; L$\ddot{\rm u}$, G.-L. et al., 2020, ApJ, 902, 125
  * [2] Ablimit I. & Li, X.-D., 2015, ApJ, 800, 98
  * [3] Ablimit I.; Maeda K., 2019, ApJ, 871, 31
  * [4] Ablimit I., 2021a, PASP, 133, 074201
  * [5] Ablimit, I., 2021b, arXiv:2110.12140,
  * [6] Abu-Saleem M. & Taani A., 2021a, Chinese Journal of Physics, 74, 53
  * [7] Abu-Saleem M. & Taani A., 2021b, AIP Advances, 11, 015309
  * [Alligood et al.(1996)] Alligood K., Sauer T.D. & Yorke J.A., 1996, Chaos. An Introduction to Dynamical Systems, Springer Verlag, New-York.
  * [8] Almusleh N., Taani A., Ozdemir S., et al., 2021. Astron. Nachr. 342, 625.
  * [9] Alpar A., Cheng F., Ruderman A., Shaham J., 1982, Nature, 300, 728
  * [10] Arzoumanian Z., Chernoff D., & Cordes J., 2002, ApJ, 568, 289
  * [11] Bagchi M., 2011, MNRAS, 413, L47
  * [12] Bhattacharya D. & van den Heuvel E., 1991, Phys. Rep., 203, 1
  * [13] Boyles J., Lorimer D. R., Turk P. J., & et al. 2011, ApJ, 742, 51
  * [14] Burgay M., Joshi B.C., DAmico N., & et al. 2006, MNRAS, 368, 283
  * [15] Cai Y., Taani A., Zhao Y.-H., Zhang C. M., 2012, Chinese Astronomy and Astrophysics, 36, 137
  * [16] Champion D. J., Ransom S. M., Lazarus P., & et al. 2008, Science, 320, 1309,
  * [17] Chen W.-C., Liu X.-W. Xu R.-X. & Li X.-D., 2011, MNRAS, 410, 1441
  * [18] Darbha S., Metzger B. D., Quataert E., & et al. 2010, MNRAS, 409, 846
  * [19] Ferrario L. & Wickramasinghe D. T., 2007, AIPC, 968, 194
  * [20] Freire P. C. C., Bassa C. G., Wex N., et al., 2011, MNRAS, 412, 2763
  * [21] Freire P. C. C. & Tauris T., 2014, MNRAS, 438, 86
  * [22] Fryxell B. A., & Arnett W. D., 1981, ApJ, 243, 994
  * [23] Gautam A., Crocker R. M., Ferrario L., 2021, arXiv:2106.00222
  * [24] Hills J. G, 1983, ApJ, 267, 322H
  * [25] Hurley J. R., Tout C. A. & Pols O. R., 2002, MNRAS, 329, 897
  * [26] Hurley J. R., Tout C. A., Wickramasinghe D. T., et al. 2010, MNRAS, 402, 1437
  * [27] Isern J. & Hernanz M., 1994, MmSAI, 65, 339
  * [28] Janka H.-T., 2017, ApJ, 837, 84
  * [29] Kalogera V., 1996, ApJ, 471, 352
  * [30] Kiel P., Hurley J. R., Bailes M. & Murray J. R., 2008, MNRAS, 388, 393
  * [31] Kiel P. & Hurley J., 2009, MNRAS, 395, 2326
  * [32] Kwiatkowski D., 2015, arXiv151200678
  * [33] Li X.D.,2008, MNRAS, 384, 16
  * [34] Lorimer D., 2009, Astroph & Spac Sci. Lib., 357, 1
  * [35] Mardini M. K., Placco V. M., Taani A., & et al. 2019a, ApJ, 882, 27
  * [36] Mardini M. K., Li H., Placco V. M., & et al., 2019b, ApJ, 875, 89
  * [37] Mardini, M.K., Placco, V. M., Meiron, Y., et al. 2020. ApJ, 903, 88
  * [38] Melatos A., 2007, AdSpR, 40, 147M
  * [39] Melatos A., Clearwater P., Suvorova S. & et al. 2021, PhRvD, 104d2003M
  * [40] Metzger B. D., Piro A. & Quataert E., 2009, MNRAS, 396, 304
  * [41] Metzger B. D. & Piro A. L., 2014, MNRAS, 439, 3916
  * [42] Nomoto, K., 1987, ApJ, 322, 206
  * [43] Pfahl E., Rappaport S., & Podsiadlowski P., 2002, ApJ, 573, 283
  * [44] Pijloo J. T., Caputo D. P., & Zwart S. P., 2012, MNRAS, 424, 2914
  * [45] Piro A. L., & Kulkarni S. R., 2012, ApJ, 762, L17
  * [46] Piro A. L. & Thompson T. A., 2014, ApJ, 794, 28
  * [47] Podsiadlowski Ph., Langer N. & Poelarends A., 2004, ApJ, 612, 1044
  * [48] Rein H. & Liu S.F. 2012, A&A, 537, 128
  * [49] Ruiter A. J. et al, 2019, MNRAS, 484, 698R
  * [50] Seoane J. M., Sanjuan M.A.F, 2013, Rep. Prog. Phys., 76, 016001
  * [51] Shapiro S. L., & Teukolsky S. A., 1983, Black Holes, White Dwarfs and Neutron Stars. Wiley, New York
  * [52] Taani A., Zhang C.M., Al-Wardat M., & Zhao Y. H., 2012a, Astronomische Nachrichten, 333, 53
  * [53] Taani A., Zhang C.M., Al-Wardat M., & Zhao Y. H. 2012b, Astrophysics and Space Science, 340, 147
  * [54] Taani A., 2015, Jordan Journal of Physic 8, 149
  * [55] Taani A., 2016, Research in Astronomy and Astrophysics, 16, 101
  * [56] Taani A. & Khasawaneh A., 2017, J. of Phy. Conf. Ser., 869, 012090
  * [57] Taani A. & Vallejo J. C., 2017, Publications of the Astronomical Society of Australia, 34, 24
  * [58] Taani A., Abushattal A. & Mardini M., 2019, Astronomische Nachrichten, 340, 847
  * [59] Taani A., Abushattal A., Khasawaneh A., Almousleh N. & Al-Wardat M., 2020. Jordan J. Phys. 13, 243
  * [60] Tauris T. M. & Takens R. J., 1998, A&A, 330, 1047
  * [61] Tauris T. M., & van den Heuvel E. P. J., 2006, in Compact stellar X-ray sources. ed. W. Lewin & M. van der Klis (Cambridge: Cambridge Univ. Press), 623
  * [62] Tauris T. M., Sanyal D., Yoon S.-C. & Langer N., 2013, A&A, 558A, 39
  * [63] Tauris T. M., 2015, arXiv:1501.03882
  * [64] Tauris T. M., Kramer M., Freire P. C. C. & et al. 2017, ApJ, 846, 170
  * [65] Tutukov A.V. & Yungelson L.R., 1993, MNRAS, 260, 675
  * [66] Vallejo J.C. & Sanjuan M.A.F., 2019, Predictability of chaotic dynamics, Second Edition, Springer Verlag, p.11
  * [67] van den Heuvel, E. P. J. 2007, AIP Conf. Proceedings, 924, 598 (astro-ph/0704.1215)
  * [68] van den Heuvel E. P. J., 2009, Astrophy. and Spa. Sci. Lib., 359, 125
  * [69] Wang B., Liu D. & Chen H., 2022, MNRAS, 510, 6011W
  * [70] Wei Y. C., Taani A., Pan Y., & et al., 2010, Chin. Phys. Lett., 27, 9801
  * [71] Wickramasinghe D. T., Hurley J. R., & et al. 2009, JPhCS., 172, 12037
  * [72] Zel’dovich Y. B., Novikov I. D., 1971, Relativistic Astrophysics Vol 1: Stars and Relativity. University of Chicago Press
  * [73] Zhang C. M. & Kojima Y., 2006, MNRAS 366, 137
  * [74] Zhang C. M. Wang J. Zhao Y.H. & et al. 2011, A&A, 527, A83


# Projective closure of Gorenstein monomial curves and the Cohen-Macaulay
property

Anargyros Katsabekis  Department of Mathematics, University of Ioannina, 45110
Ioannina, Greece<EMAIL_ADDRESS>
###### Abstract.

