# A Long-lived and Efficient Optomechanical Memory for Light

Mads Bjerregaard Kristensen  Nenad Kralj  current address: Institute for
Gravitational Physics, Leibniz Universität Hannover, Callinstraße 36, 30167
Hannover, Germany  Eric Langman  Albert Schliesser
<EMAIL_ADDRESS>Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Center for Hybrid Quantum Networks (Hy-Q), Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

###### Abstract

We demonstrate a memory for light based on optomechanically induced
transparency. We achieve a long storage time by leveraging the ultra-low
dissipation of a soft-clamped mechanical membrane resonator, which oscillates
at MHz frequencies. At room temperature, we demonstrate a lifetime
$T_{1}\approx$23\text{\,}\mathrm{ms}$$ and a retrieval efficiency $\eta\approx
40\%$ for classical coherent pulses. We anticipate storage of quantum light to
be possible at moderate cryogenic conditions ($T\approx 10\,\mathrm{K}$). Such
systems could find applications in emerging quantum networks, where they can
serve as long-lived optical quantum memories by storing optical information in
a phononic mode.

##  I Introduction

Optical quantum memories Lvovsky  _et al._ (2009) are an essential
prerequisite for many exciting prospects of quantum information science, such
as quantum cryptography Gisin  _et al._ (2002) over long distances and
quantum-coherent interconnection of quantum computers Kimble (2008).
Pioneering theoretical and experimental work in the atomic physics community
has studied light storage in long-lived internal states of atoms, notably via
electromagnetically induced transparency (EIT) Fleischhauer and Lukin (2000).
In EIT, an optical control field dynamically activates the conversion of an
optical signal to an atomic excitation, and vice versa, with the control field
switched off during storage. For the specific purpose of entanglement
distribution, the Duan-Lukin-Cirac-Zoller (DLCZ) protocol Duan  _et al._
(2001) presents an alternative. However, due to the probabilistic nature and
internal generation of excitations, it cannot be used to read an arbitrary
optical state into a memory for later retrieval.

Optical quantum memories building on these (and related) protocols have been
implemented in a range of atomic systems that include hot and cold atomic
vapors, single atoms and ions, as well as solid-state crystals doped with
rare-earth ions (see Lvovsky  _et al._ (2009) and references therein). One
disadvantage of atomic and ionic memories for light is that they are tied to
the natural transition frequencies of the employed species. This is often
imcompatible with desired operation wavelengths, e.g. within a telecom band
where optical losses in fibers are low. Whereas this shortcoming can in
principle be addressed through non-linear optical frequency conversion Zaske
_et al._ (2012), this comes at a significant experimental overhead in terms of
cost, space and power.

Optomechanical systems Aspelmeyer  _et al._ (2014) offer an alternative
implementation of optical quantum memories, the properties of which can be
freely engineered. They provide a coherent, wavelength-versatile interface
between light and a resonant mechanical mode. As a stationary quantum system
with a potentially long coherence time, the latter acts as the memory.
Optomechanical memories employing the DLCZ protocol have indeed been proposed
Galland  _et al._ (2014) and demonstrated recently Wallucks  _et al._ (2020);
Fiaschi  _et al._ (2021), achieving coherence times of $10100$
$\mathrm{\SIUnitSymbolMicro s}$ and lifetimes
$T_{1}\approx$2\text{\,}\mathrm{ms}$$ at millikelvin temperatures.

Figure 1: Steady-state OMIT. a) An optomechanical cavity mode with frequency
$\Omega_{\mathrm{cav}}$ and linewidth $\kappa$ is pumped near the red sideband
$\Delta=\Omega_{\mathrm{L}}-\Omega_{\mathrm{cav}}\approx-\Omega_{\mathrm{m}}$.
An externally applied modulation at $\Omega_{\mathrm{sig}}$ provides a signal
to store. b) The optical ($n$) and mechanical ($m$) excitations form a
$\Lambda$-system which can facilitate storage when the two-photon resonance
condition $\delta=\Omega_{\mathrm{sig}}-\Omega_{\mathrm{m}}\approx 0$ is met.
c) Experimental data (circles) for coarse, broadband (top) and fine,
narrowband (bottom) sweeps of $\Omega_{\mathrm{mod}}$, with fits (solid blue)
to OMIT theory. The broad sweep outlines the cavity response, while the narrow
sweep shows interference between the mechanical sideband and applied signal
around $\delta=0$.

Conveniently, optomechanical systems also exhibit a close analog to atomic
EIT, known as optomechanically induced transparency (OMIT) Weis  _et al._
(2010), see fig. 1. OMIT results from the beamsplitter-type process

$H_{\mathrm{int}}=-\hbar g_{0}\alpha\left(ab^{\dagger}+a^{\dagger}b\right),$
(1)

which describes interconversion of optical (ladder operators $a$,
$a^{\dagger}$) and mechanical (ladder operators $b$, $b^{\dagger}$)
excitations. This process dominates optomechanical interactions when the
system is pumped by a coherent field (amplitude $\alpha$) at the red motional
sideband, and the optical cavity linewidth $\kappa$ is smaller than the
mechanical resonance frequency $\Omega_{\mathrm{m}}$, the so-called resolved-
sideband regime. The vacuum optomechanical coupling rate $g_{0}$ is fixed by
geometry and material properties. It was recognized early on Schliesser
(2009); Weis  _et al._ (2010); Safavi-Naeini  _et al._ (2011) that OMIT
enables dynamic control of a storage and retrieval process by tuning the
coherent amplitude $\alpha$, just as in EIT. OMIT-based memories have been
demonstrated in silica microspheres Fiore  _et al._ (2011) and diamond
microdisks Lake  _et al._ (2021), albeit with relatively short memory
lifetimes (µs-timescales) and without quantifying storage efficiency. Related
experiments with propagating acoustic phonons suffer from even shorter
lifetimes Stiller  _et al._ (2020). Millisecond-level storage times have yet
only been demonstrated with microwave (instead of optical) pulses stored in
mechanical systems held at millikelvin temperatures Palomaki  _et al._ (2013).

In this work, we present an OMIT-type optical memory that operates at telecom-
wavelengths. It is based on a long-lived mechanical mode of a soft-clamped
membrane resonator Tsaturyan  _et al._ (2017) embedded in a Fabry-Pérot cavity
Jayich  _et al._ (2008). We characterize and model the performance as an
optical memory by injecting coherent state signals and measuring retrieval
efficiency as a function of bandwidth, two-photon detuning and storage time.
Ultimately, we demonstrate the longest lifetime and highest efficiency of any
optomechanical memory to date.

##  II The experimental platform

Fig. 2 shows the mechanical memory, which is based on a
$20\text{\,}\mathrm{nm}$ thin membrane of stoichiometric LPCVD
$\textrm{Si}_{3}\textrm{N}_{4}$, with a large in-plane stress
($\bar{\sigma}\approx$1.15\text{\,}\mathrm{GPa}$$ before release). It is
patterned with a honeycomb phononic crystal lattice, resulting in several
bandgaps for out-of-plane vibrational modes Tsaturyan  _et al._ (2017). A
central defect breaks the crystal symmetry and spatially localizes resonance
modes. Careful design of the defect geometry places the desired defect mode
frequency in the second bandgap, allowing a higher frequency that well
fulfills the resolved-sideband condition $\Omega_{\mathrm{m}}>\kappa$.

Figure 2:  Soft-clamped membrane resonator. a) The mechanical resonator. From
left to right, a micrograph of the device used, a simulated displacement field
of the mode we work with, and a photograph of an exemplary mechanical
resonator. b) In black, simulated band diagram, showing a fundamental (red)
and higher-order (green) bandgap; inset: first Brillouin zone. c) Mechanical
displacement spectrum around the second bandgap, showing distinct localized
modes, annotated with displacement profiles (the black peak at
$2.36\text{\,}\mathrm{MHz}$ is an external phase modulation). d) Amplitude-
decay of the mode of interest, exhibiting $Q=$100(3)\text{\times}{10}^{6}$$ at
$\Omega_{\textrm{m}}/2\pi=$2.4\text{\,}\mathrm{MHz}$$.

Thanks to the phenomena of dissipation dilution and soft clamping Tsaturyan
_et al._ (2017), the defect modes of such membranes have very low dissipation
rates $\Gamma_{\mathrm{m}}$ and correspondingly high Q-factors
$Q=\Omega_{\mathrm{m}}/\Gamma_{\mathrm{m}}$. In fig. 2d, we show a room-
temperature ringdown measurement demonstrating
$Q=$100(3)\text{\times}{10}^{6}$$ at
$\Omega_{\mathrm{m}}/2\pi=$2.4\text{\,}\mathrm{MHz}$$. Working with a
localized mode in the higher-order bandgap enables a high mechanical frequency
without compromising the size of the working area.

Figure 3: Membrane-in-the-middle cavity optomechanics. a) Simplified
experimental schematic. All beamsplitters are polarizing. b) Mechanical memory
bandwidth $\Gamma_{\textrm{eff}}$ as a function of science laser detuning
$\Delta$ at fixed input power. The black line is a single free parameter
($g_{0}$) fit to a model of dynamical backaction.

We place the membrane inside a single-sided Fabry-Pérot cavity of length
$L_{\textrm{cav}}\approx$5.5\text{\,}\mathrm{mm}$$ and high finesse
$\mathcal{F}=13600$, with a corresponding linewidth
$\kappa/2\pi=$2.1\text{\,}\mathrm{MHz}$$. By positioning the membrane between
a node and an antinode of the standing optical wave, the “membrane-in-the-
middle” linearly shifts the cavity resonance frequency upon displacement,
leading to the standard dispersive optomechanical coupling Aspelmeyer  _et
al._ (2014); Jayich  _et al._ (2008).

To use the membrane as a memory, we couple two lasers to the cavity, cf. Fig.
3. A strong “science” laser, after being frequency-shifted with an acousto-
optic modulator (AOM), provides the “control” field of amplitude $\alpha$. It
is red-detuned from a cavity mode, and drives the process of eq. (1) that
facilitates the conversion between photons and phonons. A second, weaker,
“lock” laser is stabilized to the science laser at a frequency offset near the
cavity free spectral range
$\Delta\nu_{\textrm{offset}}\approx\nu_{\textrm{FSR}}$. The cavity, in turn,
is stabilized to the lock laser with a Pound-Drever-Hall scheme. That is, the
two lasers address different longitudinal cavity modes, and additionally have
orthogonal polarizations. This setup allows us to stabilize the control field
to detunings $-\Delta\approx\Omega_{\mathrm{m}}>\kappa$ outside the cavity
bandwidth, as required for sideband-resolved optomechanics. Furthermore, using
the AOM, we can switch the control field off and on again at the same
detuning, crucially only having the weaker lock laser on during the storage.
The experimental apparatus is described in more detail in the supplementary
information.

We measure the effective memory bandwidth by recording mechanical noise
spectra as we change the science laser detuning $\Delta$. For the effective
mechanical linewidth $\Gamma_{\mathrm{eff}}$ we observe fair agreement with a
dynamical back-action (DBA) model that accounts also for the cavity
susceptibility, cf. fig. 3b (more details in supplementary). The largest
optical broadening corresponds to a classical coorperativity
$\mathcal{C}=4g_{0}^{2}\alpha^{2}/\kappa\Gamma_{\mathrm{m}}>$4\text{\times}{10}^{4}$$.
By simultaneously measuring the detuning, transmitted power and the optical
broadening, we can extract $g_{0}/2\pi=$1.0(1)\text{\,}\mathrm{Hz}$$ from the
DBA fit.

##  III Light storage and retrieval

To characterize the optomechanical quantum memory, we perform several
experiments with optical input pulses, see fig. 4. Using the AOM, the control
field is switched on for a first (write) and second (read) pulses of variable
millisecond duration, separated by a storage time $T_{\mathrm{delay}}$, during
which the control field is off. During the write pulse, an electro-optic
modulator (EOM) generates a “signal” field as a sideband of the control field.
The signal field falls on cavity resonance, and therefore to the center of the
OMIT window.

Figure 4:  Optomechanical light storage. a) Pulse sequence diagram. We drive
the EOM with exponentially rising sinusoids, while the science laser is on.
After waiting a time $T_{\mathrm{delay}}$, we switch the AOM on again and look
at the retrieved mechanical signal. b) By lock-in detection around the
mechanical sideband, we record the magnitude of the reflected science laser.
Here we show data for fixed $\Gamma_{\mathrm{eff}}$ and increasing
$\Gamma_{\mathrm{sig}}$ from light to dark blue, the legend showing
$\Gamma_{\mathrm{eff}}/\Gamma_{\mathrm{sig}}$.

This is the field that will be stored as a mechanical excitation. We choose
exponentially rising sinusoid waveforms

$V_{\mathrm{EOM}}(t)=V_{\mathrm{max}}\sin(\Omega_{\mathrm{sig}}t)e^{(\Gamma_{\mathrm{sig}}t/2)}H(-t)$
(2)

with programmable amplitude $V_{\textrm{max}}$, bandwidth
$\Gamma_{\textrm{sig}}$, frequency $\Omega_{\textrm{sig}}$, and the Heavyside
step function $H(t)$. This gives rise to a signal field

$s_{\mathrm{sig}}^{\mathrm{in}}(t)=i\beta
s_{0}\sin(\Omega_{\mathrm{sig}}t)\mathrm{e}^{\Gamma_{\mathrm{sig}}t/2}H(-t),$
(3)

impinging on the cavity. Here $\beta=\pi V_{\mathrm{max}}/V_{\mathrm{\pi}}\ll
1$ is the modulation depth, $V_{\mathrm{\pi}}$ denoting the modulator half-
wave voltage.