Let $C({\bf a})$ be a Gorenstein non-complete intersection monomial curve in
the 4-dimensional affine space. There is a vector ${\bf v}\in\mathbb{N}^{4}$
such that for every integer $m\geq 0$, the monomial curve $C({\bf a}+m{\bf
v})$ is Gorenstein non-complete intersection whenever the entries of ${\bf
a}+m{\bf v}$ are relatively prime. In this paper, we study the arithmetically
Cohen-Macaulay property of the projective closure of $C({\bf a}+m{\bf v})$.

###### Key words and phrases:

Cohen-Macaulayness; Projective closure; Gorenstein monomial curve; Gröbner
basis

###### 1991 Mathematics Subject Classification:

13F65, 13P10, 14M25

##  1\. Introduction

Let ${\bf a}=(a_{1},\ldots,a_{n})$ be a sequence of positive integers with
${\rm gcd}(a_{1},\ldots,a_{n})=1$. Consider the polynomial ring
$R=K[x_{1},\ldots,x_{n}]$ in $n$ variables over a field $K$. Let $C({\bf a})$
be the monomial curve in the $n$-dimensional affine space $\mathbb{A}^{n}(K)$
defined parametrically by $x_{1}=t^{a_{1}},\ldots,x_{n}=t^{a_{n}}$. The toric
ideal of $C({\bf a})$, denoted by $I({\bf a})$, is the kernel of the
$K$-algebra homomorphism $\phi:R\rightarrow K[t]$ given by

$\phi(x_{i})=t^{a_{i}}\ \ \textrm{for all}\ \ 1\leq i\leq n.$

It is known that $I({\bf a})$ is a prime ideal generated by binomials
$x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}-x_{1}^{v_{1}}\cdots x_{n}^{v_{n}}$ such
that $u_{1}a_{1}+\cdots+u_{n}a_{n}=v_{1}a_{1}+\cdots+v_{n}a_{n}$, see [14,
Lemma 4.1].

Let us assume that $a_{n}>a_{i}$ for all $i<n$. The homogenization of $I({\bf
a})$ with respect to the variable $x_{0}$, denoted by $I^{h}({\bf a})$, is the
kernel of the $K$-algebra homomorphism $\psi:R[x_{0}]\rightarrow K[s,t]$ given
by $\psi(x_{0})=s^{a_{n}}$ and $\psi(x_{i})=t^{a_{i}}s^{a_{n}-a_{i}}$, for al
$1\leq i\leq n$. The projective monomial curve $\overline{C({\bf a})}$ in
$\mathbb{P}^{n}$ is given parametrically by
$[s:t]\mapsto[s^{a_{n}}:t^{a_{1}}s^{a_{n}-a_{1}}:\cdots:t^{a_{n-1}}s^{a_{n}-a_{n-1}}:t^{a_{n}}]$.
If $K$ is algebraically closed, then $I^{h}({\bf a})$ is the vanishing ideal
of $\overline{C({\bf a})}$. The projective monomial curve $\overline{C({\bf
a})}$ is called arithmetically Cohen-Macaulay if its vanishing ideal
$I^{h}({\bf a})$ is a Cohen-Macaulay ideal. Arithmetically Cohen–Macaulayness
of projective monomial curves has been studied by many authors; see, for
instance, [2], [6], [10], [11], [12].

Recently, there has been an increased interest in studying the behaviour of
the Betti numbers of a shifted toric ideal $I({\bf a}+m{\bf u})$, where $m\geq
0$ is an integer and ${\bf u}\in\mathbb{N}^{n}$, see [3], [4], [7], [8], [13].
That’s mainly due to a conjecture of J. Herzog and H. Srinivasan saying that
the Betti numbers of $I(a_{1}+m,\ldots,a_{n}+m)$ are eventually periodic in
$m$. The conjecture was proved by T. Vu in [13]. From Theorem 5.7 in [13] it
follows that the monomial curve $\overline{C(a_{1}+m,\ldots,a_{n}+m)}$ is
arithmetically Cohen-Macaulay for all $m>>0$.

Given a Gorenstein non-complete intersection monomial curve $C({\bf a})$ in
$\mathbb{A}^{4}(K)$, P. Gimenez and H. Srinivasan [4, Theorem 4] showed that
there is a vector $(1,1,1,1)\neq{\bf v}\in\mathbb{N}^{4}$ such that for every
$m\geq 0$, the monomial curve $C({\bf a}+m{\bf v})$ is Gorenstein non-complete
intersection whenever the entries of ${\bf a}+m{\bf v}$ are relatively prime.

In this article our aim is to provide necessary and sufficient conditions for
the arithmetically Cohen–Macaulayness of $\overline{C({\bf a}+m{\bf v})}$ by
using a minimal generating set for $I({\bf a}+m{\bf v})$, see Theorems 2.7 and
2.12. This information will allow us to check the arithmetically
Cohen–Macaulay property by just computing a minimal generating set of the
ideal. Our results apply also in the case $m=0$, thus providing necessary and
sufficient conditions for the arithmetically Cohen–Macaulayness of
$\overline{C({\bf a})}$. A basic step towards our goal is to find a Gröbner
basis for the ideal $I({\bf a}+m{\bf v})$ with respect to a degree reverse
lexicographic order, see Propositions 2.5, 2.11. We further determine a
generating set for $I^{h}({\bf a}+m{\bf v})$, see Corollary 2.9 and Corollary
2.16. Finally we provide families of monomial curves $\overline{C({\bf
a}+m{\bf v})}$ which are not arithmetically Cohen-Macaulay, see Examples 2.8,
2.13.

##  2\. Cohen-Macaulay criteria for $\overline{C({\bf a}+m{\bf v})}$

In this section we first recall Bresinsky’s theorem, which gives the explicit
description of $I({\bf a})$ when $C({\bf a})$ is a Gorenstein non-complete
intersection monomial curve in $\mathbb{A}^{4}(K)$.

######  Theorem 2.1.

([1, Theorem 3]) Let $C({\bf a})$ be a monomial curve having the
parametrization

$x_{1}=t^{a_{1}},x_{2}=t^{a_{2}},x_{3}=t^{a_{3}},x_{4}=t^{a_{4}}.$

Then $C({\bf a})$ is a Gorenstein non-complete intersection monomial curve if
and only if $I({\bf a})$ is minimally generated by the set

$\\{x_{1}^{d_{1}}-x_{3}^{d_{13}}x_{4}^{d_{14}},x_{2}^{d_{2}}-x_{1}^{d_{21}}x_{3}^{d_{23}},x_{3}^{d_{3}}-x_{2}^{d_{32}}x_{4}^{d_{34}},x_{1}^{d_{41}}x_{2}^{d_{42}}-x_{4}^{d_{4}},x_{2}^{d_{42}}x_{3}^{d_{13}}-x_{1}^{d_{21}}x_{4}^{d_{34}}\\}$

where the above binomials are unique up to isomorphism, $d_{ij}>0$ and also

$d_{1}=d_{21}+d_{41},d_{2}=d_{32}+d_{42},d_{3}=d_{13}+d_{23},d_{4}=d_{14}+d_{34}.$

######  Remark 2.2.

Bresinsky [1, Theorem 4] also showed that $C({\bf a})$ and $I({\bf a})$ are as
in the previous theorem if and only if
$a_{1}=d_{2}d_{4}d_{13}+d_{42}d_{14}d_{23}$,
$a_{2}=d_{3}d_{4}d_{21}+d_{41}d_{34}d_{23}$,
$a_{3}=d_{1}d_{2}d_{34}+d_{32}d_{21}d_{14}$,
$a_{4}=d_{1}d_{3}d_{42}+d_{13}d_{32}d_{41}$ with ${\rm
gcd}(a_{1},a_{2},a_{3},a_{4})=1$, $d_{i}>1$, $0<d_{ij}<d_{j}$ for $1\leq i\leq
4$, and $d_{1}=d_{21}+d_{41}$, $d_{2}=d_{32}+d_{42}$, $d_{3}=d_{13}+d_{23}$,
$d_{4}=d_{14}+d_{34}$.