When the read control light pulse probes the cavity, the EOM is off. Yet the
control field acquires a sideband due to the mechanical motion, regenerating a
signal field $s_{\mathrm{sig}}^{\mathrm{out}}(t)$ at the cavity output. A
detector measuring optical power emerging from the cavity then retrieves a
transient beat
$V_{\mathrm{ret}}(t)\propto|\alpha^{*}s_{\mathrm{sig}}^{\mathrm{out}}(t)|$
oscillating at the modulation frequency $\Omega_{\textrm{sig}}$. On the time
scale of the state-swap process, the cavity responds fast,
$g_{0}\alpha\ll\kappa$, meaning we can adiabatically eliminate the mode $a$
and describe $s_{\mathrm{sig}}^{\mathrm{out}}$ in terms of $b$ using a
Heisenberg-Langevin equation and input-output relation:

$\displaystyle\dot{b}(t)=$
$\displaystyle-\left(\frac{\Gamma_{\mathrm{eff}}}{2}-i\delta\right)b(t)+\sqrt{\eta_{\mathrm{c}}\Gamma_{\mathrm{opt}}}s_{\mathrm{sig}}^{\mathrm{in}}(t),$
(4) $\displaystyle s_{\mathrm{sig}}^{\mathrm{out}}(t)=$ $\displaystyle
s_{\mathrm{sig}}^{\mathrm{in}}(t)-i\sqrt{\eta_{\mathrm{c}}\Gamma_{\mathrm{opt}}}b(t).$
(5)

Here, $\Gamma_{\mathrm{eff}}=\Gamma_{\mathrm{m}}+\Gamma_{\mathrm{opt}}$ with
$\Gamma_{\mathrm{opt}}$ the optomechanical broadening, which is zero during
the storage period. The signal input $s_{\mathrm{sig}}^{\mathrm{in}}$ is non-
zero only during the write pulse, and
$\delta=\Omega_{\mathrm{sig}}-\Omega_{\mathrm{m}}$ the two-photon detuning. We
can therefore obtain the magnitude of the retrieved signal field
$|s_{\mathrm{sig}}^{\mathrm{out}}(t)|$ by demodulating the photodetector
signal $V_{\mathrm{ret}}(t)$ at the frequency $\Omega_{\textrm{sig}}$ with a
lock-in amplifier. Exemplary time traces of such storage experiments are shown
in fig. 4b.

To quantify the retrieval efficiency $\eta$ of our memory, we systematically
vary the signal field parameters in a range of experiments. We define the
memory efficiency $\eta\equiv E_{\mathrm{out}}/E_{\mathrm{in}}$ as the ratio
of the time-integrated energy content of the retrieved field,
$E_{\mathrm{out}}$, relative to that of the input field $E_{\mathrm{in}}$.
Both the cavity over-coupling $\eta_{\mathrm{c}}$ and an internal efficiency
$\eta_{\mathrm{int}}$ contribute to the memory efficiency as in
$\eta=\eta_{\mathrm{c}}^{2}\cdot\eta_{\mathrm{int}}$, where the cavity
overcoupling is squared accounting for the fact that light has to enter and
leave the cavity. Here, the cavity overcoupling $\eta_{\mathrm{c}}=63\%$ is
limited by excess intracavity loss due to the membrane. Upon detection, we
incur additional losses due to light lost along the optical path
($\eta_{\mathrm{loss}}=60\%$) and the quantum efficiency of our photodetector
($\eta_{\mathrm{QE}}=83\%$), measured with a power meter and taken from the
specifications respectively, totalling
$\eta_{\mathrm{det}}=\eta_{\mathrm{loss}}\cdot\eta_{\mathrm{QE}}=50\%$. In the
following we study the internal efficiency $\eta_{\mathrm{int}}$, cf. fig. 5.

Figure 5:  Storage efficiency contributions. a) Normalized retrieved amplitude
subject to increasing readout delay. The gray hatched area indicates the
thermomechanical noise floor. b) Blue points: retrieved energy
$E_{\mathrm{out}}$ normalized to signal energy $E_{\mathrm{in}}$ for varying
$\Gamma_{\mathrm{sig}}$ and fixed $\Gamma_{\mathrm{eff}}$. In dashed gray, the
theoretical prediction eq. (6) corresponding to unity efficiency. The black
curve is a fit to the same model with one free multiplicative constant, with
the shaded region indicating $\pm$ twice the error on the fit. The red hatched
area indicates the limit to efficiency predicted by the cavity over-coupling,
in fair agreement with the fit. c) Two-photon detuning. For fixed
$\Gamma_{\mathrm{sig}}\approx\Gamma_{\mathrm{eff}}$ the signal frequency
$\Omega_{\mathrm{sig}}$ is tuned across the mechanical resonance. The
normalized, retrieved energy (blue points) is well-modelled by a Lorentzian,
as predicted by eq.(7). The gray hatched area indicates for the mechanical
decay during the storage time. Accounting for the limitations of over-
coupling, and the mechanical decay during the storage, we find good
quantitative agreement between measurements and the expected bound. All
errorbars indicate twice the standard error on the mean.

We use time traces like those in fig. 4b to measure the retrieved energy
$E_{\mathrm{out}}\propto\int_{T_{\mathrm{delay}}}^{\infty}\left|V_{\mathrm{ret}}\right|^{2}dt$.
We estimate the signal energy
$E_{\mathrm{in}}\propto\int_{-\infty}^{0}\left|V_{\mathrm{in}}\right|^{2}dt$,
when the signal does not probe the mechanics but remains near cavity
resonance, $\kappa\gg|\delta|\gg\Gamma_{\mathrm{eff}}$.

We first study the effect of mechanical decay, cf. fig. 5a, by varying
$T_{\mathrm{delay}}$. Ideally, this is only limited by the bare mechanical
decay time, unmodified by dynamical backaction. In practice, the lowest power
of the lock laser we can operate at is around
$\approx$2\text{\,}\mathrm{\SIUnitSymbolMicro W}$$. Since we also slightly
red-detune it to $\Delta_{\mathrm{lock}}\approx-0.05\kappa$ for stability
reasons, it gives rise to a small amount of DBA-induced damping. For each
$T_{\mathrm{delay}}$ we compare our retrieved amplitude $A_{\mathrm{out}}$ to
the drive signal amplitude $A_{\mathrm{in}}$ from a corresponding reference
measurement. In fig. 5a we plot these measurements and observe an exponential
amplitude decay at a rate $1/2T_{1}$ with
$T_{1}\approx$23\text{\,}\mathrm{ms}$$. We attribute this increase over the
intrinsic value $\Gamma_{\mathrm{m}}/2$ to the presence of the lock laser,
even in the dark, when the science laser is off. We corroborate this by
observing an decrease in $T_{1}$ with increasing lock laser power.
Consequently, in eq. (4), we have $\Gamma_{\mathrm{eff}}\mapsto
1/T_{1}+\Gamma_{\mathrm{opt}}$.

Next we study the effect of bandwidth matching, by fixing
$\Gamma_{\mathrm{eff}}$, setting $T_{\mathrm{delay}}=\delta=0$, and varying
$\Gamma_{\mathrm{sig}}$, cf. fig. 5b. From equations (3,4,5) we calculate in
this case

$\eta_{\mathrm{int}}=\frac{4\Gamma_{\mathrm{sig}}\Gamma_{\mathrm{eff}}}{(\Gamma_{\mathrm{sig}}+\Gamma_{\mathrm{eff}})^{2}}.$
(6)

Fitting the data with the model of eq. (6), we obtain a multiplicative
constant of $\eta=0{.}45\cdot\eta_{\mathrm{int}}$, reasonably compatible with
the expected $\eta_{\mathrm{c}}^{2}=40\%$.

Furthermore, we study the effect of two-photon detuning $\delta$, cf. fig. 5c.
For a fixed delay $T_{\mathrm{delay}}\approx$4.4\text{\,}\mathrm{ms}$$, and
matched bandwidth $\Gamma_{\mathrm{sig}}\approx\Gamma_{\mathrm{eff}}$, we vary
$\Omega_{\mathrm{sig}}$ and accordingly $\delta$. From eq. (4) we expect the
mechanics to respond with a Lorentzian susceptibility, leading to

$\eta=\eta_{\mathrm{c}}^{2}\cdot\frac{\Gamma_{\mathrm{eff}}^{2}}{\delta^{2}+\Gamma_{\mathrm{eff}}^{2}}\cdot
e^{-\frac{T_{\mathrm{delay}}}{T_{1}}}.$ (7)

We observe good agreement between our data and model, both at the qualitative
and quantitative level. Accounting for mechanical decay during storage, and
the limit imposed by the overcoupling, we achieve the expected efficiency
predicted by eq. (7).

##  IV Discussion

Taken together, our measurements demonstrate a retrieval efficiency of
$\eta\approx 40\%$, with well-understood scalings. This is limited by
intracavity optical losses. We also achieve an already long
$T_{1}\approx$23\text{\,}\mathrm{ms}$$ lifetime, limited by the smallest lock
laser power we can operate with. Upgrading to a more sensitive detector should
push this number closer to
$\Gamma_{\mathrm{m}}^{-1}\approx$6.3\text{\,}\mathrm{s}$$. Taken together,
this makes the present work the first simultaneous demonstration of a telecom
wavelength, efficient, and long-lived OMIT-based memory for light. In fact,
our room-temperature experiment already shows the slowest decay time $T_{1}$
and highest efficiency $\eta$ of any optomechanical memory to date.

In addition, the prospects for quantum-coherent operation are promising. For a
future cryogenic experiment we extrapolate a heating-limited coherence time of
$1/\Gamma_{\mathrm{m}}\bar{n}_{\mathrm{th}}\approx 0.2\,\mathrm{ms}$,
conservatively assuming thermalization to $10\text{\,}\mathrm{K}$ and a three-
fold improvement in $Q$-factor. Furthermore, potential effects of pure
dephasing should be investigated, even though our membrane resonators have
previously not shown dephasing beyond that caused by heating by the thermal
bath Rossi  _et al._ (2018, 2019). With modest improvements in coupling rate
$g_{0}$ and optical losses $\kappa$, we envision storage and retrieval with
larger bandwidth $\Gamma_{\mathrm{eff}}$, enabling shorter (sub-millisecond)
optical pulses.

In terms of storage of quantum states of light, single-photon storage is of
great interest due to the quantum network applications Kimble (2008). Based on
our findings, the main limitation to efficiency in such an experiment would be
the bandwidth ratio, where we find a
$\eta\propto\Gamma_{\mathrm{eff}}/\Gamma_{\mathrm{sig}}$ scaling for
$\Gamma_{\mathrm{eff}}\ll\Gamma_{\mathrm{sig}}$. In this vein, a range of
approaches for generating narrowband single photons is being pursued, with
spontaneous parametric downconversion Rambach  _et al._ (2016), non-linear
optical bandwidth compression Sośnicki  _et al._ (2023), and charged-exciton
Raman transitions in quantum dot sources Béguin  _et al._ (2018) as prominent
examples. Regarding the latter, recent progress in controlling the charge
noise Uppu  _et al._ (2020) and nuclear spin bath in such devices bodes well
for transform-limited, temporally longer single photons Zaporski  _et al._
(2023), enabled by spin coherence times improved from nano- to micro-seconds.
Such sources would facilitate storage and retrieval of single photons in our
OMIT memory.

## Acknowledgements

The authors acknowledge helpful discussions with Anders S. Sørensen regarding
the theoretical modelling. This work was supported by the European Research
Council project PHOQS (grant no. 101002179), the Novo Nordisk Foundation
(grant no. NNF20OC0061866), the Danish National Research Foundation (Center of
Excellence “Hy-Q”), as well as the Independent Research Fund Denmark (grant
no. 1026-00345B).

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##  Appendix A Supplementary Information

###  A.1 Mechanical Resonator

Here we describe the design of the mechanical resonator. First we optimize the
phononic crystal to give the widest second-order bandgap. For a fixed lattice
constant $a$ (see fig. 6 left), we sweep the hole radius $r$ and we find an
optimal value of $r/a\approx 0.22$, cf. fig. 6 right.

Figure 6: Unit cell design. Left: Parametrization of the unit cell. Right:
Simulated relative second-order bandgap for varying hole radius.

With the phononic crystal design in place we turn to the defect. Our design
goal is to simultaneously achieve a large defect and a high mechanical
frequency $\Omega_{\mathrm{m}}$. This is untenable by geometric rescaling of
the device since $\Omega_{\mathrm{m}}\propto 1/a$ while
$r_{\mathrm{defect}}\propto a$. Thankfully we can both utilize the fact the
tensile stress in our perforated devices is anisotropic and that we have
engineered a higher order band gap. Tweaking the perforations near the defect
lets us tailor the stress field, fine-tuning the defect mode frequencies, and
the higher order bandgap gives us a factor $\approx 2$ over the first bandgap,
for the same defect size. In fig. 7 we show the result of our optimization
(right), with our so-far usual design for comparison (left).

Figure 7: Defect design. Left we show the traditional “Generation 2 Dahlia”
design, which was used in eg. Rossi  _et al._ (2018), with the defect size
indicated by the largest inscribed circle of radius $r_{\mathrm{D}}$ shown in
blue. Right we show the new design used in this work, exhibiting a
significantly larger defect $r_{\mathrm{K}}\approx 1.45\times r_{\mathrm{D}}$,
for the same lattice constant $a$.

The work presented in the main text was conducted using the new “Generation K
Sunflower” design. We confirmed that the new design preserves the high
$Q$-factors associated with soft-clamped membrane resonators by fabricating
devices with a range of sizes and conducting ringdown measurements, with a
sample measurement shown in the main text. In figure 8 we catalogue measured
$Q$ factors and $Qf$ products for 24 different membranes.

Figure 8: Top: $Q$ factors for the first (burgundy) and second bandgap (blue)
modes of the “K” design. Bottom: corresponding $Qf$-products, highlighting the
advantage of the second bandgap mode.

While both modes have high $Q$, the higher $Qf$ products shown for the second
bandgap mode means we choose this mode for our experiments where simultaneous
high frequency and $Q$ are important.

###  A.2 Experimental Setup

The optical and electronic elements of the experimental setup is shown in fig.
9. As illustrated in fig. 9, we switch the science laser on and off using an
acousto-optic modulator and inject the two lasers with orthogonal
polarizations and directly detect the reflection of each laser independently,
as well as their joint transmission. The lock laser reflection is processed
for the PDH lock, whereas a phase-sensitive measurement of the science laser
reflection is done with a lock-in amplifier.