Let ${\bf a}=(a_{1},\ldots,a_{4})$ be a sequence of positive integers. A
binomial $B=M-N\in I({\bf a})$ is called indispensable of $I({\bf a})$ if
every system of binomial generators of $I({\bf a})$ contains $B$ or $-B$,
while a monomial $M$ is called indispensable of $I({\bf a})$ if every system
of binomial generators of $I({\bf a})$ contains a binomial $B$ such that $M$
is a monomial of $B$. If $C({\bf a})$ is a Gorenstein non-complete
intersection monomial curve, then $I({\bf a})$ is generated by the
indispensable binomials, see [9, Corollary 3.15].

######  Theorem 2.3.

([4, Theorem 4], [5, Theorem 8]) Let $C({\bf a})$ be a Gorenstein non-complete
intersection monomial curve in $\mathbb{A}^{4}(K)$. Consider the vector ${\bf
v}=(v_{1},\ldots,v_{4})\in\mathbb{N}^{4}$, where
$v_{1}=d_{2}d_{3}-d_{23}d_{32}$, $v_{2}=d_{21}d_{3}+d_{23}d_{34}$,
$v_{3}=d_{2}d_{34}+d_{21}d_{32}$ and $v_{4}=d_{2}d_{3}-d_{23}d_{32}$. For
every $m\geq 0$ the monomial curve $C({\bf a}+m{\bf v})$ is Gorenstein non-
complete intersection whenever the entries of ${\bf a}+m{\bf v}$ are
relatively prime. Furthermore, $I({\bf a}+m{\bf v})$ is minimally generated by
the binomials
$f_{1}:=x_{1}^{d_{1}+m}-x_{3}^{d_{13}}x_{4}^{d_{14}+m},f_{2}:=x_{2}^{d_{2}}-x_{1}^{d_{21}}x_{3}^{d_{23}},f_{3}:=x_{3}^{d_{3}}-x_{2}^{d_{32}}x_{4}^{d_{34}},f_{4}:=x_{1}^{d_{41}+m}x_{2}^{d_{42}}-x_{4}^{d_{4}+m},f_{5}:=x_{2}^{d_{42}}x_{3}^{d_{13}}-x_{1}^{d_{21}}x_{4}^{d_{34}}$.

Throughout this paper $C({\bf a})$ will always be a Gorenstein non-complete
intersection monomial curve in $\mathbb{A}^{4}(K)$. Also we will use the
notation introduced in Theorem 2.3.

######  Remark 2.4.

If $d_{1}\geq d_{13}+d_{14}$ and $d_{21}+d_{34}\leq d_{42}+d_{13}$, then
$d_{4}\leq d_{41}+d_{42}$ since $d_{14}+d_{34}\leq
d_{1}-d_{13}+d_{34}=d_{21}+d_{41}-d_{13}+d_{34}\leq d_{41}+d_{42}$.

######  Proposition 2.5.

Suppose that the following conditions hold.

  1. (1)

$d_{1}\geq d_{13}+d_{14}$.

  2. (2)

$d_{2}\geq d_{21}+d_{23}$.

  3. (3)

$d_{3}\geq d_{32}+d_{34}$.

  4. (4)

$d_{21}+d_{34}\leq d_{42}+d_{13}$.

If the entries of ${\bf a}+m{\bf v}$ are relatively prime, then
$G:=\\{f_{i}|1\leq i\leq 5\\}$ is the reduced Gröbner basis for $I({\bf
a}+m{\bf v})$ with respect to the degree reverse lexicographic order $<$ with
$x_{2}>x_{1}>x_{3}>x_{4}$.

Proof. We have that ${\rm in}_{<}(f_{1})=x_{1}^{d_{1}+m}$, ${\rm
in}_{<}(f_{2})=x_{2}^{d_{2}}$, ${\rm in}_{<}(f_{3})=x_{3}^{d_{3}}$, ${\rm
in}_{<}(f_{4})=x_{1}^{d_{41}+m}x_{2}^{d_{42}}$ and ${\rm
in}_{<}(f_{5})=x_{2}^{d_{42}}x_{3}^{d_{13}}$. Since ${\rm in}_{<}(f_{1})$ and
${\rm in}_{<}(f_{2})$ are relatively prime, we get
$S(f_{1},f_{2})\stackrel{{\scriptstyle G}}{{\longrightarrow}}0$. Similarly
$S(f_{1},f_{3})\stackrel{{\scriptstyle G}}{{\longrightarrow}}0$,
$S(f_{1},f_{5})\stackrel{{\scriptstyle G}}{{\longrightarrow}}0$,
$S(f_{2},f_{3})\stackrel{{\scriptstyle G}}{{\longrightarrow}}0$ and
$S(f_{3},f_{4})\stackrel{{\scriptstyle G}}{{\longrightarrow}}0$. We have that
$S(f_{1},f_{4})=x_{1}^{d_{21}}x_{4}^{d_{4}+m}-x_{2}^{d_{42}}x_{3}^{d_{13}}x_{4}^{d_{14}+m}\stackrel{{\scriptstyle
f_{5}}}{{\longrightarrow}}0$. Also
$S(f_{2},f_{4})=x_{2}^{d_{32}}x_{4}^{d_{4}+m}-x_{1}^{d_{1}+m}x_{3}^{d_{23}}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}x_{2}^{d_{32}}x_{4}^{d_{4}+m}-x_{3}^{d_{3}}x_{4}^{d_{14}+m}\stackrel{{\scriptstyle
f_{3}}}{{\longrightarrow}}0$. Additionally
$S(f_{2},f_{5})=x_{1}^{d_{21}}x_{3}^{d_{3}}-x_{1}^{d_{21}}x_{2}^{d_{32}}x_{4}^{d_{34}}\stackrel{{\scriptstyle
f_{3}}}{{\longrightarrow}}0$. Furthermore,
$S(f_{3},f_{5})=x_{1}^{d_{21}}x_{3}^{d_{23}}x_{4}^{d_{34}}-x_{2}^{d_{2}}x_{4}^{d_{34}}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0$. Finally
$S(f_{4},f_{5})=x_{1}^{d_{1}+m}x_{4}^{d_{34}}-x_{3}^{d_{13}}x_{4}^{d_{4}+m}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}0$. $\square$

In this paper we will make extensive use of the following result.

######  Theorem 2.6.

([6, Theorem 2.2]) Let ${\bf a}=(a_{1},\ldots,a_{4})$ be a sequence of
positive integers with $a_{4}>a_{i}$ for all $i<4$. Denote $<$ any reverse
lexicographic order on $R=K[x_{1},\ldots,x_{4}]$ such that $x_{4}$ is the
smallest variable. The following conditions are equivalent:

  1. (1)

the projective monomial curve $\overline{C({\bf a})}$ is arithmetically Cohen-
Macaulay.

  2. (2)

$x_{4}$ does not divide any minimal generator of ${\rm in}_{<}(I({\bf a}))$.

First we consider the case that $d_{2}\geq d_{21}+d_{23}$.

######  Theorem 2.7.

Let $d_{2}\geq d_{21}+d_{23}$ and $a_{4}+mv_{4}>a_{i}+mv_{i}$ for all $i<4$.
Suppose that the entries of ${\bf a}+m{\bf v}$ are relatively prime. Then
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay if and only
if the following conditions hold:

  1. (1)

$d_{1}\geq d_{13}+d_{14}$;

  2. (2)

$d_{3}\geq d_{32}+d_{34}$;

  3. (3)

$d_{21}+d_{34}\leq d_{42}+d_{13}$.

Proof. ($\Leftarrow$) Suppose that (1), (2) and (3) are true. By Proposition
2.5, $G$ is the reduced Gröbner basis for $I({\bf a}+m{\bf v})$ with respect
to the degree reverse lexicographic order $<$ with $x_{2}>x_{1}>x_{3}>x_{4}$.
Since $x_{4}$ does not divide ${\rm in}_{<}(f_{i})$ for every $1\leq i\leq 5$,
we have, from Theorem 2.6, that $\overline{C({\bf a}+m{\bf v})}$ is
arithmetically Cohen-Macaulay.
($\Rightarrow$) Suppose that $\overline{C({\bf a}+m{\bf v})}$ is
arithmetically Cohen-Macaulay. The binomials $f_{i}$, $1\leq i\leq 5$, are
indispensable of $I({\bf a}+m{\bf v})$, so they belong to the reduced Gröbner
basis for $I({\bf a}+m{\bf v})$ with respect to $<$. But $x_{4}$ does not
divide ${\rm in}_{<}(f_{i})$ for every $1\leq i\leq 5$, thus ${\rm
in}_{<}(f_{1})=x_{1}^{d_{1}+m}$, ${\rm in}_{<}(f_{3})=x_{3}^{d_{3}}$ and ${\rm
in}_{<}(f_{5})=x_{2}^{d_{42}}x_{3}^{d_{13}}$. Therefore (1), (2) and (3) are
true. $\square$

######  Example 2.8.

Let ${\bf a}=(2a^{2}+a-2,2a+1,2a+3,2a^{2}+a-1)$, where $a\geq 2$ is an
integer. Then $C({\bf a})$ is a Gorenstein non-complete intersection monomial
curve and $I({\bf a})$ is minimally generated by $x_{1}^{2}-x_{3}^{a-1}x_{4}$,
$x_{2}^{a+1}-x_{1}x_{3}$, $x_{3}^{a}-x_{2}x_{4}$, $x_{1}x_{2}^{a}-x_{4}^{2}$
and $x_{2}^{a}x_{3}^{a-1}-x_{1}x_{4}$. Here ${\bf
v}=(a^{2}+a-1,a+1,a+2,a^{2}+a-1)$. Let $g$ be the greatest common divisor of
the entries of ${\bf a}+m{\bf v}$. Then $g$ divides
$(2a^{2}+a-1)-(2a^{2}+a-2)$, so $g=1$. By Theorem 2.7 the monomial curve
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay if and only
if $a=2$. Consequently the projective monomial curve
$\overline{C(8+5m,5+3m,7+4m,9+5m)}$ is arithmetically Cohen-Macaulay for every
$m\geq 0$. If $a>2$, then the monomial curve $\overline{C({\bf a}+m{\bf v})}$
is not arithmetically Cohen-Macaulay for every $m\geq 0$.

Given a polynomial $f\in K[x_{1},\ldots,x_{4}]$, we shall denote by $f^{h}$
the homogenization of $f$ with respect to the variable $x_{0}$. The next
corollary provides a generating set for $I^{h}({\bf a}+m{\bf v})$ under the
conditions of Theorem 2.7.

######  Corollary 2.9.

Let $d_{2}\geq d_{21}+d_{23}$ and $a_{4}+mv_{4}>a_{i}+mv_{i}$ for all $i<4$.
Suppose that the entries of ${\bf a}+m{\bf v}$ are relatively prime. If
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay, then
$G^{h}:=\\{f_{i}^{h}|1\leq i\leq 5\\}$ is the reduced Gröbner basis for
$I^{h}({\bf a}+m{\bf v})$ with respect to the degree reverse lexicographic
order $<_{0}$ on $R[x_{0}]$ extended from $R$ with
$x_{2}>_{0}x_{1}>_{0}x_{3}>_{0}x_{4}>_{0}x_{0}$.

Proof. By Proposition 2.5, $G$ is the reduced Gröbner basis for $I({\bf
a}+m{\bf v})$ with respect to $<$. From [6, Lemma 1.1] the set

$G^{h}=\\{x_{1}^{d_{1}+m}-x_{0}^{d_{1}-d_{13}-d_{14}}x_{3}^{d_{13}}x_{4}^{d_{14}+m},x_{2}^{d_{2}}-x_{0}^{d_{2}-d_{21}-d_{23}}x_{1}^{d_{21}}x_{3}^{d_{23}},$
$x_{3}^{d_{3}}-x_{0}^{d_{3}-d_{32}-d_{34}}x_{2}^{d_{32}}x_{4}^{d_{34}},x_{1}^{d_{41}+m}x_{2}^{d_{42}}-x_{0}^{d_{41}+d_{42}-d_{4}}x_{4}^{d_{4}+m},$
$x_{2}^{d_{42}}x_{3}^{d_{13}}-x_{0}^{d_{13}+d_{42}-d_{21}-d_{34}}x_{1}^{d_{21}}x_{4}^{d_{34}}\\}$

is the reduced Gröbner basis for $I^{h}({\bf a}+m{\bf v})$ with respect to
$<_{0}$. $\square$

Next we consider the case that $d_{2}<d_{21}+d_{23}$.

######  Notation 2.10.

We shall denote by $w$ the minimum element of

$\\{l\in\mathbb{N}|d_{1}+m-ld_{21}\leq
0\\}\cup\\{l\in\mathbb{N}|d_{3}-ld_{23}\leq 0\\}.$

Notice that $w\geq 2$.

Consider the binomials

$p_{i}=x_{2}^{(i+1)d_{2}-d_{32}}x_{3}^{d_{3}-(i+1)d_{23}}-x_{1}^{(i+1)d_{21}}x_{4}^{d_{34}},\
0\leq i\leq w-2,$
$q_{i}=x_{1}^{d_{1}+m-(i+1)d_{21}}x_{2}^{(i+1)d_{2}-d_{32}}-x_{3}^{id_{23}}x_{4}^{d_{4}+m},\
0\leq i\leq w-2,$

and

$r=\begin{cases}x_{2}^{wd_{2}-d_{32}}-x_{1}^{wd_{21}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}},\
\ \ \ \ \ \ \ \ \ \ \ \ \textrm{if}\ d_{1}+m-wd_{21}>0\\\
x_{2}^{wd_{2}-d_{32}}-x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-1)d_{23}}x_{4}^{d_{4}+m},\
\textrm{if}\ d_{1}+m-wd_{21}\leq 0\\\ \end{cases}$

Note that $p_{0}=f_{5}$ and $q_{0}=f_{4}$.

######  Proposition 2.11.

Let $d_{2}<d_{21}+d_{23}$ and suppose that the following conditions hold.

  1. (1)

$d_{1}\geq d_{13}+d_{14}$.

  2. (2)

$d_{3}\geq d_{32}+d_{34}$.