A small fraction is tapped off each laser, such that we can measure their beat
note on a fast $\approx$35\text{\,}\mathrm{GHz}$$ bandwidth photodiode, which
faithfully resolves the $\approx$26\text{\,}\mathrm{GHz}$$ beat note. We
downmix the measured beatnote to baseband and use a digital phase-frequency
detector to derive an error signal that is PI processed and fed to the lock
laser Prevedelli  _et al._ (1995). The lock laser injection current (input
$I$) has a high bandwidth (MHz) whereas the temperature (input $T$)
counteracts long term drifts. In this way we stabilize the beatnote of the two
lasers to a full-width-half-max
$\Gamma_{\mathrm{beat}}/2\pi\approx$2\text{\,}\mathrm{Hz}$$.

Figure 9: Detailed optical and electronic schematic of the experiment, see
text for elaboration. AOM: acousto-optic modulator, EOM: electro-optic
modulator, FR: Faraday rotator, $\lambda/2$: half-wave plate, $\lambda/4$:
quarter-wave plate, PI: feedback controller with proportional and integral
gain.

With the two-lasers offset-locked we can PDH lock the cavity to the lock
laser, and by extension effectively to the science laser. This is implemented
using a commercial analog lock-box, homebuilt sum- and difference-amplifier,
and a commercial multichannel, high-voltage piezo amplifier with independently
variable gains $G_{1}$ and $G_{2}$. The two gains $G_{1}$ and $G_{2}$ are
chosen such that they fulfill $G_{1}L_{1}=G_{2}L_{2}$, with
$L_{1}\approx$4.7\text{\,}\mathrm{mm}$$ and
$L_{2}\approx$0.8\text{\,}\mathrm{mm}$$ denoting the sub-cavity lengths. In
this manner the $2kz$ position is to first order insensitive to the common
mode input $X$, which let’s us stabilize the cavity length. Experimentally we
observed improved stability by augmenting the gain-balancing with a secondary
slow (through the second integrator) feedback using the differential input.

###  A.3 Measuring $\Delta$, $\kappa$, and $g_{0}$

Here we describe our process to calibrate the parameters of our optomechanical
cavity, namely detuning $\Delta$, linewidth $\kappa$, and vacuum coupling rate
$g_{0}$. For the detuning and linewidth we rely on OMIT sweeps, where the
sweep step size is deliberately chosen
$\delta_{\mathrm{step}}\gg\Gamma_{\mathrm{eff}}$. In this way we can get a
clean measurement of the bare cavity response, which we robustly can fit with
an OMIT model with the coupling rate $g=0$. This procedure is illustrated in
fig. 10 where hollow circles are measured data and solid lines are fits. The
colours encode the extracted detuning as detailed on the color. This color-
encoding is shared throughout the manuscript. Each fit also give us a value
for $\kappa$. We take the average of these (the value quoted in the main
text), and utilise it to extract the input power to our cavity. For each of
the measurements in fig. 10, we also record the transmitted DC power level,
which is shown in fig. 11. Using the OMIT-extracted cavity linewidth and
knowledge of our mirror transmission coefficents ($280\text{\times}{10}^{-6}$
and $10\text{\times}{10}^{-6}$ respectively), we perform a single free
parameter fit to the transmitted powers at a given detuning, from which we
extract the input power. With simultaneous knowledge of $P_{\mathrm{in}}$,
$\eta_{\mathrm{c}}$, $\Delta$, $\Omega_{\mathrm{m}}$, and $\kappa$, we can fit
our measured effective mechanical bandwidths $\Gamma_{\mathrm{eff}}$ to the
following dynamical backaction model, as we vary the detuning Aspelmeyer  _et
al._ (2014):

$\displaystyle\Gamma_{\mathrm{eff}}(\Delta)$
$\displaystyle=\Gamma_{\mathrm{m}}+\bar{n}_{\mathrm{cav}}(\Delta)g_{0}^{2}\kappa\left(\frac{1}{(\kappa/2)^{2}+(\Delta+\Omega_{\mathrm{m}})^{2}}-\frac{1}{(\kappa/2)^{2}+(\Delta-\Omega_{\mathrm{m}})^{2}}\right),$
(8a) $\displaystyle\bar{n}_{\mathrm{cav}}(\Delta)$
$\displaystyle=\frac{P_{\mathrm{in}}}{\hbar\Omega_{\mathrm{L}}}\frac{\eta_{\mathrm{c}}\kappa}{(\kappa/2)^{2}+\Delta^{2}}.$
(8b)

With our other characterizations, only $\Gamma_{\mathrm{m}}$ and $g_{0}$ are
left as free fit parameters. Since $\Gamma_{\mathrm{m}}$ is small we can
further restrict the fit by setting $\Gamma_{\mathrm{m}}=0$. This is the model
we use in fig. 3 to extract a value for $g_{0}$.

Figure 10: OMIT cavity spectroscopy. A fast and coarse sweep of a phase-
modulation traces out the cavity response, enabling robust fitting from which
we extract $\Delta$ and $\kappa$. Figure 11: Transmitted power as a function
of science laser detuning. The gray line is a one-parameter fit with a fixed
linewidth from which we extract the input optical power.


# Arithmetic aspects of the Jouanolou foliation

Wodson Mendson

00footnotetext: 2010 _Mathematics Subject Classification._ 32S65; 13N15;
13A3500footnotetext: _Keywords._ Foliations; Foliations over positive
characteristic; Reduction modulo $p$; Jouanolou foliation;
$p$-divisor00footnotetext: This work was developed at Université de Rennes 1.
W. Mendson acknowledges support of CAPES

######  Abstract.

We investigate the structure of the $p$-divisor for the Jouanolou foliation
where we show, under some conditions, that it can be irreducible or has a
$p$-factor. We study the reduction modulo $p$ of foliations on the projective
plane and its applications to the problems of holomorphic foliations. We give
new proof, via reduction modulo $2$, of the fact that the Jouanolou foliation
on the complex projective plane of odd degree, under some arithmetic
conditions, has no algebraic solutions.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Foliations on the projetive plane
  4. 4 Algebraicity and the $p$-divisor
  5. 5 Automorphism and the $p$-divisor
  6. 6 Non-algebraicity via reduction modulo $2$

##  1 Introduction

On the work [13] the author introduces the notion of the $p$-divisor for
foliations on a smooth algebraic surface defined over a field characteristic
$p>0$. This $p$-divisor appears as the degeneration divisor of the curvature
morphism: this morphism takes a local section of the tangent bundle of the
foliation and send it to its $p$-power. The notion of $p$-divisor has been
shown to be relevant to undestand the nature of the algebraic curves that are
invariant by the foliation with applications to the study of codimension one
holomorphic foliations on projective spaces. For instance, the struture of the
$p$-divisor for generic foliations on
$\mathbb{P}_{\operatorname{k}}^{1}\times\mathbb{P}_{\operatorname{k}}^{1}$ was
a important step in the construction of new irreducible components of the
space of codimension one holomorphic foliations on projective spaces done in
the work [14] (see [13, Theorem B] and [14, Theorem 11.7]).

A fundamental property is the following: any algebraic invariant curve should
appear in the support of the $p$-divisor and conversely, any prime divisor on
the support of the $p$-divisor with multiplicity not zero modulo $p$ should be
an invariant curve (see [13, Proposition 3.8]).

In the study of the $p$-divisor appears naturally the following questions:

######  Problem 1.

What we can say about the structure of the $p$-divisor for generic foliations?
Is it irreducible, reduced?

######  Problem 2.

How many algebraic solutions can a foliation on a smooth projective surface
have?

A simple instance of the Problem 2 occurs when we consider $p$-closed
foliations. In this case, there are infinitely many solutions and the
$p$-divisor is the zero divisor. In the work [13], the author gives an answer
to the Problem 1 in the case of foliations on the projective plane and in the
Hirzebruch surfaces. It is explored the structure of the $p$-divisor where it
is shown, under some conditions, that it is reduced, generically. On the
projective plane, this is the following theorem (see [13, Theorem A]).

######  Theorem.

Let $\operatorname{k}$ be a field of characteristic $p>0$. A generic foliation
on the projective plane $\mathbb{P}_{\operatorname{k}}^{2}$ of degree $d\geq
1$ with $p\nmid d-1$ has reduced $p$-divisor.

In this work, we investigate the struture of the $p$-divisor for the Jouanolou
foliations where we give results in the direction of the Problem 1 and Problem
2. The Jouanolou foliation of degree $d\geq 2$, $\mathcal{J}_{d}$, on the
projective plane over a field $K$ is given by the planar vector field:

$v_{d}=(xy^{d}-1)\partial_{x}-(x^{d}-y^{d+1})\partial_{y}.$

The interesting fact about $\mathcal{J}_{d}$ is that when $K=\mathbb{C}$ the
foliation $\mathcal{J}_{d}$ has no algebraic solutions for every
$d\in\mathbb{Z}_{>1}$. This was proved by Jouanolou in his monograph [9]. On
the other hand, when $K$ has characteristic $p>0$ it follows from [13,
Corollary 3.8] that the Jouanolou foliation has an algebraic solution for
every $d\in\mathbb{Z}_{>1}$ such that $p\nmid d+2$. The main result is the
following theorem.

######  Theorem A.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $d$ be a positive integer such that

  * •

$p<d$ and $p\not\equiv 1\mod 3$;

  * •

$d^{2}+d+1$ is prime.

Then the Jouanolou foliation $\mathcal{J}_{d}$ on
$\mathbb{P}_{\operatorname{k}}^{2}$ has irreducible $p$-divisor or
$\Delta_{\mathcal{J}_{d}}=C+pR$ with $(\mathcal{J}_{d},C)$ special and
$R=\sum_{i}\alpha_{i}P_{i}$ with $P_{i}$ not $\mathcal{J}_{d}$-invariant for
every $i$.

The notion of special pair $(\mathcal{J}_{d},C)$ is introduced in the Section
5 (see Definition 5.3). Under some arithmetic conditions on the characteristic
of $\operatorname{k}$ we prove that the $p$-divisor of the Jouanolou foliation
of degree two is always irreducible.

######  Theorem B.

Let $p>2$ be a prime number such that $7\nmid p+4$ and such that $p\not\equiv
1\mod 3$. Then, the Jouanolou foliation of degree two, $\mathcal{J}_{2}$,
defined over a field of characteristic $p$ has irreducible $p$-divisor.

As a consequence of the Theorem A and Theorem B we give a answer to the
Problem 2 to the Jouanolou foliations (see Corollary 5.18).

######  Proposition A.

In the conditions of Theorem A, the Jouanolou foliation of degree $d\geq 2$ in
characteristic $p>0$ has an unique invariant curve and that curve has degree
$pl+d+2$, with $0<l\leq d-1$.

The proof of the Theorem A and Theorem B envolves some study of the influence
of the automorphism group of a foliation on the structure of their
$p$-divisor. This is explored in the Section 5.

In the last part of the article, we present a simple proof of the fact that
the Jouanolou foliation of odd degree, under some arithmetic conditions, has
no algebraic solutions. This implies, in particular, the following result.

######  Theorem C.

A very generic foliation of degree odd $d\geq 2$, with $d\not\equiv 1\mod 3$
on the projetive plane has no algebraic solutions.

Recall that the term very generic means that there exists a countable union of
closed sets $S$ in the space of holomorphic foliations of degree $d$ such that
any foliations lying outside $S$ has no algebraic solutions.

The Theorem C is well-known (see [9] and [17, Theorem 1]). The new content in
this work is the method of the proof which uses only reduction to
characteristic two and some irreducibility test for polynomials in two
variables due to S.Gao (see [8, Corollary 4.12]).

###  1.1 Organization of the paper

In Section 2 we fix the notations that will be used in the paper. In Section 3
we recall the definition of foliation on the projective plane, the notion of
the $p$-divisor, automorphisms of foliation and we discuss some topics around
reduction modulo $p$. In Section 4, we discuss some relations between
foliations on the complex projective plane that has no algebraic solutions and
foliations defined over a field of characteristic $p>0$ that has irreducible
$p$-divisor. In Section 5, we explore the automorphism group of foliations and
its influence on the struture of the $p$-divisor where we present proofs of
Theorem A and Theorem B. In the last section, we investigate the Jouanolou
foliations in characteristic two and we present the proof of Theorem C.

##  2 Notation

  * •

$\mathcal{J}_{d}=$ the Jouanolou foliation;

  * •

$\Delta_{\mathcal{F}}$ = the $p$-divisor associated to a foliation
$\mathcal{F}$;

  * •

$\operatorname{\textbf{Spm}}(R)=$ the collection of maximal ideals of a domain
$R$;

  * •

$\mathbb{F}_{\mathfrak{q}}=$ the residue fied of
$\mathfrak{q}\in\operatorname{\textbf{Spm}}(R)$;

  * •

$\mathbb{Z}[\mathcal{F}]=$ the $\mathbb{Z}$-algebra obtained by adjunction of
all coefficients which appear in $\mathcal{F}$;

  * •

$\mathcal{F}_{\mathfrak{p}}=$ the reduction modulo $\mathfrak{p}$ of
$\mathcal{F}$;

  * •

$\operatorname{k}[x,y,z]_{e}=$ $\operatorname{k}$-vector space of homogeneous
polynomials of degree $e$ in $x,y,z$;

  * •

$m_{Q}(C)=$ algebraic multiplicity of the curve $C\subset X$ at $Q\in X$;

  * •

$O(2)=$ terms of the order at least $2$;

##  3 Foliations on the projetive plane

Let $d\in\mathbb{Z}_{>0}$ and $\operatorname{k}$ be an algebraically closed
field. A foliation $\mathcal{F}$ on the projective plane
$\mathbb{P}_{\operatorname{k}}^{2}$ is given, modulo $\operatorname{k}^{*}$,
by a non-zero global section
$\omega\in\operatorname{H^{0}}(\mathbb{P}_{\operatorname{k}}^{2},\Omega_{\mathbb{P}_{\operatorname{k}}^{2}}^{1}(d+2))$
with finite singular locus. The degree of $\mathcal{F}$ has a geometric
content: it is the number of tangencies between $\omega$ and a generic line
$l\cong\mathbb{P}_{\operatorname{k}}^{1}$ in
$\mathbb{P}_{\operatorname{k}}^{2}$. Using the Euler exact sequence, we can
see $\omega$ as a homogeneous $1$-form on $\mathbb{A}_{\operatorname{k}}^{3}$:
$\omega=Adx+Bdy+Cdz$ where

$Ax+By+Cz=0\quad\mbox{and}\quad\operatorname{sing}(\mathcal{F})=\mathcal{Z}(A,B,C)\mbox{
is finite}.$

###  3.1 $p$-closed and non-$p$-closed foliations

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $\mathcal{F}$ be a foliation of degree $d$ on
$\mathbb{P}_{\operatorname{k}}^{2}$ and assume that $p$ does not divide $d+2$.
Suppose that $\mathcal{F}$ is given by the projective $1$-form:
$\omega=Adx+Bdy+Cdz$ and write

$d\omega=(d+2)(Ldy\wedge dz-Mdx\wedge dz+Ndx\wedge dy).$

Let $v_{\omega}$ be the homogeneous vector field on
$\mathbb{A}_{\operatorname{k}}^{3}$ defined by:

$v_{\omega}=L\partial_{x}+M\partial_{y}+N\partial_{z}.$

Observe that by construction we have

$\mbox{{div}}(v_{\omega})=\partial_{x}L+\partial_{y}M+\partial_{z}N=0.$

We recall the following proposition form [9].