  3. (3)

$d_{21}+d_{34}\leq d_{13}+d_{42}$.

  4. (4)

$(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq 0$.

  5. (5)

$(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq 0$.

  6. (6)

$w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq 0$ when $d_{1}+m-wd_{21}>0$.

  7. (7)

$w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq 0$ when
$d_{1}+m-wd_{21}\leq 0$.

If the entries of ${\bf a}+m{\bf v}$ are relatively prime, then

$T=G\cup\\{p_{i}|1\leq i\leq w-2\\}\cup\\{q_{i}|1\leq i\leq w-2\\}\cup\\{r\\}$

is a Gröbner basis for $I({\bf a}+m{\bf v})$ with respect to the degree
reverse lexicographic order $<$ on $R$ with $x_{2}>x_{1}>x_{3}>x_{4}$.

Proof. Notice that the polynomials $p_{i}$, $q_{i}$, $r$ are in the ideal
$I({\bf a}+m{\bf v})$ by computation, for every $1\leq i\leq w-2$. Here ${\rm
in}_{<}(f_{1})=x_{1}^{d_{1}+m}$, ${\rm
in}_{<}(f_{2})=x_{1}^{d_{21}}x_{3}^{d_{23}}$ and ${\rm
in}_{<}(f_{3})=x_{3}^{d_{3}}$. For every $0\leq i\leq w-2$ we have that
$i+1\leq w-1$, thus
$(i+1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}$
since $d_{2}-d_{21}-d_{23}<0$. So
$(i+1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq 0$ and therefore
$(i+1)d_{2}-d_{32}+d_{3}-(i+1)d_{23}\geq(i+1)d_{21}+d_{34}$. Thus ${\rm
in}_{<}(p_{i})=x_{2}^{(i+1)d_{2}-d_{32}}x_{3}^{d_{3}-(i+1)d_{23}}$, for every
$0\leq i\leq w-2$. Since $i+1\leq w-1$ for every $0\leq i\leq w-2$, we have
that
$(i+1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}$
and therefore $(i+1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq 0$.
Thus $d_{1}-(i+1)d_{21}+(i+1)d_{2}-d_{32}\geq id_{23}+d_{4}$, so ${\rm
in}_{<}(q_{i})=x_{1}^{d_{1}+m-(i+1)d_{21}}x_{2}^{(i+1)d_{2}-d_{32}}$ for every
$0\leq i\leq w-2$. Finally ${\rm in}_{<}(r)=x_{2}^{wd_{2}-d_{32}}$. Since
${\rm in}_{<}(f_{1})$ and ${\rm in}_{<}(f_{3})$ are relatively prime, we
deduce that $S(f_{1},f_{3})\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$.
Similarly $S(f_{1},p_{i})\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$, for
every $0\leq i\leq w-2$, $S(f_{1},r)\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, $S(f_{2},r)\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, $S(f_{3},q_{i})\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, for every $0\leq i\leq w-2$, and
$S(f_{3},r)\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$.

We have that
$S(f_{1},f_{2})=x_{1}^{d_{41}+m}x_{2}^{d_{2}}-x_{3}^{d_{3}}x_{4}^{d_{14}+m}\stackrel{{\scriptstyle
f_{4}}}{{\longrightarrow}}x_{2}^{d_{32}}x_{4}^{d_{4}+m}-x_{3}^{d_{3}}x_{4}^{d_{14}+m}\stackrel{{\scriptstyle
f_{3}}}{{\longrightarrow}}0$. We will show that
$S(f_{1},q_{i})\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$, for every
$0\leq i\leq w-2$. It holds that

$S(f_{1},q_{i})=x_{1}^{(i+1)d_{21}}x_{3}^{id_{23}}x_{4}^{d_{4}+m}-x_{2}^{(i+1)d_{2}-d_{32}}x_{3}^{d_{13}}x_{4}^{d_{14}+m}\stackrel{{\scriptstyle
f_{5}}}{{\longrightarrow}}$
$x_{1}^{(i+1)d_{21}}x_{3}^{id_{23}}x_{4}^{d_{4}+m}-x_{1}^{d_{21}}x_{2}^{id_{2}}x_{4}^{d_{4}+m}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.$

Also
$S(f_{2},f_{3})=x_{1}^{d_{21}}x_{2}^{d_{32}}x_{4}^{d_{34}}-x_{2}^{d_{2}}x_{3}^{d_{13}}\stackrel{{\scriptstyle
f_{5}}}{{\longrightarrow}}0$. Next we will show that
$S(f_{2},p_{i})\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$, for every
$0\leq i\leq w-2$. If $0\leq i\leq w-3$, then
$S(f_{2},p_{i})=x_{1}^{(i+2)d_{21}}x_{4}^{d_{34}}-x_{2}^{(i+2)d_{2}-d_{32}}x_{3}^{d_{3}-(i+2)d_{23}}\stackrel{{\scriptstyle
p_{i+1}}}{{\longrightarrow}}0$. Suppose that $i=w-2$ and let
$d_{3}-wd_{23}\leq 0$. Then
$S(f_{2},p_{w-2})=x_{1}^{wd_{21}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}-x_{2}^{wd_{2}-d_{32}}$.
Suppose first that $d_{1}+m-wd_{21}>0$. Then $S(f_{2},p_{w-2})=-r$ and
therefore $S(f_{2},p_{w-2})\stackrel{{\scriptstyle r}}{{\longrightarrow}}0$.
Suppose now that $d_{1}+m-wd_{21}\leq 0$, then
$S(f_{2},p_{w-2})\stackrel{{\scriptstyle
r}}{{\longrightarrow}}x_{1}^{wd_{21}-d_{1}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}f_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}0$. If $d_{3}-wd_{23}>0$, then $d_{1}+m-wd_{21}\leq
0$ by definition of $w$ and therefore
$S(f_{2},p_{w-2})=x_{1}^{wd_{21}}x_{4}^{d_{34}}-x_{2}^{wd_{2}-d_{32}}x_{3}^{d_{3}-wd_{23}}\stackrel{{\scriptstyle
r}}{{\longrightarrow}}x_{1}^{wd_{21}}x_{4}^{d_{34}}-x_{1}^{wd_{21}-d_{1}-m}x_{3}^{d_{13}}x_{4}^{d_{4}+m}=x_{1}^{wd_{21}-d_{1}-m}x_{4}^{d_{34}}f_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}0$. We will show that
$S(f_{2},q_{i})\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$. If $0\leq
i\leq w-3$, then
$S(f_{2},q_{i})=x_{1}^{d_{1}+m-(i+2)d_{21}}x_{2}^{(i+2)d_{2}-d_{32}}-x_{3}^{(i+1)d_{23}}x_{4}^{d_{4}+m}\stackrel{{\scriptstyle
q_{i+1}}}{{\longrightarrow}}0$. Suppose that $i=w-2$. If $d_{1}+m-wd_{21}\leq
0$, then
$S(f_{2},q_{w-2})=x_{2}^{wd_{2}-d_{32}}-x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-1)d_{23}}x_{4}^{d_{4}+m}\stackrel{{\scriptstyle
r}}{{\longrightarrow}}0$. If $d_{1}+m-wd_{21}>0$, then
$S(f_{2},q_{w-2})=x_{3}^{(w-1)d_{23}}x_{4}^{d_{4}+m}-x_{1}^{d_{1}+m-wd_{21}}x_{2}^{wd_{2}-d_{32}}\stackrel{{\scriptstyle
r}}{{\longrightarrow}}-x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}f_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}0$. For every $0\leq i\leq w-2$ we have that