######  Proposition 3.1.

The map which associes $\omega$ to $v_{\omega}$ gives a bejection between the
set
$\operatorname{H^{0}}(\mathbb{P}_{\operatorname{k}}^{2},\Omega_{\mathbb{P}_{\operatorname{k}}^{2}}^{1}(d+2))$
and the set of homogeneous vector fields of degree $d$ on
$\mathbb{A}_{\operatorname{k}}^{3}$ with zero divergent.

###### Proof.

See [9, Proposition 1.1.4]. ∎

We say that $\mathcal{F}$ is $p$-closed if $i_{v_{\omega}^{p}}\omega=0$, where
$i_{v_{\omega}^{p}}$ denotes the contraction map. If $\mathcal{F}$ is not
$p$-closed then we obtain a divisor
$\Delta_{\mathcal{F}}=\\{i_{v_{\omega}^{p}}\omega=0\\}\in\operatorname{Div}(\mathbb{P}_{\operatorname{k}}^{2})$
of degree $p(d-1)+d+2$. This is called the $p$-divisor associated to
$\mathcal{F}$. Recall that a reduced algebraic curve $C=\\{F=0\\}$ is
$\mathcal{F}$-invariant if there is $\Theta$, a $2$-homogeneous form of degree
$d-1$, on $\mathbb{A}_{\operatorname{k}}^{3}$ such that

$dF\wedge\omega=F\Theta.$

A fundamental property of this notion is contained in the following
proposition.

######  Proposition 3.2.

Let $\mathcal{F}$ be a non-$p$-closed foliation on
$\mathbb{P}_{\operatorname{k}}^{2}$ and $C$ be an irreducible curve in
$\mathbb{P}_{\operatorname{k}}^{2}$. Then:

  * •

if $C$ is $\mathcal{F}$-invariant then
$\operatorname{ord}_{\Delta_{\mathcal{F}}}(C)>0$;

  * •

if $\operatorname{ord}_{\Delta_{\mathcal{F}}}(C)\not\equiv 0\mod p$ then $C$
is $\mathcal{F}$-invariant.

###### Proof.

See [13, Proposition 3.8]. ∎

Let $\mathcal{F}$ be a foliation on $\mathbb{P}_{\operatorname{k}}^{2}$ and
$q\in\operatorname{sing}(\mathcal{F})$ be a singular point. Recall (see [13,
Definition 3.10]) that the point $q$ is called $p$-reduced singularity if
there is a affine open subset $U\subset\mathbb{P}_{\operatorname{k}}^{2}$
which contains $q$ such that $\mathcal{F}|_{U}$ has linear part at $q$ of the
form $ydx+\alpha xdy$ with $\alpha\not\in\mathbb{F}_{p}$. The foliation
$\mathcal{F}$ is called $p$-reduced foliation if all singular point
$q\in\operatorname{sing}(\mathcal{F})$ is $p$-reduced. If $\mathcal{F}$ has at
lest a $p$-reduced singularity then $\mathcal{F}$ is not $p$-closed by [13,
Lemma 3.11].

######  Remark 3.3.

Let $X$ be a smooth surface defined over $\operatorname{k}$. A foliation
$\mathcal{F}$ on $X$ can be defined as a system
$\\{(U_{i},\omega_{i},v_{i})\\}_{i}$ where $\\{U_{i}\\}$ is an open cover of
$X$, $\omega_{i}\in\Omega_{X}^{1}(U_{i})$ and $v_{i}\in T_{X}(U_{i})$
satisfying compatibility conditions. The collection $\\{\omega_{i}\\}_{i}$ and
$\\{v_{i}\\}_{i}$ define the normal bundle, $N_{\mathcal{F}}$, and the tangent
bundle, $T_{\mathcal{F}}$, of the foliation. The foliation is not $p$-closed
if $i_{v_{i}^{p}}\omega_{i}\neq 0$ for every $i$. In that case, the collection
$\\{i_{v_{i}^{p}}\omega_{i}\\}_{i}$ defines a non-zero global section
$s_{\mathcal{F}}$ of $(T_{\mathcal{F}}^{*})^{p}\otimes N_{\mathcal{F}}$ and
the zero divisor of $s_{\mathcal{F}}$ is the $p$-divisor of $\mathcal{F}$ (see
[13, Section 3]).

###  3.2 Automorphism of foliations

Let $\operatorname{k}$ be an algebraically closed field and $\mathcal{F}$ a
foliation on $\mathbb{P}_{\operatorname{k}}^{2}$ defined by a projective
$1$-form
$\omega\in\operatorname{H^{0}}(\mathbb{P}_{\operatorname{k}}^{2},\Omega_{\mathbb{P}_{\operatorname{k}}^{2}}^{1}(d+2))$.
An automorphism of $\mathcal{F}$ is an automorphism of the projective plane
$\Phi\in\operatorname{Aut}(\mathbb{P}_{\operatorname{k}}^{2})$ which preserves
$\mathcal{F}$. By preserves $\mathcal{F}$ we mean the following: if the
foliation is given by the projective $1$-form
$\omega\in\operatorname{H^{0}}(\mathbb{P}_{\operatorname{k}}^{2},\Omega_{\mathbb{P}_{\operatorname{k}}^{2}}^{1}(d+2))$
then $\Phi^{*}\omega=\sigma(\Phi)\omega$ for some
$\sigma(\Phi)\in\operatorname{k}^{*}$.

######  Remark 3.4.

By the definition of $p$-divisor it follows that if $\mathcal{F}$ is not
$p$-closed and $\Phi\in\operatorname{Aut}(\mathcal{F})$ then
$\Phi^{*}\Delta_{\mathcal{F}}=\Delta_{\mathcal{F}}$.

The Jouanolou foliation defined on the projective plane
$\mathbb{P}_{\operatorname{k}}^{2}$ of degree $d>1$ is given by the projective
$1$-form:

$\mathcal{J}_{d}:\quad\omega=(x^{d}z-y^{d+1})dx+(xy^{d}-z^{d+1})dy+(z^{d}y-x^{d+1})dz.$

Suppose that the characteristic of $\operatorname{k}$ does not divide
$d^{2}+d+1$. Let $\mu(\operatorname{k})$ the group of the $(d^{2}+d+1)$th
roots of the unity and let $\gamma$ be a generator of this group. Consider the
automorphism
$\Phi\colon\mathbb{P}_{\operatorname{k}}^{2}\longrightarrow\mathbb{P}_{\operatorname{k}}^{2}$
which associe $[x:y:z]\mapsto[\gamma^{d^{2}+1}x:\gamma y:z]$. Then
$\Phi^{*}\omega=\gamma\omega$ so that $\Phi$ is an automorphism of
$\mathcal{F}$. Note that $\Phi$ has order $d^{2}+d+1$. This automorphism will
be used in the Section 5 to study the struture of the $p$-divisor of
$\mathcal{J}_{d}$.

###  3.3 Reduction modulo $p$

Let $\mathcal{F}$ be a foliation of degree $d$ on the complex projective plane
given by the projective $1$-form:

$\omega=Adx+Bdy+Cdz$

where $A,B,C\in\mathbb{C}[x_{0},x_{1},x_{2}]_{d+1}$. Let
$\mathbb{Z}[\mathcal{F}]$ the $\mathbb{Z}$-algebra of finite type obtained by
adjunction of all constant coefficients which appears on $A,B$ and $C$. Let
$\mathfrak{p}\in\operatorname{\textbf{Spm}}(\mathbb{Z}[\mathcal{F}])$ be a
maximal ideal and let $p\in\mathbb{Z}_{>0}$ the characteristic of the residue
field $\mathbb{F}_{\mathfrak{p}}=\mathbb{Z}[\mathcal{F}]/\mathfrak{p}$. Note
that $\mathbb{F}_{\mathfrak{p}}$ is a finite field by [18, Tag 00GB] and, in
particular, it has characteristic $p>0$. Denote by
$\omega_{\mathfrak{p}}=\omega\otimes\mathbb{F}_{\mathfrak{p}}$ the section of
$\Omega_{\mathbb{P}_{\overline{\mathbb{F}}_{\mathfrak{p}}}^{2}}^{1}(d+2)$
obtained by reduction modulo $\mathfrak{p}$ of the coefficients of $\omega$.

######  Definition 3.5.

The foliation $\mathcal{F}_{\mathfrak{p}}$ defined by $\omega_{\mathfrak{p}}$
is called the reduction modulo $\mathfrak{p}$ of $\mathcal{F}$.

Let $\mathcal{F}$ be a foliation on the complex projective plane and P be an
abstract property. We say that P holds to $\mathcal{F}_{\mathfrak{p}}$ for
infinitely many primes $\mathfrak{p}$ if there is a dense subset
$S\subset\operatorname{\textbf{Spm}}(\mathbb{Z}[\mathcal{F}])$ such that for
any $\mathfrak{m}\in S$ the property P holds to $\mathcal{F}_{\mathfrak{m}}$.
Similary, we say that P holds to $\mathcal{F}_{\mathfrak{p}}$ for almost all
primes $\mathfrak{p}$ if there is a non-empty open subset
$U\subset\operatorname{\textbf{Spm}}(\mathbb{Z}[\mathcal{F}])$ such that for
any $\mathfrak{m}\in U$ the property P holds to $\mathcal{F}_{\mathfrak{m}}$.

The property P can be for example: $\mathcal{F}_{\mathfrak{p}}$ has an
invariant curve of degree $e$, the foliation $\mathcal{F}_{\mathfrak{p}}$ is
not $p$-closed or the $p$-divisor of $\mathcal{F}_{\mathfrak{p}}$ is
irreducible.

##  4 Algebraicity and the $p$-divisor

In this section, we investigate the relation between the class of foliations
on the complex projective plane without algebraic solution and the class of
foliations on the projective plane over characteristic $p>0$ that has
irreducible $p$-divisor. We begin the study with the following well-known
basic lemma.

######  Lemma 4.1.

Let $X=\mathcal{Z}(f_{1},\ldots,f_{r})\subset\mathbb{A}_{\mathbb{C}}^{m}$ be
an affine variety and $A$ be the $\mathbb{Z}$-algebra of finite type obtained
by adjunction of all coefficients which appers on
$f_{0},\ldots,f_{r}\in\mathbb{C}[x_{1},\ldots,x_{m}]$. Then,
$X(\mathbb{C})\neq\varnothing$ if and only if
$X(\overline{\mathbb{F}}_{\mathfrak{q}})\neq\varnothing$ for infinitely many
primes $\mathfrak{q}\in\operatorname{\textbf{Spm}}(A)$

###### Proof.

This is well-known, but we will present a simple proof here of the non-trivial
part (see [15, Proposition 2.6]). Suppose by contradiction that
$X(\overline{\mathbb{F}}_{\mathfrak{q}})\neq\varnothing$ for infinitely many
primes $\mathfrak{q}\in\operatorname{\textbf{Spm}}(R)$ but
$X(\mathbb{C})=\varnothing$. Let $S\subset\operatorname{\textbf{Spm}}(A)$ the
set of that maximal ideals. By the Nullstellensatz there is a relation:

$1=\alpha_{1}f_{1}+\dots+\alpha_{n}f_{n}$

for some $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{C}[x_{1},\ldots,x_{m}]$. Let
$R$ the $A$-algebra of finite type obtained by inverting all constant
coefficients of $\alpha_{1},\ldots,\alpha_{n}$. Since, for almost all maximal
ideal of $\mathbb{Z}$ is contraction of some maximal ideal of $R$ (see [19,
Proposition 4.1.5]) we conclude that there are infinity of maximal ideals
$\mathfrak{p}\in\operatorname{\textbf{Spm}}(R)$ such that $\mathfrak{p}\cap
A\in S$. So, for infinitely many ideals
$\mathfrak{q}\in\operatorname{\textbf{Spm}}(R)$ we have
$1\equiv\alpha_{1}f_{1}+\dots+\alpha_{n}f_{n}\mod\mathfrak{q}$, but this
contradicts the fact $X(\overline{\mathbb{F}}_{\mathfrak{q}})\neq\varnothing$.
∎

######  Proposition 4.2.

Let $\mathcal{F}$ be a foliation on the complex projective plane and let
$d\in\mathbb{Z}_{>0}$. Then $\mathcal{F}$ has a $\mathcal{F}$-invariant curve
of degree $d$ if and only if $\mathcal{F}_{\mathfrak{p}}$ has a
$\mathcal{F}_{\mathfrak{p}}$-invariant curve of degree $d$ for infinitely many
primes $\mathfrak{p}$.

###### Proof.

Let $\operatorname{k}$ be an algebraically closed field and $\mathcal{G}$ a
foliation on $\mathbb{P}_{\operatorname{k}}^{2}$ defined by a $1$-projective
form
$\Omega\in\operatorname{H^{0}}(\mathbb{P}_{\operatorname{k}}^{2},\Omega^{1}_{\mathbb{P}_{\operatorname{k}}^{2}}(e+2))$.
Note that the set of all $\mathcal{G}$-invariant curves of degree $d$ on
$\mathbb{P}_{\operatorname{k}}^{2}$ is an projective variety defined over
$\operatorname{k}$. Indeed, consider the following algebraic set:

$\mathcal{S}_{d}(\mathcal{G})=\\{([F],[\Theta])\in\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}(d))\times\mathbb{P}(V_{e-1})\mid\Omega\wedge
dF-F\Theta=0\\}$

where $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}(d))$ denotes the projective
space which parametrizes all homogeneous polynomial of degree $d$ and
$\mathbb{P}(V_{e-1})$ parametrizes all homogeneous $3$-forms on
$\mathbb{A}_{\mathbb{C}}^{3}$ of degree $e-1$. Then the set of all
$\mathcal{F}$-invariant curves of degree $d$ is the set
$S_{d}(\mathcal{G}):=\pi_{1}(\mathcal{S}_{d}(\mathcal{G}))$ where
$\pi_{1}\colon\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}(d))\times\mathbb{P}(V_{e-1})\longrightarrow\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}(d))$
is the projection to the first factor. Since the projection is a proper map,
we conclude that $S_{d}(\mathcal{G})$ is a projective variety.

By Lemma 4.1, we conclude $S_{d}(\mathcal{F})(\mathbb{C})\neq\varnothing$ if
and only if
$S_{d}(\mathcal{F}_{\mathfrak{q}})(\overline{\mathbb{F}}_{\mathfrak{q}})\neq\varnothing$
for infinitely many primes $\mathfrak{q}$ and this proves the lemma. ∎

######  Proposition 4.3.

Let $\mathcal{F}$ be a foliation on the complex projective plane of degree
$d>1$. Then if the $p$-divisor of the foliation $\mathcal{F}_{\mathfrak{p}}$
is irreducible for infinitely many primes $p$ then $\mathcal{F}$ has no
algebraic solutions.

###### Proof.

Suppose that $\mathcal{F}_{\mathfrak{q}}$ has irreducible $q$-divisor for
infinitely many primes $\mathfrak{q}$. Assume by contradiction that there is
$C$ an irreducible curve that is invariant by $\mathcal{F}$. Since the
invariance is preseved by reduction modulo $\mathfrak{q}$, we have that
$C_{\mathfrak{q}}$ is a $\mathcal{F}_{\mathfrak{q}}$-invariant for infinitely
many primes $\mathfrak{q}$. By [13, Proposition 3.8] we know that
$C_{\mathfrak{q}}\leq\Delta_{\mathcal{F}_{\mathfrak{q}}}$. But, since
$\Delta_{\mathcal{F}_{\mathfrak{q}}}$ is irreducible for infinitely many
primes, we concude that $\Delta_{\mathcal{F}_{\mathfrak{q}}}=C_{\mathfrak{q}}$
for infinitely many primes $\mathfrak{q}$, a contradiction by degree
comparison. ∎

######  Remark 4.4.

There are foliations on the complex projective plane that has reduced
$p$-divisor for almost all primes $\mathfrak{p}$. Indeed, consider the
foliation $\mathcal{F}$ given in $\mathbb{A}^{2}_{\mathbb{C}}$ by the
polynomial $1$-form:

$\omega=ydx-xdy+a(x,y)dx+b(x,y)dy=ydx-xdy+\omega_{2}$

where $a(x,y),b(x,y)\in\mathbb{C}[x,y]$ are generic homogeneous polynomials of
degree two. The foliation induced on $\mathbb{P}_{\mathbb{C}}^{2}$ by $\omega$
has four invariant lines: the line at infinity $l_{\infty}=\\{z=0\\}$ and
$l_{1},l_{2},l_{3}$ the factors of the polynomial
$i_{R}\omega_{2}=l_{1}l_{2}l_{3}$. Then for almost primes $\mathfrak{p}$ the
$p$-divisor of $\mathcal{F}_{\mathfrak{p}}$ has the form

$\Delta_{\mathcal{F}_{\mathfrak{p}}}=l_{1}+l_{2}+l_{3}+l_{\infty}+C_{\mathfrak{p}}$

where $C_{\mathfrak{p}}$ is irreducible of degree $p$ (see [13, Proposition
4.3]).

The next proposition follows from results that are contained in the paper [14]
but we present a direct proof here (compare with [14, Proposition 5.5]).

######  Proposition 4.5.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$ and $X$ and $Y$ smooth projective surfaces defined over
$\operatorname{k}$. Consider $\Phi\colon X\longrightarrow Y$ be a finite
dominant separable morphism with ramification divisor $D$ and suppose that the
reduced part of $\Phi(D)$ is not $\mathcal{F}$-invariant. Let $\mathcal{F}$ be
a foliation on $Y$ with irreducible $p$-divisor and let
$\mathcal{G}=\Phi^{*}\mathcal{F}$ the induced folition on $X$. Then
$\mathcal{G}$ is not $p$-closed with $p$-divisor given by
$\Delta_{\mathcal{G}}=\Phi^{*}\Delta_{\mathcal{F}}+pD$, where $D$ denotes the
the ramification divisor $D$.

###### Proof.

Let $\omega\in N_{\mathcal{F}}^{*}$ be a local section defining $\mathcal{F}$
in an open set $U$. Since the reduced part $\Phi(D)$ is not
$\mathcal{F}$-invariant we have $\Phi^{*}\omega$ a local section of
$N_{\mathcal{G}}^{*}$ on $\Phi^{-1}(U)$, so that
$\Phi^{*}N_{\mathcal{F}}=N_{\mathcal{G}}$. Now, the formula
$K_{X}=\Phi^{*}K_{Y}+D$ and the adjunction formula implies:
$K_{\mathcal{G}}+N_{\mathcal{G}}^{*}=K_{X}=\Phi^{*}K_{\mathcal{F}}+\Phi^{*}N_{\mathcal{F}}^{*}+D$
so that $K_{\mathcal{G}}=\Phi^{*}K_{\mathcal{F}}+D.$ So,

$\Delta_{\mathcal{G}}=pK_{\mathcal{G}}+N_{\mathcal{G}}=p(\Phi^{*}K_{\mathcal{F}}+D)+\Phi^{*}N_{\mathcal{F}}=\Phi^{*}\Delta_{\mathcal{F}}+pD$

and this proves the result. ∎

######  Example 4.6.

As an explicit example we consider the Jouanolou foliation $\mathcal{F}$ of
degree $d=2$ over a field $\operatorname{k}$ of characteristic $p=3$. The
foliation $\mathcal{F}$ is defined by the projective $1$-form:

$\Omega=(x^{2}z-y^{3})dx+(xy^{2}-z^{3})dy+(z^{2}y-x^{3})dz.$

The foliation $\mathcal{F}$ is not $3$-closed with irreducible $3$-divisor
given by

$\Delta_{\mathcal{F}}=\\{x^{7}+y^{7}+z^{7}+2x^{4}yz^{2}+2x^{2}y^{4}z+2xy^{2}z^{4}=0\\}\in\operatorname{Div}(\mathbb{P}_{\operatorname{k}}^{2}).$

Now consider the finite map
$\Phi\colon\mathbb{P}_{\operatorname{k}}^{2}\longrightarrow\mathbb{P}_{\operatorname{k}}^{2}$
which associe $[x_{0}:x_{1}:x_{2}]\mapsto[x_{0}^{2}:x_{1}^{2}:x_{2}^{2}]$ and
consider the induced foliation $\mathcal{G}=\Phi^{*}\mathcal{F}$. Then
$\mathcal{G}$ is defined by the projective $1$-form

$\Phi^{*}\Omega=2x(x^{4}z^{2}-y^{6})dx+2y(x^{2}y^{4}-z^{6})dy+2z(z^{4}y^{2}-x^{6})dz$

with $3$-divisor given by

$\Delta_{\mathcal{G}}=\\{x^{14}+y^{14}+z^{14}+2x^{8}y^{2}z^{4}+2x^{4}y^{8}z^{2}+2x^{2}y^{4}z^{8}=0\\}+3\\{x=0\\}+3\\{y=0\\}+3\\{z=0\\}$
$=\Phi^{*}\Delta_{\mathcal{F}}+3D.$

######  Corollary 4.7.

There are foliations on the projective plane $\mathbb{P}_{\mathbb{C}}^{2}$
that have no algebraic solutions and are non-$p$-closed for almost all primes
$\mathfrak{p}$ with $p$-divisor having a $p$-power.

###### Proof.

Let
$\Phi\colon\mathbb{P}_{\mathbb{C}}^{2}\longrightarrow\mathbb{P}_{\mathbb{C}}^{2}$
be a finite morphism and $\mathcal{F}$ be a foliation on
$\mathbb{P}_{\mathbb{C}}^{2}$ that has no algebraic solutions. Assume that
foliation has at least a singular point with eigenvalue
$\alpha\not\in\overline{\mathbb{Q}}$. Then $\mathcal{G}=\Phi^{*}\mathcal{F}$
is not $p$-closed for an almost all primes $p$ and does not have any algebraic
solution. But, if $D$ is the ramification divisor of $\Phi$ then by the
Proposition 4.5 it follows that the $p$-divisor,
$\Delta_{\mathcal{G}_{\mathfrak{p}}}$, has $D\mod\mathfrak{p}$ as a $p$-power.
In particular, it is not reduced for almost all primes $p$ and having a
$p$-power. ∎

######  Remark 4.8.

The Corollary 4.7 shows that the converse of the Proposition 4.3 does not
holds in general.

Let $\mathcal{F}$ be a foliation on $\mathbb{P}_{\operatorname{k}}^{2}$ and
$C$ be a $\mathcal{F}$-invariant curve. In positive characteristic, the
singular locus of the invariant curve $C$ is not always contained in the
singular locus of the foliation.

######  Example 4.9.

Let $\mathcal{F}$ be the foliation on the complex projective plane given by
the projective 1-form (see [1, Theorem 4]):

$\Omega=-(xyz+y^{3}+z^{3})dx+(x^{2}z+y^{2}x-yz^{2})dy+z(xz+y^{2})dz.$

Consider the foliation $\mathcal{G}=\mathcal{F}\otimes 5\mathbb{Z}$ the
projective plane over $\overline{\mathbb{F}}_{5}$ obtained by reduction modulo
$5\mathbb{Z}$ of the coefficients of $\Omega$. Then the $5$-divisor of
$\mathcal{G}$ is irreducible given by:

$\Delta_{\mathcal{G}}=\\{-2x^{3}z^{6}-2x^{2}y^{2}z^{5}+xy^{7}z+2xyz^{7}+y^{9}+y^{6}z^{3}+y^{3}z^{6}-z^{9}=0\\}\in\operatorname{Div}(\mathbb{P}_{\overline{\mathbb{F}}_{5}}^{2}).$

The foliation $\mathcal{G}$ has a unique singular point given by $[0:1:0]$. On
the other hand, the curve $\Delta_{\mathcal{G}}$ has $[1:2:1]$ as singular
point so that
$\operatorname{sing}(\Delta_{\mathcal{G}})\nsubseteq\operatorname{sing}(\mathcal{G})$.
Note that $\deg(\Delta_{\mathcal{G}})=9\equiv 4\mod 5$.

We can say some information about the structure of the points in
$\operatorname{sing}(C)-\operatorname{sing}(\mathcal{F})$. This is the content
of the following proposition.

######  Proposition 4.10.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p\geq 2$ and $\mathcal{F}$ be a non $p$-closed foliation on
$\mathbb{P}_{\operatorname{k}}^{2}$. Suppose that there is an irreducible
curve, $C$, that is $\mathcal{F}$-invariant with degree $\deg(C)<p$. Then,
$\operatorname{sing}(C)\subset\operatorname{sing}(\mathcal{F})$ or every
$q\in\operatorname{sing}(C)-\operatorname{sing}(\mathcal{F})$ is a hypercusp.

###### Proof.

Suppose that there is a point
$q\in\operatorname{sing}(C)-\operatorname{sing}(\mathcal{F})$. By the [11,
Corollary 2.4] we know that there is a formal coordinate system $x,y$ on the
completion of the ring $\mathcal{O}_{\mathbb{P}^{2},q}$ such that the
foliation is given by the vector field:

$v=\partial_{x}+x^{p-1}g(x,y)\partial_{y}$

with $g(x,y)\in\operatorname{k}[[x^{p},y]]$. The curve in this coordinates
system has form $f=f_{e}+f_{e+1}+\cdots\in\operatorname{k}[[x,y]]$ where
$f_{i}$ is a homogeneous polynomial of degree $i$ in $x,y$. We need to show
$f_{e}=l^{e}$ for some $l$, polynomial of degree $1$. Indeed, by definition of
invariance we have $v(f)=fg$ for some $g\in\operatorname{k}[[x,y]]$ and
considering the smallest term we obtain $\partial_{x}(f_{e})=0$. Since,
$e<\deg(C)<p$ we conclude that $f_{e}=y^{e}$. ∎

The difficult of the author in construct examples with
$\operatorname{sing}(C)\not\subset\operatorname{sing}(\mathcal{F})$ and
$\deg(C)<p$ leads to the following question.

######  Problem 3.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p\geq 2$ and $\mathcal{F}$ be a non $p$-closed foliation on
$\mathbb{P}_{\operatorname{k}}^{2}$. Suppose that there is an irreducible
curve, $C$, that is $\mathcal{F}$-invariant with degree $\deg(C)<p$. Is it
true that $\operatorname{sing}(C)\subset\operatorname{sing}(\mathcal{F})$?

######  Remark 4.11.

The Example 4.9 shows that the analogue of the Problem 3 with the condition
$\deg(C)\neq 0\mod p$ is not true.

######  Proposition 4.12.

Assume that the answer for the Problem 3 is YES. Let $\mathcal{F}$ be a
holomorphic foliation on $\mathbb{P}_{\mathbb{C}}^{2}$ of degree two and
suppose that it is non-degenerate with singularities having all eigenvalues
not in $\mathbb{Q}$. Then, $\mathcal{F}$ has no algebraic solution if and only
if $\mathcal{F}_{\mathfrak{p}}$ has irreducible $p$-divisor for infinitely
many primes $\mathfrak{p}$.