$S(f_{3},p_{i})=x_{1}^{(i+1)d_{21}}x_{3}^{(i+1)d_{23}}x_{4}^{d_{34}}-x_{2}^{(i+1)d_{2}}x_{4}^{d_{34}}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.$

Next we will show that $S(p_{i},p_{j})\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, where $0\leq i\leq w-2$ and $0\leq j\leq w-2$ with
$i<j$. We have that
$S(p_{i},p_{j})=x_{1}^{(i+1)d_{21}}x_{4}^{d_{34}}(x_{1}^{(j-i)d_{21}}x_{3}^{(j-i)d_{23}}-x_{2}^{(j-i)d_{2}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0$. We will show that
$S(p_{i},r)\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$. If
$d_{1}+m-wd_{21}>0$, then

$S(p_{i},r)=x_{1}^{(i+1)d_{21}}x_{4}^{d_{34}}(x_{1}^{(w-i-1)d_{21}}x_{3}^{(w-i-1)d_{23}}-x_{2}^{(w-i-1)d_{2}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.$

Suppose that $d_{1}+m-wd_{21}\leq 0$. Then

$S(p_{i},r)=x_{1}^{wd_{21}-d_{1}-m}x_{4}^{d_{34}}(x_{3}^{d_{3}+(w-i-2)d_{23}}x_{4}^{d_{14}+m}-x_{1}^{d_{1}+m-(w-i-1)d_{21}}x_{2}^{(w-i-1)d_{2}})\stackrel{{\scriptstyle
q_{w-i-2}}}{{\longrightarrow}}$
$x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-i-2)d_{23}}x_{4}^{d_{4}+m}(x_{3}^{d_{3}}-x_{2}^{d_{32}}x_{4}^{d_{34}})\stackrel{{\scriptstyle
f_{3}}}{{\longrightarrow}}0.$

Next we will show that $S(q_{i},q_{j})\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, where $0\leq i\leq w-2$ and $0\leq j\leq w-2$ with
$i<j$. We have that
$S(q_{i},q_{j})=x_{3}^{id_{23}}x_{4}^{d_{4}+m}(x_{1}^{(j-i)d_{21}}x_{3}^{(j-i)d_{23}}-x_{2}^{(j-i)d_{2}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0$. We will prove that
$S(q_{i},r)\stackrel{{\scriptstyle T}}{{\longrightarrow}}0$. If
$d_{1}+m-wd_{21}\leq 0$, then

$S(q_{i},r)=x_{3}^{id_{23}}x_{4}^{d_{4}+m}(x_{1}^{(w-i-1)d_{21}}x_{3}^{(w-i-1)d_{23}}-x_{2}^{(w-i-1)d_{2}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.$

Suppose that $d_{1}+m-wd_{21}>0$. Then

$S(q_{i},r)=x_{1}^{d_{1}+m+(w-i-1)d_{21}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}-x_{2}^{(w-i-1)d_{2}}x_{3}^{id_{23}}x_{4}^{d_{4}+m}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}$
$x_{3}^{id_{23}}x_{4}^{d_{4}+m}(x_{1}^{(w-i-1)d_{21}}x_{3}^{(w-i-1)d_{23}}-x_{2}^{(w-i-1)d_{2}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.$

Finally we show that $S(p_{i},q_{j})\stackrel{{\scriptstyle
T}}{{\longrightarrow}}0$, where $0\leq i\leq w-2$ and $0\leq j\leq w-2$.
Suppose that $i<j$. Then

$S(p_{i},q_{j})=x_{3}^{d_{3}+(j-i-1)d_{23}}x_{4}^{d_{4}+m}-x_{1}^{d_{1}+m-(j-i)d_{21}}x_{2}^{(j-i)d_{2}}x_{4}^{d_{34}}\stackrel{{\scriptstyle
f_{3}}}{{\longrightarrow}}$
$x_{2}^{d_{32}}x_{4}^{d_{34}}(x_{3}^{(j-i-1)d_{23}}x_{4}^{d_{4}+m}-x_{1}^{d_{1}+m-(j-i)d_{21}}x_{2}^{(j-i)d_{2}-d_{32}})\stackrel{{\scriptstyle
q_{j-i-1}}}{{\longrightarrow}}0.$

Assume that $i\geq j$, then

$S(p_{i},q_{j})=x_{2}^{(i-j)d_{2}}x_{3}^{d_{3}-(i-j+1)d_{23}}x_{4}^{d_{4}+m}-x_{1}^{d_{1}+m+(i-j)d_{21}}x_{4}^{d_{34}}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}$
$x_{3}^{d_{3}-(i-j+1)d_{23}}x_{4}^{d_{4}+m}(x_{2}^{(i-j)d_{2}}-x_{1}^{(i-j)d_{21}}x_{3}^{(i-j)d_{23}})\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}0.\ \ \ \ \ \square$

######  Theorem 2.12.

Let $d_{2}<d_{21}+d_{23}$. Suppose that $a_{4}+mv_{4}>a_{i}+mv_{i}$ for all
$i<4$ and the entries of ${\bf a}+m{\bf v}$ are relatively prime. Then
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay if and only
if the following conditions hold:

  1. (1)

$d_{1}\geq d_{13}+d_{14}$;

  2. (2)

$d_{3}\geq d_{32}+d_{34}$;

  3. (3)

$d_{21}+d_{34}\leq d_{13}+d_{42}$;

  4. (4)

$(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq 0$;

  5. (5)

$(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq 0$;

  6. (6)

$w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}\geq 0$ when $d_{1}+m-wd_{21}>0$;

  7. (7)

$w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}\geq 0$ when
$d_{1}+m-wd_{21}\leq 0$.

Proof. ($\Leftarrow$) Suppose that (1), (2), (3), (4), (5), (6) and (7) are
true. By Proposition 2.11, $T$ is a Gröbner basis for $I({\bf a}+m{\bf v})$
with respect to the degree reverse lexicographic order $<$ with
$x_{2}>x_{1}>x_{3}>x_{4}$. Since $x_{4}$ does not divide the initial monomial
of any binomial in $T$, we have, from Theorem 2.6, that $\overline{C({\bf
a}+m{\bf v})}$ is arithmetically Cohen-Macaulay.
($\Rightarrow$) Suppose that $\overline{C({\bf a}+m{\bf v})}$ is
arithmetically Cohen-Macaulay. Since the binomials $f_{1}$, $f_{3}$ and
$f_{5}$ are indispensable of $I({\bf a}+m{\bf v})$, they belong to the reduced
Gröbner basis of $I({\bf a}+m{\bf v})$ with respect to $<$. Consequently (1),
(2) and (3) are true. Suppose that
$(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}<0$. Then the binomial
$p_{w-2}$ belongs to $I({\bf a}+m{\bf v})$ and ${\rm
in}_{<}(p_{w-2})=x_{1}^{(w-1)d_{21}}x_{4}^{d_{34}}$. Note that
$(w-1)d_{21}<d_{1}+m$ by definition of $w$. Since $\overline{C({\bf a}+m{\bf
v})}$ is arithmetically Cohen-Macaulay, there is a binomial $B\in I({\bf
a}+m{\bf v})$ such that ${\rm in}_{<}(B)$ divides $x_{1}^{(w-1)d_{21}}$, a
contradiction to the fact that the monomial $x_{1}^{d_{1}+m}$ is
indispensable. Assume that
$(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}<0$. Then the binomial
$q_{w-2}$ belongs to $I({\bf a}+m{\bf v})$ and ${\rm
in}_{<}(q_{w-2})=x_{3}^{(w-2)d_{23}}x_{4}^{d_{4}+m}$. Note that
$(w-2)d_{23}<d_{3}$ by definition of $w$. Since $\overline{C({\bf a}+m{\bf
v})}$ is arithmetically Cohen-Macaulay, there is a binomial $u\in I({\bf
a}+m{\bf v})$ such that ${\rm in}_{<}(u)$ divides $x_{3}^{(w-2)d_{23}}$, a
contradiction to the fact that the monomial $x_{3}^{d_{3}}$ is indispensable.
Suppose that $d_{1}+m-wd_{21}>0$ and let
$w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}<0$. Then the binomial $r$ belongs
to $I({\bf a}+m{\bf v})$ and ${\rm
in}_{<}(r)=x_{1}^{wd_{21}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}$. Since
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay, there is a
binomial $B\in I({\bf a}+m{\bf v})$ such that ${\rm in}_{<}(B)$ divides
$x_{1}^{wd_{21}}x_{3}^{wd_{23}-d_{3}}$. Thus
$B=x_{1}^{b_{1}}x_{3}^{b_{3}}-x_{1}^{c_{1}}x_{2}^{b_{2}}x_{3}^{c_{3}}x_{4}^{b_{4}}$
with ${\rm in}_{<}(B)=x_{1}^{b_{1}}x_{3}^{b_{3}}$, where $b_{2}\neq 0$ or/and
$b_{4}\neq 0$, $b_{1}\leq wd_{21}$ and $b_{3}\leq wd_{23}-d_{3}$. Note that
$wd_{21}<d_{1}+m$ and $wd_{23}-d_{3}<d_{23}$ by definition of $w$, so
$b_{1}<d_{1}+m$ and $b_{3}<d_{23}$. Moreover $b_{3}<d_{3}$ since
$d_{23}<d_{3}$. But $B\in I({\bf a}+m{\bf v})$ and $I({\bf a}+m{\bf v})$ is
generated by the set $\\{f_{1},\ldots,f_{5}\\}$, therefore there are
polynomials $A_{i}\in K[x_{1},\ldots,x_{4}]$, $1\leq i\leq 5$, such that

$B=A_{1}(x_{1}^{d_{1}+m}-x_{3}^{d_{13}}x_{4}^{d_{14}+m})+A_{2}(x_{2}^{d_{2}}-x_{1}^{d_{21}}x_{3}^{d_{23}})+A_{3}(x_{3}^{d_{3}}-x_{2}^{d_{32}}x_{4}^{d_{34}})+$
$A_{4}(x_{1}^{d_{41}+m}x_{2}^{d_{42}}-x_{4}^{d_{4}+m})+A_{5}(x_{2}^{d_{42}}x_{3}^{d_{13}}-x_{1}^{d_{21}}x_{4}^{d_{34}}).$

In the previous equation let $x_{2}=0$ and $x_{4}=0$. Then
$x_{1}^{b_{1}}x_{3}^{b_{3}}=A_{1}(x_{1},0,x_{3},0)x_{1}^{d_{1}+m}-A_{2}(x_{1},0,x_{3},0)x_{1}^{d_{21}}x_{3}^{d_{23}}+A_{3}(x_{1},0,x_{3},0)x_{3}^{d_{3}}$,
thus ${\rm in}_{<}(B)$ is divided by at least one of the monomials
$x_{1}^{d_{1}+m}$, $x_{1}^{d_{21}}x_{3}^{d_{23}}$ and $x_{3}^{d_{3}}$, a
contradiction. Suppose that $d_{1}+m-wd_{21}\leq 0$ and let
$w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}<0$. Then $r$ belongs to
$I({\bf a}+m{\bf v})$ and ${\rm
in}_{<}(r)=x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-1)d_{23}}x_{4}^{d_{4}+m}$. Since
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay, there is a
binomial $u\in I({\bf a}+m{\bf v})$ such that ${\rm in}_{<}(u)$ divides
$x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-1)d_{23}}$. Thus ${\rm
in}_{<}(u)=x_{1}^{e_{1}}x_{3}^{e_{3}}$ where $e_{1}\leq wd_{21}-d_{1}-m$ and
$e_{3}\leq(w-1)d_{23}$. Note that $wd_{21}-d_{1}-m<d_{21}$ and
$(w-1)d_{23}<d_{3}$ by definition of $w$, so $e_{1}<d_{21}$ and $e_{3}<d_{3}$.
Moreover $e_{1}<d_{1}+m$ since $d_{21}<d_{1}+m$. But $u\in I({\bf a}+m{\bf
v})$ and $I({\bf a}+m{\bf v})$ is generated by the set
$\\{f_{1},\ldots,f_{5}\\}$, therefore ${\rm in}_{<}(u)$ is divided by at least
one of the monomials $x_{1}^{d_{1}+m}$, $x_{1}^{d_{21}}x_{3}^{d_{23}}$ and
$x_{3}^{d_{3}}$, a contradiction. $\square$

######  Example 2.13.

Let ${\bf a}=(19,29,26,43)$, then $C({\bf a})$ is a Gorenstein non-complete
intersection monomial curve and $I({\bf a})$ is minimally generated by
$x_{1}^{5}-x_{3}^{2}x_{4},x_{2}^{4}-x_{1}^{2}x_{3}^{3},x_{3}^{5}-x_{2}^{3}x_{4},x_{1}^{3}x_{2}-x_{4}^{2},x_{2}x_{3}^{2}-x_{1}^{2}x_{4}$.
Here ${\bf v}=(11,13,10,11)$, $w=2$ and $d_{1}+m-wd_{21}=m+1>0$. Suppose first
that $m\leq 7$, then $43+11m={\rm max}\\{19+11m,29+13m,26+10m,43+11m\\}$. We
have that $w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}=-1<0$, so from Theorem
2.12 the monomial curve $\overline{C({\bf a}+m{\bf v})}$ is not arithmetically
Cohen-Macaulay for $m\leq 7$ whenever ${\rm
gcd}(19+11m,29+13m,26+10m,43+11m)=1$. Assume that $m>7$, then $29+13m={\rm
max}\\{19+11m,29+13m,26+10m,43+11m\\}$. Let ${\bf b}=(b_{1},\ldots,b_{4})$
with $b_{1}=26+10m$, $b_{2}=19+11m$, $b_{3}=43+11m$, $b_{4}=29+13m$. Note that
$b_{4}={\rm max}\\{b_{1},\ldots,b_{4}\\}$. Then $I({\bf b})$ is minimally
generated by
$x_{1}^{5}-x_{3}x_{4}^{3},x_{2}^{5+m}-x_{1}^{2}x_{3}^{1+m},x_{3}^{2+m}-x_{2}^{3+m}x_{4},x_{1}^{3}x_{2}^{2}-x_{4}^{4},x_{2}^{2}x_{3}-x_{1}^{2}x_{4}$.
Since $x_{3}^{2+m}-x_{2}^{3+m}x_{4}\in I({\bf b})$, we have, from Theorem 2.7,
that for every $m>7$ the monomial curve $\overline{C({\bf b})}$ is not
arithmetically Cohen-Macaulay whenever ${\rm gcd}(b_{1},b_{2},b_{3},b_{4})=1$.

######  Example 2.14.

Let ${\bf a}=(a^{2}+5a,7a+6,6a+1,3a^{2}+3a-2)$ where $a>2$ is an even integer.
Then $C({\bf a})$ is a Gorenstein non-complete intersection monomial curve and
$I({\bf a})$ is minimally generated by
$x_{1}^{3}-x_{3}^{2}x_{4},x_{2}^{a}-x_{1}x_{3}^{a},x_{3}^{a+2}-x_{2}x_{4}^{2},x_{1}^{2}x_{2}^{a-1}-x_{4}^{3},x_{2}^{a-1}x_{3}^{2}-x_{1}x_{4}^{2}$.
Here ${\bf v}=(a^{2}+a,3a+2,2a+1,a^{2}+a)$, $w=2$ and $d_{1}+m-wd_{21}=m+1>0$.
It holds that $w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}=a-3>0$ and
$(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}=a-2>0$, so, from Theorem
2.12, for every $m\geq 0$ the monomial curve $\overline{C({\bf a}+m{\bf v})}$
is arithmetically Cohen-Macaulay whenever the entries of ${\bf a}+m{\bf v}$
are relatively prime.