###### Proof.

If the $p$-divisor is irreducible for infinitely many primes $\mathfrak{p}$
then the result follows from Proposition 4.3. So, suppose that $\mathcal{F}$
does not have algebraic solutions. Since $\mathcal{F}$ is non-degenerate with
all eigenvalues not a element of $\mathbb{Q}$ we conclude that
$\mathcal{F}_{\mathfrak{p}}$ is not $p$-closed for an infinitely many primes
$\mathfrak{p}$. Suppose by contradiction that
$\Delta_{\mathcal{F}_{\mathfrak{p}}}$ is not irreducible for infinitely many
primes $\mathfrak{p}$. Then for each that prime $\mathfrak{p}$ there is
$C_{\mathfrak{p}}$ an irreducible factor of
$\Delta_{\mathcal{F}_{\mathfrak{p}}}$ that is
$\mathcal{F}_{\mathfrak{p}}$-invariant with
$\deg(C_{\mathfrak{p}})<\deg(\Delta_{\mathcal{F}_{\mathfrak{p}}})$. There are
two possibilities:

  * •

$\deg(C_{\mathfrak{p}})\leq 4$ for infinitely many primes $\mathfrak{p}$, or

  * •

$\deg(C_{\mathfrak{p}})>4$ for infinitely many primes $\mathfrak{p}$: in that
case, since $\deg(\Delta_{\mathcal{F}_{\mathfrak{p}}})=p+4$ there is a
irreducible $\mathcal{F}_{\mathfrak{p}}$-invariant curve $D_{\mathfrak{p}}$ of
degree less than $p$.

In any case we conclude that for infinitely many primes $\mathfrak{p}$ there
is a $\mathcal{F}_{\mathfrak{p}}$-invariant curve $D_{\mathfrak{p}}$ of degree
$\deg(D_{\mathfrak{p}})<p$. Since we are assuming the Problem 3 we conclude
that for an infinitely of many primes $\mathfrak{p}$ we have
$\operatorname{sing}(D_{\mathfrak{p}})\subset\operatorname{sing}(\mathcal{F}_{\mathfrak{p}})$.
In particular, since each eigenvalue of $\mathcal{F}$ is not a rational
number, we have that $\alpha\mod\mathfrak{p}$ is not an element of
$\mathbb{F}_{p}$ for infinitely many primes $\mathfrak{p}$. In particular, the
singularities of the curve $D_{\mathfrak{p}}$ are nodes. By the [5, Theorem
13] we conclude that $\deg(D_{\mathfrak{p}})\leq 4$. So, for infinitely many
primes $\mathfrak{p}$ the foliation $\mathcal{F}_{\mathfrak{p}}$ admits an
invariant curve of degree $\leq 4$. By Proposition 4.2 it follows that
$\mathcal{F}$ has an invariant curve of degree $\leq 4$, a contradiction. ∎

##  5 Automorphism and the $p$-divisor

In this section, motivated by the work [17] we introduce the notion of special
pair and study its main properties. We investigate the influence of the
automorphism group of foliations on the degree of invariant curves and we
prove Theorem A and Theorem B.

######  Proposition 5.1.

Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ non-$p$-closed on
$\operatorname{\mathbb{P}_{k}^{2}}$ over a field of characteristic $p>0$.
Suppose that $p<d$ and that there exist $\Phi$ an automorphism of
$\mathcal{F}$ of order $e$. Let $C$ be an irreducible curve and invariant by
$\mathcal{F}$ such that $\Phi^{l}C\neq\Phi^{h}C$ if $l\neq h$ for every
$l,h\in\\{0,\ldots,e-1\\}$. If $e\geq d^{2}$ then $\deg(C)=1$.

###### Proof.

Let $\psi=\Phi^{*}$ the isomorphism induced by $\Phi$ on the space of curves
on $\mathbb{P}_{\operatorname{k}}^{2}$ of degree $\deg(C)$. This action is
given via the pull-back by $\Phi$ of the polynomial equation defining the
curve. Since $\Phi\in\operatorname{Aut}(\mathcal{F})$ and
$\psi^{h}C\neq\psi^{l}C$ for $h\neq l\in\\{0,\ldots,e-1\\}$ we have by [13,
Proposition 3.8] that

$\Delta_{\mathcal{F}}\geq
C+\psi(C)+\cdots+\psi^{e-1}(C)=\sum_{j=0}^{e-1}\psi^{j}C.$

In particular, $\deg(\Delta_{\mathcal{F}})\geq\sum_{j=0}^{e-1}\deg(\psi^{j}C)$
so that $\deg(\Delta_{\mathcal{F}})\geq e\deg(C)$. Since
$\deg(\Delta_{\mathcal{F}})=p(d-1)+d+2$, $e\geq d^{2}$ and $p<d$ we get
$d^{2}+2=d(d-1)+d+2>p(d-1)+d+2\geq d^{2}\deg(C)$. In particular,
$2>d^{2}(\deg(C)-1)$ so that $\deg(C)=1$. ∎

We recall the following lemma (see [17, Lemma 1]).

######  Lemma 5.2.

Let $C$ be a $\mathcal{F}$-invariant curve. If $\operatorname{k}$ has
characteristic $p>0$ suppose that $p$ does not divide $d+2$. Then there is an
unique form $\beta_{\mathcal{F},C}$ such that

$dF\wedge\omega=F\left(\left(\frac{\deg(C)}{d+2}\right)d\omega+i_{R}\beta_{\mathcal{F},C}\right)$

where $i_{R}$ denotes the contraction of $\beta_{\mathcal{F},C}$ with the
radial vector field $R=x\partial_{x}+y\partial_{y}+z\partial_{z}$

###### Proof.

We recall the proof. The sequence $(x,y,z)$ of the polynomial ring
$R=\operatorname{k}[x,y,z]$ is a regular sequence. From this, we conclude by
[18, Tag 062F] that the Koszul complex induced by the radial vector field

${0}$${\bigwedge^{3}R^{3}}$${\bigwedge^{2}R^{3}}$${R^{3}}$${R}$${0}$$\scriptstyle{i_{R}}$$\scriptstyle{i_{R}}$$\scriptstyle{i_{R}}$

is exact. Now, consider the element
$\sigma:=dF\wedge\omega-(\frac{\deg(C)}{d+2})Fd\omega\in\bigwedge^{2}R^{3}$
and note that $i_{R}\sigma=0$, so that there is an unique
$\beta\in\bigwedge^{3}R^{3}\cong R$ such that $i_{R}\beta=\sigma$. ∎

######  Definition 5.3.

Let $\mathcal{F}$ be a foliation on $\operatorname{\mathbb{P}_{k}^{2}}$ and
$C$ be a $\mathcal{F}$-invariant curve. We say that the pair $(\mathcal{F},C)$
is special if $\beta_{\mathcal{F},C}=0$.

The degree of the pair $(\mathcal{F},C)$ is denoted by $(d,e)$ where $d$ is
the degree of $\mathcal{F}$ and $e$ is the degree of $C$.

######  Example 5.4.

If $\operatorname{k}$ is a field of characteristic $p>0$ and $\mathcal{F}$ is
a non-$p$-closed foliation on $\mathbb{P}_{\operatorname{k}}^{2}$ with reduced
$p$-divisor then $(\mathcal{F},\Delta_{\mathcal{F}})$ is special. This follows
from [3, Theorem 6.2]

######  Proposition 5.5.

Let $(\mathcal{F},C)$ be a special pair of degree $(d,e)$ and suppose that
$\mathcal{F}$ admits only $p$-reduced singularities. Then
$\operatorname{sing}(\mathcal{F})\subset C$ and $m_{q}(C)=2$ for all
$q\in\operatorname{sing}(\mathcal{F})$.

###### Proof.

Let $q\in\operatorname{sing}(\mathcal{F})$ and suppose by contradiction that
$q\not\in C$. By a change of coordinates we can assume that $q=[0:0:1]$. Let
$\Omega$ be a projective $1$-form defining $\mathcal{F}$ and
$F\in\operatorname{k}[x,y,z]_{e}$ a homogeneous polynomial defining $C$. Since
$(\mathcal{F},C)$ is special there is a relation: $(d+2)\cdot
dF\wedge\Omega=e\cdot Fd\Omega$. Since we are assuming that $q\not\in C$ in
the affine open subset $D_{+}(z)\cong\mathbb{A}_{\operatorname{k}}^{2}$ the
have the equation $(d+2)\cdot df\wedge\omega=e\cdot fd\omega$ with

$\omega=\omega_{1}+\omega_{2}+\cdots\qquad,\qquad f=f_{0}+f_{1}+f_{2}+\cdots$

where $\omega_{i}=a_{i}(x,y)dx+b(x,y)dy$ with $a_{i},b_{i}$ homogeneous
polynomials of degree $i$, $f_{h}$ homogeneous polynomial of degree $h$,
$\omega_{1}=ydx+\alpha xdy$, $f_{0}\in\operatorname{k}^{*}$ and
$\alpha\in\operatorname{k}-\mathbb{F}_{p}$. In particular,
$\sigma:=df/f\in\Omega_{\mathbb{P}_{\mathbb{C}}^{2},q}^{1}$, i.e., it is
regular at point $q$. So, if
$\mathcal{M}_{\mathbb{P}_{\operatorname{k}}^{2},q}$ is the maximal ideal of
the point $q$ in $\mathbb{P}_{\operatorname{k}}^{2}$ we have

$0=(d+2)\cdot\sigma\wedge\omega\mod\mathcal{M}_{\mathbb{P}_{\mathbb{C}}^{2},q}=e\cdot
f_{0}d\omega_{1}\mod\mathcal{M}_{\mathbb{P}_{\mathbb{C}}^{2},q}=e\cdot
f_{0}(\alpha-1)dx\wedge dy\mod\mathcal{M}_{\mathbb{P}_{\mathbb{C}}^{2},q}$

so that $\alpha=1$. But, this is a contradiction since we are assuming that
$q$ is $p$-reduced. So, $q\in C$. Now, suppose that $f_{1}\neq 0$, that is,
$q$ is a regular point of $C$ and write $f_{1}=ux+vy$ where
$u\in\operatorname{k}^{*}$ or $v\in\operatorname{k}^{*}$. Considering the
smallest term in the equation $(d+2)\cdot df\wedge\omega=e\cdot fd\omega$ we
get

$(d+2)\cdot df_{1}\wedge\omega_{1}=e\cdot f_{1}d\omega_{1}$

and by expansion we get two equations:

$u\alpha(d+2)=ue(\alpha-1)\qquad,\qquad-v(d+2)=ev(\alpha-1).$

Now, if $u\neq 0$ then we conclude that $1/\alpha=(1-(d+2)/e)$. In particular,
$\alpha\in\mathbb{F}_{p}$, a contradiction. If $v\neq 0$ then
$\alpha=1-(d+2)/e\in\mathbb{F}_{p}$, a contradiction. So, we conclude that
$u=v=0$ so that $f_{1}=0$. In particular,
$\operatorname{sing}(\mathcal{F})\subset\operatorname{sing}(C)$. Now since $q$
is $p$-reduced by [13, Lemma 3.11] we conclude that
$m_{q}(\Delta_{\mathcal{F}})=2$ and since $C\leq\Delta_{\mathcal{F}}$ we get
$m_{q}(C)=2$. This ends the proof of the proposition. ∎

######  Corollary 5.6.

Assume that $\operatorname{k}$ has characteristic $p>2$. Let $(\mathcal{F},C)$
be a special pair of degree $(d,e)$ and suppose that $\mathcal{F}$ admits at
least a $p$-reduced singularity. Then $\deg(C)\equiv d+2\mod p$.

###### Proof.

Let $q$ be a $p$-reduced singularity of $\mathcal{F}$. Without loss of
generality, we can assume that $q=[0:0:1]$. In the affine open set $D_{+}(z)$
the foliation is given by a $1$-form $\omega=\omega_{1}+\omega_{2}+O(2)$ and
by Proposition 5.5 the curve given by $f=f_{2}+O(3)$ with
$f_{2}=a_{1}x^{2}+a_{2}xy+a_{3}y^{2}$ with $a_{i}\neq 0$ for some $i$.
Considering the smallest term in the equation $(d+2)\cdot
df\wedge\omega=e\cdot fd\omega$ we obtain:

$(d+2)df_{2}\wedge\omega_{1}=e\cdot f_{2}d\omega_{1}$

and expanding that equation we obtain the following equations over the field
$\operatorname{k}$:

$2a_{1}\alpha(d+2)=a_{1}e(\alpha-1)\quad,\quad
a_{2}e(\alpha-1)=(d+2)a_{2}(\alpha-1)\quad\mbox{and}\quad
a_{3}e(\alpha-1)=-2(d+2)a_{3}.$

If $a_{1}\neq 0$ or $a_{3}\neq 0$ then we get a contradiction since
$\alpha\not\in\mathbb{F}_{p}$. So, $a_{1}=a_{3}=0$ and we conclude that
$a_{2}\neq 0$. In particular, we obtain $d+2=e$ identity over the field
$\operatorname{k}$. This proves the corollary. ∎

######  Remark 5.7.

The proof of the Corollary 5.6 shows that all singularities of $C$ along
$\operatorname{sing}(\mathcal{F})$ are nodal.

######  Remark 5.8.

It follows from Corollary 5.6 that a foliation, $\mathcal{F}$, defined over a
field of characteristic $p>0$ has a $p$-reduced singular point then
$\mathcal{F}$ does not have any invariant lines.

In the next proposition, we use the fact that a $p$-reduced foliation of
degree $d$ on the projective plane has $d^{2}+d+1$ distict singularities. This
follows from the computations in [2, Proposition 2.1].

######  Lemma 5.9.

Assume that $\operatorname{k}$ has characteristic $p>2$. Let $\mathcal{F}$ be
a non $p$-closed foliation of degree two on the projective plane
$\mathbb{P}_{\operatorname{k}}^{2}$ and suppose that $\mathcal{F}$ is
$p$-reduced. Let $C=\\{F=0\\}$ be an irreducible $\mathcal{F}$-invariant curve
with $\deg C<\deg\Delta_{\mathcal{F}}$. Then $(\mathcal{F},C)$ is not special.

###### Proof.

Suppose by contradiction that $(\mathcal{F},C)$ is special. The $p$-divisor
has the form $\Delta_{\mathcal{F}}=C+R$ and by the Corollary 5.6 we know that
$\deg(C)\in\\{4,4+p\\}$ and since $\deg(C)<\deg(\Delta_{\mathcal{F}})$ we
conclude $\deg(C)=4$. Proposition 5.5 ensures that
$\operatorname{sing}(\mathcal{F})\subset\operatorname{sing}(C)$ and
$m_{q}(C)=2$ for all $q\in\operatorname{sing}(\mathcal{F})$. Now, by [6,
Theorem 2, page 60] we have

$7=\\#\operatorname{sing}(\mathcal{F})\leq\sum_{q\in\operatorname{sing}(C)}\frac{m_{q}(C)(m_{q}(C)-1)}{2}\leq\frac{(4-1)(4-2)}{2}=3$

a contradiction. So, $(\mathcal{F},C)$ is not special. ∎

######  Lemma 5.10.

Let $p>2$ be a prime number such that $p\not\equiv 1\mod 3$. Let
$d\in\mathbb{Z}_{>1}$ be an integer and suppose that $p\nmid
d(d+2)(d^{2}+d+1)$. Then the Jouanolou foliation of degree $d$,
$\mathcal{J}_{d}$, over $\operatorname{k}$ is not $p$-closed.

###### Proof.

Note that the Jouanolou foliation over $\operatorname{k}$ is just the
reduction modulo $p$ of the Jouanolou foliation defined on the complex
projective plane. So, by the [13, Lemma 3.11] it is sufficient to show that
the reduction modulo $p$ of the eigenvalues of $\mathcal{J}_{d}$ is not an
element of $\mathbb{F}_{p}$. By the properties of the Jouanolou foliation, we
know that the automorphism $\Phi_{d}$ acts transitively on the set
$\operatorname{sing}(\mathcal{J}_{d})$ and this implies that all the
singularities of $\mathcal{J}_{d}$ has the same eigenvalues
$\alpha\in\mathbb{C}$. The Baum-Bott formula (see [9, Theorem 3.1]) implies
$(d+2)^{2}=(d^{2}+d+1)(\alpha+\alpha^{-1}+2)$ and this implies that $\alpha$
and $\alpha^{-1}$ are the roots of the polynomial
$p(x)=x^{2}+((d^{2}-2d-2)/(d^{2}+d+1))x+1\in\mathbb{C}[x]$.

The polynomial $p(x)$ has discriminant equal to
$\delta=-3d^{2}(d+2)^{2}/(d^{2}+d+1)^{2}$. In particular, $\alpha\mod p$ is
not an element of $\mathbb{F}_{p}$ if and only if $-3$ is not square modulo
$p$. By the law of quadratic reciprocity, we conclude that $\mathcal{J}_{d}$
is not $p$-closed if and only if $p\not\equiv 1\mod 3$. ∎

######  Theorem 5.11.

Let $p>2$ be a prime number such that $7\nmid p+4$ and such that $p\not\equiv
1\mod 3$. Then, the Jouanolou foliation of degree two $\mathcal{J}_{2}$
defined over a field of characteristic $p$ has irreducible $p$-divisor.

###### Proof.

The Jouanolou foliation on the $\mathbb{P}_{\operatorname{k}}^{2}$ is given by
the projective $1$-form:

$\omega=(x^{2}z-y^{3})dx+(xy^{2}-z^{3})dy+(z^{2}y-x^{3})dz.$

Let $\gamma\in\operatorname{k}$ be a primitive $7$th root of unity. Note that
since $p\neq 7$ the morphism

$\Phi\colon\mathbb{P}_{\operatorname{k}}^{2}\longrightarrow\mathbb{P}_{\operatorname{k}}^{2}\qquad[x:y:z]\mapsto[\gamma^{5}x:\gamma
y:z]$

is a automorphism of the foliation $\mathcal{J}_{2}$ with
$\Phi^{*}\omega=\gamma\omega$.

The condition $p\not\equiv 1\mod 3$ implies that the Jouanolou foliation is
not $p$-closed by Lemma 5.10. Suppose by contradiction that $\mathcal{J}_{2}$
has non-irreducible $p$-divisor. In particular, there is a irreducible
$\mathcal{J}_{2}$-invariant curve $C=\\{F=0\\}$ with
$\deg(C)<\deg(\Delta_{\mathcal{J}_{2}})$. Lemma 5.9 ensures that the pair
$(\mathcal{J}_{2},C)$ is not special. In particular, there is a
$3$-homogeneous form $\beta$ of degree one on
$\mathbb{A}_{\operatorname{k}}^{3}$ such that

$dF\wedge\omega=F\Theta\quad\mbox{where}\quad\Theta=\frac{\deg(C)}{4}d\omega+i_{R}\beta.$

Now we will consider two cases:

  * •

some power $\Phi^{l}$ fixes the curve $C$: In that case, we conclude that
$(\Phi^{l})^{*}\beta=\gamma^{l}\beta$. If $\beta=H(x,y,z)dx\wedge dy\wedge dz$
with $H(x,y,z)=ax+by+cz\in\operatorname{k}[x,y,z]_{1}$ we get the equations:

$\gamma^{10l}a=a\quad,\quad\gamma^{6l}b=b\quad,\quad\gamma^{5l}c=c$

so that $a=b=c=0$, a contradiction since $C$ is not special.

  * •

$\Phi^{l}$ does not fix the curve $C$ for any $l\in\\{0,\ldots,6\\}$: In
particular, $(\Phi^{*})^{l}C\neq(\Phi^{*})^{k}C$ for every $k\neq
l\in\\{0,\ldots,6\\}$: Since $\deg(\Delta_{\mathcal{J}_{2}})=p+4$ and $\Phi$
has order $7$ we observe that $\Delta_{\mathcal{J}_{2}}$ does not have
$p$-factor. So, there are two possibilities: There is a prime divisor $P$ in
the support of $\Delta_{\mathcal{J}_{2}}$ that is fixed by some power of
$\Phi$ or there is not that prime. In the first case, we use the previous
argument in order to get a contradiction. In the last case, we conclude that
$7\mid p+4$, a contradiction.

This ends the proof of the theorem. ∎

######  Corollary 5.12.

In the conditions of the Theorem 5.11, a generic foliation of degree two on
the projective plane over characteristic $p>0$ has irreducible $p$-divisor.

######  Corollary 5.13.

The Jouanolou foliation of degree two $\mathcal{J}_{2}$ on
$\mathbb{P}_{\mathbb{C}}^{2}$ has no algebraic solutions.

###### Proof.

By Theorem 5.11 we know that the reduction modulo $p$ of $\mathcal{J}_{2}$ has
irreducible $p$-divisor for infinitely many primes $p$. Now, the result
follows from Proposition 4.3. ∎

######  Remark 5.14.

If $p=7$ the argument used in the corollary above does not works because in
that case there is no primitive $7$th root. Using the computer algebra system
[4, Singular] we check that the $7$-divisor of the Jouanolou foliation
$\mathcal{J}_{2}$ is given by

$\Delta_{\mathcal{J}_{2}}=\\{-3x^{3}z+xy^{3}+2yz^{3}=0\\}+7\\{2x+y-3y=0\\}.$

######  Proposition 5.15.

Let $\mathcal{F}$ be a $p$-reduced foliation on
$\operatorname{\mathbb{P}_{k}^{2}}$ and let $C$ be an irreducible
$\mathcal{F}$-invariant curve. Suppose that $\mathcal{F}$ is not $p$-closed
and that $\deg(C)<\deg(\Delta_{\mathcal{F}})$. Let
$\Phi\in\operatorname{Aut}(\mathcal{F})-\\{1\\}$ be an automorphism of
$\mathcal{F}$. If $(\mathcal{F},C)$ is special then there is a
$\mathcal{F}$-invariant irreducible curve $D$ such that $(\mathcal{F},D)$ is
not special or $\Phi$ fixes the curve $C$.

###### Proof.

Suppose that $\Phi$ does not fix the curve $C$. Write
$\Delta_{\mathcal{F}}-C=\sum_{i}\alpha_{i}P_{i}$ where $P_{i}$’s are prime
divisors. By Corollary 5.6 we know that $\deg(C)=d+2+pl$ for some
$l\in\\{0,\ldots,d-2\\}$ so that $\sum_{i}\alpha\deg(P_{i})=p(d-1-l)$. We have
two possibilities:

  * •

$\alpha_{i}\equiv 0\mod p$ for all $i$: In that case, if $\alpha_{i}=pl_{i}$
for some $l_{i}\in\mathbb{Z}_{\geq 0}$ then we conclude
$\sum_{i}l_{i}\deg(P_{i})=d-1-l<d$. In particular, $\deg(P_{i})<d$. But this
is a contradiction since
$\Delta_{\mathcal{F}}=\Phi^{*}\Delta_{\mathcal{F}}=\Phi^{*}C+\sum_{i}\alpha_{i}\Phi^{*}P_{i}$
with $C\neq\Phi^{*}C$ and $C<\Delta_{\mathcal{F}}$.

  * •

$\alpha_{i}\not\equiv 0\mod p$ for some $i$: In that case, $P_{i}$ is a
$\mathcal{F}$-invariant divisor by [13, Proposition 3.8]. Now, note that
$(\mathcal{F},P_{i})$ is not special. Indeed, assume by contradiction that
$(\mathcal{F},P_{i})$ is special. Then, by Proposition 5.5 we know that
$\operatorname{sing}(\mathcal{F})\subset\operatorname{sing}(P_{i})$. In
particular, for every $q\in\operatorname{sing}(\mathcal{F})$ we have
$2=m_{q}(\Delta_{\mathcal{F}})\geq m_{q}(C)+m_{q}(P_{i})\ =4$, a
contradiction.

This ends the proof of the proposition. ∎

######  Theorem 5.16.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $d>2$ be a positive integer such that

  * •

$p<d$ and $p\not\equiv 1\mod 3$;

  * •

$d^{2}+d+1$ is prime.

Then the Jouanolou foliation $\mathcal{J}_{d}$ on
$\mathbb{P}_{\operatorname{k}}^{2}$ has irreducible $p$-divisor or
$\Delta_{\mathcal{J}_{d}}=C+pR$ with $(\mathcal{J}_{d},C)$ special,
$\deg(C)=pl+d+2$ with $l>0$ and $R=\sum_{i}\alpha_{i}P_{i}$ with $P_{i}$ not
$\mathcal{J}_{d}$-invariant for every $i$.

###### Proof.

The condition $p\not\equiv 1\mod 3$ implies that $\mathcal{J}_{d}$ is not
$p$-closed by Lemma 5.10. Suppose that there is $C$ a $\mathcal{F}$-invariant
curve such that $\deg(C)<\deg(\Delta_{\mathcal{J}_{d}})$. Consider the pair
$(\mathcal{J}_{d},C)$ and the automorphism:
$\Phi_{d}\colon\mathbb{P}_{\operatorname{k}}^{2}\longrightarrow\mathbb{P}_{\operatorname{k}}^{2}\in\operatorname{Aut}(\mathcal{J}_{d})$
which associes $[x:y:z]\mapsto[\gamma^{d^{2}+1}x:\gamma y:z].$ We have two
cases:

  * •

$(\mathcal{J}_{d},C)$ is not special: in this case we consider two subcases:
suppose first that $\Phi_{d}^{l}$ does not fix $C$ for every
$l\in\\{0,\ldots,d^{2}+d\\}$. In this case, by Proposition 5.1 we conclude
that $\deg(C)=1$, a contradiction since $\mathcal{J}_{d}$ does not have
invariant lines (see Remark 5.8). Consider the case where $\Phi_{d}^{l}$ fixes
$C$ for some $l$ and consider to the identity:

$dF\wedge\omega=F\Theta_{C}\quad\mbox{where}\quad\Theta_{C}=\frac{\deg(C)}{d+2}d\omega+i_{R}\beta.$

Applying the automorphism $\Phi_{d}^{l}$ on the equation above we conclude
that

$(\Phi_{d}^{l})^{*}\beta=\gamma^{l}\beta.$

Write $\beta=b(x,y,z)dx\wedge dy\wedge dz$ and $b(x,y,z)=\sum_{0\leq j<i\leq
d-1}b_{ij}x^{j}y^{i-j}z^{d-1-i}$. Then, since $\Phi_{d}^{l}$ associes
$[x:y:z]\mapsto[\gamma^{l(d^{2}+1)}x:\gamma^{l}y:z]$ we conclude that
$b(x,y,z)=\gamma^{l(d^{2}+1)}b(\gamma^{l(d^{2}+1)}x,\gamma^{l}y,z)=\gamma^{-ld}b(\gamma^{l(d^{2}+1)}x,\gamma^{l}y,z)$,
so that if $b_{ij}\neq 0$ then

$(\gamma^{j(d^{2}+1)+i-j-d})^{l}=(\gamma^{-dj+i-j-d})^{l}=1\Longrightarrow(\gamma^{dj-i+j+d})^{l}=1.$

Since $d^{2}+d+1$ is prime we conclude that $d^{2}+d+1\mid dj-i+j+d$. But,
$dj-i+j+d<dj+j+d\leq d(d-1)+d-1+d=d^{2}+d-1<d^{2}+d+1$, a contradiction.

  * •

$(\mathcal{J}_{d},C)$ is special: by Proposition 5.15 we know that there is a
$\mathcal{J}_{d}$-invariant prime divisor $D$ such that $(\mathcal{J}_{d},D)$
is not special or the automorphism $\Phi_{d}$ fixes the curve $C$. By the
precedent case, it follows that for every $\mathcal{J}_{d}$-invariant prime
divisor $D$ with $\deg(D)<\deg(\Delta_{\mathcal{J}_{d}})$ the pair
$(\mathcal{J}_{d},D)$ is special. So, it follows that $\Phi_{d}$ fixes $C$.
Write

$\Delta=C+\sum_{i}\alpha_{i}P_{i}$

and observe that $P_{i}^{\prime}s$ are not $\mathcal{J}_{d}$-invariant for
every $i$. Indeed, if $P_{i}$ is $\mathcal{J}_{d}$-invariant for some $i$,
then the pair $(\mathcal{J}_{d},P_{i})$ is special. By Proposition 5.5 we
conclude that $2=m_{q}(\Delta_{\mathcal{J}_{d}})\geq m_{q}(C)+m_{q}(P_{i})\geq
2+2$ for all $q\in\operatorname{sing}(\mathcal{F})$, a contradiction. In
particular, by [13, Proposition 3.8] we conclude that $\alpha_{i}\equiv 0\mod
p$ for every $i$ and the $p$-divisor has the form
$\Delta_{\mathcal{J}_{d}}=C+pR$ with $(\mathcal{J}_{d},C)$ special and
$R=\sum_{i}\alpha_{i}P_{i}$ with $P_{i}$ not $\mathcal{J}_{d}$-invariant for
every $i$.