######  Example 2.15.

Let ${\bf a}=(1191,1239,582,2303)$, then $C({\bf a})$ is a Gorenstein non-
complete intersection monomial curve and $I({\bf a})$ is minimally generated
by $x_{1}^{16}-x_{3}^{9}x_{4}^{6}$, $x_{2}^{11}-x_{1}^{9}x_{3}^{5}$,
$x_{3}^{14}-x_{2}x_{4}^{3}$, $x_{1}^{7}x_{2}^{10}-x_{4}^{9}$ and
$x_{2}^{10}x_{3}^{9}-x_{1}^{9}x_{4}^{3}$. Notice that ${\bf
v}=(149,141,42,149)$. Suppose first that $m\leq 2$, then $w=2$ and
$d_{1}+m-wd_{21}=m-2\leq 0$. We have that
$w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}=5>0$ and
$(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}=7>0$. By Theorem 2.12, for
every $m\leq 2$ the monomial curve $\overline{C({\bf a}+m{\bf v})}$ is
arithmetically Cohen-Macaulay whenever the entries of ${\bf a}+m{\bf v}$ are
relatively prime. Suppose now that $m>2$, then $w=3$ and
$d_{1}+m-wd_{21}=m-11$. Assume that $m\leq 11$, then $d_{1}+m-wd_{21}\leq 0$.
We have that $w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}=2>0$ and
$(w-1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}=4>0$. By Theorem 2.12, the
monomial curve $\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-
Macaulay for every $2<m\leq 11$ whenever the entries of ${\bf a}+m{\bf v}$ are
relatively prime. Suppose that $m>11$, then $d_{1}+m-wd_{21}>0$. We have that
$w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}=1>0$ and
$(w-1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}=5>0$. By Theorem 2.12,
the monomial curve $\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-
Macaulay for every $m>11$ whenever the entries of ${\bf a}+m{\bf v}$ are
relatively prime.

Finally we provide a generating set for $I^{h}({\bf a}+m{\bf v})$ under the
conditions of Theorem 2.12.

######  Corollary 2.16.

Let $d_{2}<d_{21}+d_{23}$. Suppose that $a_{4}+mv_{4}>a_{i}+mv_{i}$ for all
$i<4$ and the entries of ${\bf a}+m{\bf v}$ are relatively prime. If
$\overline{C({\bf a}+m{\bf v})}$ is arithmetically Cohen-Macaulay, then
$T^{h}=G^{h}\cup\\{p_{i}^{h}|1\leq i\leq w-2\\}\cup\\{q_{i}^{h}|1\leq i\leq
w-2\\}\cup\\{r^{h}\\}$ is a Gröbner basis for $I^{h}({\bf a}+m{\bf v})$ with
respect to the degree reverse lexicographic order $<_{0}$ on $R[x_{0}]$
extended from $R$ with $x_{2}>_{0}x_{1}>_{0}x_{3}>_{0}x_{4}>_{0}x_{0}$.

Proof. We have that

$r^{h}=x_{2}^{wd_{2}-d_{32}}-x_{0}^{w(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}}x_{1}^{wd_{21}-d_{1}-m}x_{3}^{(w-1)d_{23}}x_{4}^{d_{4}+m}$

when $d_{1}+m-wd_{21}\leq 0$, while

$r^{h}=x_{2}^{wd_{2}-d_{32}}-x_{0}^{w(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}}x_{1}^{wd_{21}}x_{3}^{wd_{23}-d_{3}}x_{4}^{d_{34}}$

when $d_{1}+m-wd_{21}>0$. From Proposition 2.11 the set $T$ is a Gröbner basis
for $I({\bf a}+m{\bf v})$ with respect to $<$. By [6, Lemma 1.1],
$T^{h}=\\{f_{i}^{h}|1\leq i\leq
5\\}\cup\\{r^{h}\\}\cup\\{x_{2}^{(i+1)d_{2}-d_{32}}x_{3}^{d_{3}-(i+1)d_{23}}-x_{0}^{(i+1)(d_{2}-d_{21}-d_{23})+d_{3}-d_{32}-d_{34}}x_{1}^{(i+1)d_{21}}x_{4}^{d_{34}}|1\leq
i\leq
w-2\\}\cup\\{x_{1}^{d_{1}+m-(i+1)d_{21}}x_{2}^{(i+1)d_{2}-d_{32}}-x_{0}^{(i+1)(d_{2}-d_{21}-d_{23})+d_{1}+d_{23}-d_{4}-d_{32}}x_{3}^{id_{23}}x_{4}^{d_{4}+m}|1\leq
i\leq w-2\\}$ is a Gröbner basis for $I^{h}({\bf a}+m{\bf v})$ with respect to
$<_{0}$. $\square$

Acknowledgment. The author would like to thank the referee for careful reading
and helpful comments.

## References

  * [1] H. Bresinsky. Symmetric semigroups of integers generated by 4 elements, Manuscripta Math. 17 (3) (1975), 205–219.
  * [2] M. P. Cavaliere, G. Niesi. On monomial curves and Cohen-Macaulay type, Manuscripta Math. 42 (1983), no. 2-3, 147–159.
  * [3] R. Conaway, F. Gotti, J. Horton, C. O’Neill, R. Pelayo, M. Pracht, B. Wissman. Minimal presentations of shifted numerical monoids, Internat. J. Algebra Comput. 28 (2018), 53-68.
  * [4] P. Gimenez, H. Srinivasan. A note on Gorenstein monomial curves, Bull. Braz. Math. Soc. (N.S.) 45 (2014), no. 4, 671–678.
  * [5] P. Gimenez, H. Srinivasan. Structure of some numerical semigroup rings, Banach Center Publications 121 (2020), 53-61.
  * [6] J. Herzog, D. Stamate. Cohen–Macaulay criteria for projective monomial curves via Gröbner bases, Acta Math. Vietnam. 44 (2019), 51–64.
  * [7] A.V. Jayanthan, H. Srinivasan. Periodic occurence of complete intersection monomial curves, Proc. Amer. Math. Soc. 141 (2013), 4199–4208.
  * [8] A. Katsabekis. Complete intersection monomial curves and the Cohen-Macaulayness of their tangent cones, Algebra Colloq. 26 (2019), no. 4, 629–642.
  * [9] A. Katsabekis, I. Ojeda. An indispensable classification of monomial curves in $\mathbb{A}^{4}(\mathbb{K})$, Pacific J. Math. 268 (2014), 95–116.
  * [10] L. Reid, L. G. Roberts. Non-Cohen-Macaulay projective monomial curves, J. Algebra 291 (2005), no. 1, 171–186.
  * [11] J. Saha, I. Sengupta, P. Srivastava. Betti sequence of the projective closure of affine monomial curves, J. Symbolic Comput. 119 (2023), 101–111.
  * [12] T. Se, S. J. Grant. The Cohen-Macaulay property of affine semigroup rings in dimension 2, Comm. Algebra 47 (2019), no. 7, 2979–2994.
  * [13] T. Vu. Periodicity of Betti numbers of monomial curves, J. Algebra 418 (2014), 66–90.
  * [14] B. Sturmfels. Gröbner Bases and Convex Polytopes, University Lecture Series Vol. 8. Providence, RI, USA: American Mathematical Society, 1996.