This ends the proof of the theorem. ∎

######  Example 5.17.

Consider the Jouanolou foliation of degree $d\leq 100$ over characteristic
$p=5$. Using the computer algebra system [4, Singular] we can check that the
$5$-divisor is irreducible if $d\in\\{2,14,24,54,59,69,89,99\\}$. Otherwise,
$\mathcal{J}_{d}$ is $5$-closed or $\Delta_{\mathcal{J}_{d}}=C+pR$ with
structure given by the following table:

$d$ |  $d^{2}+d+1$ |  $\deg(C)$ |  $l$ |  $R$
---|---|---|---|---
$6$ |  $43$ |  $18$ |  $2$ |  $\\{xyz=0\\}$
$12$ |  $157$ |  $39$ |  $5$ |  $\\{xyz=0\\}$
$17$ |  $307$ |  $54$ |  $7$ |  $\\{xyz=0\\}$
$21$ |  $463$ |  $63$ |  $8$ |  $\\{xyz=0\\}$
$27$ |  $757$ |  $84$ |  $11$ |  $\\{xyz=0\\}$
$41$ |  $1723$ |  $123$ |  $16$ |  $\\{xyz=0\\}$
$57$ |  $3307$ |  $174$ |  $23$ |  $\\{xyz=0\\}$
$62$ |  $3907$ |  $189$ |  $25$ |  $\\{xyz=0\\}$
$66$ |  $4423$ |  $198$ |  $26$ |  $\\{xyz=0\\}$
$71$ |  $5113$ |  $213$ |  $28$ |  $\\{xyz=0\\}$
$77$ |  $6007$ |  $234$ |  $31$ |  $\\{xyz=0\\}$

where $(\mathcal{J}_{d},C)$ is a special pair of degree $(d,5l+d+2)$. Note
that $p$ and $d$ satisfy the conditions of the Theorem 5.16.

######  Corollary 5.18.

In the conditions of Theorem 5.16 the Jouanolou foliation on the projective
plane $\operatorname{\mathbb{P}_{k}^{2}}$ has an unique irreducible invariante
curve of degree $pl+d+2$ with $0<l\leq d-1$.

######  Remark 5.19.

It is expected that for infinitely many of $d\in\mathbb{Z}_{>0}$ the integer
$d^{2}+d+1$ is a prime number. This is a particular case of the Bunyakovsky
conjecture which says that, under some conditions, given a polynomial
$p(x)\in\mathbb{Z}[x]$ we have $p(d)$ prime for infinitely many
$d\in\mathbb{Z}_{>0}$ (see [16]).

##  6 Non-algebraicity via reduction modulo $2$

In this section, we prove a particular case of a well-known theorem of
Jouanolou which says that a very generic foliation of degree $d>1$ on the
projective plane does not have any algebraic solutions. In this section, we
present a simple proof of this result in the case where $d$ is odd with
$d\not\equiv 1\mod 3$. The main topic in this proof is that we use reduction
modulo two and some irreducibility criterion for polynomials in two variables
via Newton polytopes due to S.Gao.

Let $\mathcal{S}=\\{P_{1},\ldots,P_{n}\\}$ be a collection of points on
$\mathbb{R}^{2}$. Recall that the convex hull of $\mathcal{S}$ denoted by
$\langle\mathcal{S}\rangle=\langle P_{1},P_{2},\ldots,P_{n-1},P_{n}\rangle$ is
the smallest convex subset of $\mathbb{R}^{2}$ which contains $S$. This is
equivalent to say that $\langle\mathcal{S}\rangle$ is the intersection of all
convex subsets of $\mathbb{R}^{2}$ which contains $S$. It is possible to check
that

$\langle\mathcal{S}\rangle=\\{t_{1}P_{1}+t_{2}P_{2}+\cdots+t_{n-1}P_{n-1}+t_{n}P_{n}\in\mathbb{R}^{2}\mid\sum_{i}t_{i}=0\\}.$

Let $\operatorname{k}$ be a field and
$f(x,y)=\sum_{i,j}a_{ij}x^{i}y^{j}\in\operatorname{k}[x,y]$ a polynomial.
Recall that the Newton polytope associated to $f(x,y)$ is the convex hull os
the set $\mathcal{S}(f)=\\{(i,j)\in\mathbb{Z}^{2}\mid a_{ij}\neq 0\\}$.

######  Example 6.1.

Let $f(x,y)=x^{2}y+xy^{4}+y^{2}+xy^{2}\in\operatorname{k}[x,y]$. In this case,
$\mathcal{S}(f)=\\{(2,1),(1,4),(0,2),(1,2)\\}$. The Newton polytope associated
to $f$ is the triangle with vertices $A=(2,1)$, $B=(1,4)$ and $C=(1,2)$.

We recall the following theorem from [8].

######  Proposition (Shuhong Gao).

Let $\operatorname{k}$ be a field and
$f=ax^{m}+by^{n}+\sum_{i,j}c_{ij}x^{i}y^{j}\in\operatorname{k}[x,y]$ with
$a,b\in\operatorname{k}^{*}$ and $(i,j)$ different from $(m,0),(0,n)$. Suppose
that the Newton polytope of $f$ is contained in the triangle with vertices
$(m,0),(0,n)$ and $(u,v)$ for some point $(u,v)\in\mathbb{R}^{2}$. If
$gcd(m,n)=1$ then $f$ is absolutely irreducible over $\operatorname{k}$.

###### Proof.

see [8, Corollary 4.12]. ∎

As a particular case, we have the following corollary.

######  Corollary 6.2.

Let $\operatorname{k}$ be any field, $d\in\mathbb{Z}_{>1}$ and consider the
following polynomial of degree $d$:
$f(x,y)=x^{d-1}+y^{2d+1}+x^{d}y^{d}+x^{2d+1}y^{d-1}\in\operatorname{k}[x,y]$.
Then $f$ is absolutely irreducible.

###### Proof.

Let $A=(d-1,0)$, $B=(0,2d+1)$, $C=(d,d)$ and $D=(2d+1,d-1)$ be points in
$\mathbb{R}^{2}$ and consider its convex hull: $\textbf{R}:=\langle
A,B,C,D\rangle\subset\mathbb{R}^{2}$. Recall that by definition this is the
set: $\mbox{{R}}=\\{aA+bB+cC+dD\mid a+b+c+d=1\\}.$ The convex hull of the
points $A,B,D$ is the triangle with vertices $A,B,C$. Denote this triangle by
$\textbf{T}:=\langle A,B,D\rangle$. We claim that $\textbf{R}=\textbf{T}$.
Indeed, it is sufficient to show that $C\in\textbf{T}$. Note that,

$C=(d,d)=\frac{1}{3}(d-1,0)+\frac{1}{3}(0,2d+1)+\frac{1}{3}(2d+1,d-1)\in\textbf{T}.$

Now, consider the ideal generated $I=\langle
d-1,2d+1\rangle\subset\mathbb{Z}$. Note that the condition $d\not\equiv 1\mod
3$ implies that the ideal $I=\mathbb{Z}$. Indeed, we have: $I=\langle
d-1,2d+1\rangle=\langle d-1,2d+1-(d-1)\rangle=\langle d-1,d+2\rangle=\langle
d-1,3\rangle=\mathbb{Z}.$ In particular, by the proposition of S.Gao we
conclude that the polynomial $f$ is absolutely irreducible. ∎

######  Proposition 6.3.

Let $\operatorname{k}$ be a algebraically closed field of characteristic two
and $\mathcal{J}_{d}$ be the Jouanolou foliation of degree $d>1$ on
$\operatorname{\mathbb{P}_{k}^{2}}$. If $d\equiv 0\mod 2$ then
$\mathcal{J}_{d}$ is $2$-closed and if $d\equiv 1\mod 2$ then
$\mathcal{J}_{d}$ is not $2$-closed with irreducible $2$-divisor.

###### Proof.

The Jouanolou foliation of degree $d>1$ defined over a field $K$ is given on
$D_{+}(z)\cong\mathbb{A}_{\operatorname{k}}^{2}$ by the following vector field
$v=(xy^{d}-1)\partial_{x}-(x^{d}-y^{d+1})\partial_{y}$ and by computations we
have:

$v^{2}=((d+1)xy^{2d}-y^{d}-dx^{d+1}y^{d-1})\partial_{x}+((d+1)y^{2d+1}-(2d+1)x^{d}y^{d}+dx^{d-1})\partial_{y}$

and $D_{2}(v)=\Delta_{\mathcal{F}}|_{D_{+}(z)}=v^{2}(y)v(x)-v^{2}(x)v(y)$ is
given by

$D_{2}(v)=(2d+2)xy^{3d+1}-(d+2)y^{2d+1}-(4d+2)x^{d+1}y^{2d}+(3d+2)x^{d}y^{d}+dx^{2d+1}y^{d-1}-dx^{d-1}.$

In particular, if $K=\operatorname{k}$ we have that $\mathcal{J}_{d}$ is
$2$-closed if $d\equiv 0\mod 2$, since $D_{2}(v)=0$ in that case. On the other
hand, if $d\equiv 1\mod 2$ then $v\wedge
v^{2}=f(x,y)\partial_{x}\wedge\partial_{y}$ where $f(x,y)$ is the local
equation to the $2$-divisor associated to $\mathcal{J}_{d}$ given by
$f(x,y)=y^{2d+1}+x^{d}y^{d}+x^{2d+1}y^{d-1}+x^{d-1}$ and by the Corollary 6.2
we conclude that the $2$-divisor is irreducible of degree $3d$. ∎

######  Lemma 6.4.

Let $\mathcal{J}_{d}$ be the Jouanolou foliation on the complex projective
plane of degree $d\not\equiv 1\mod 3$. If $\mathcal{J}_{d}$ has a reduced
$\mathcal{J}_{d}$-invariant algebraic curve then there is a
$\mathcal{J}_{d}$-invariant reduced curve $D$ such that $\deg(D)=d+2$.

###### Proof.

Let $D$ the curve obtained by conjugation of $C$ via
$\operatorname{Aut}(\mathcal{J}_{d})$. This curve is reduced and it is
invariant by $\operatorname{Aut}(\mathcal{J}_{d})$. By the properties of the
automorphism group of $\mathcal{J}_{d}$, it follows that the pair
$(\mathcal{J}_{d},D)$ is special (see [17, Lemma 3]). The argument now is very
similar to the argument on Corollary 5.6, but with computations over the
complex numbers (see also the argument after [17, Lemma 2]). ∎

Now we can present the main result of this section theorem.

######  Theorem 6.5.

A very generic foliation of odd degree $d>1$ with $d\not\equiv 1\mod 3$ on the
complex projective plane does not have algebraic solutions.

###### Proof.

It is sufficient to show that the Jouanolou foliation has no algebraic
solution. Let $\mathcal{J}_{d}$ be the Jouanolou foliation with $d\equiv 1\mod
2$ and $d\not\equiv 1\mod 3$. By the Proposition 6.3 we know that the
foliation $\mathcal{F}_{d}:=\mathcal{J}_{d}\otimes\mathbb{F}_{2}$ is not
$2$-closed with $2$-divisor $\Delta_{\mathcal{F}_{d}}$ irreducible. Suppose by
contradiction that there is $C=\\{F=0\\}$ an irreducible invariant curve
defined by a reduced polynomial $F\in\mathbb{C}[x,y,z]$. We can assume that
$C$ is defined by a irreducible polynomial over $\mathbb{Z}$. Indeed, let $K$
be the smallest field extension of $\mathbb{Q}$ where all coefficients of $F$
are defined. Clearing denominators we can assume $F\in\mathcal{O}_{K}[x,y,z]$.
Let $G_{K/\mathbb{Q}}$ the Galois group of $K/\mathbb{Q}$ and for $\sigma\in
G_{K/\mathbb{Q}}$ denote by $F^{\sigma}$ the natural action of $\sigma$ on the
coefficients of $F$. Consider the polynomial $H=\prod_{\sigma\in
G_{K/\mathbb{Q}}}F^{\sigma}$. By this construction we conclude that
$H\in\mathbb{Z}[x,y,z]$ and it is $\mathcal{J}_{d}$-invariant. In particular,
some irreducible factor $Q$ of $H$ gives a $\mathcal{J}_{d}$-invariant curve
defined over $\mathbb{Z}$. In particular, it is reduced over $\mathbb{C}$. By
the Lemma 6.4 we know that the conjugation of $Q$, via
$\operatorname{Aut}(\mathcal{J}_{d})$, defines a $\mathcal{J}_{d}$-invariant
curve $\tilde{C}=\\{\tilde{Q}=0\\}$ such that $\deg(\tilde{Q})=d+2\equiv 1\mod
2$. Let $L$ the smallest field extension of $\mathbb{Q}$ such that all
coefficients of $\tilde{Q}$ are defined. By clearing denominators we can
assume that $\tilde{Q}\in\mathcal{O}_{L}[x,y,z]$. Let
$\mathfrak{m}\in\operatorname{\textbf{Spm}}(\mathcal{O}_{L})$ be a maximal
ideal over the maximal ideal $2\mathbb{Z}$, i.e.
$2\mathbb{Z}=\mathfrak{m}\cap\mathbb{Z}$. Since $\tilde{Q}$ is not a
$2$-factor and since reduction modulo $\mathfrak{m}$ preserves the invariance
we can find $Q_{2}$ a irreducible fator of
$\tilde{Q}\otimes\mathbb{F}_{\mathfrak{m}}$ which defines a
$\mathcal{F}_{d}$-invariant curve. By the irreducibility of
$\Delta_{\mathcal{F}_{d}}$ we conclude that $Q_{2}=\Delta_{\mathcal{F}_{d}}$
and so $d+2\geq\deg(Q_{2})=\deg(\mathcal{F}_{d})=3d$, contradiction since
$d>1$. ∎

Acknowledgements. W.Mendson thanks J.V. Pereira for conversations about some
topics related to Section 4. The author acknowledges support from
CAPES/COFECUB.

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