# A hide-and-seek game: Looking for Population III stars during the Epoch of
Reionization through the HeII$\lambda$1640 line

Alessandra Venditti Dipartimento di Fisica, Sapienza, Universit$\grave{a}$ di
Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy  INFN, Sezione di Roma I,
Piazzale Aldo Moro 2, 00185, Roma, Italy  INAF-Osservatorio Astronomico di
Roma, Via di Frascati 33, 00078, Monte Porzio Catone, Italy  Dipartimento di
Fisica, Tor Vergata, Universit$\grave{a}$ di Roma, Via Cracovia 50, 00133,
Roma, Italy  Volker Bromm Department of Astronomy, University of Texas at
Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA  Weinberg Institute
for Theoretical Physics, University of Texas at Austin, Austin, TX 78712, USA
Steven L. Finkelstein Department of Astronomy, University of Texas at Austin,
2515 Speedway, Stop C1400, Austin, TX 78712, USA  Antonello Calabrò INAF-
Osservatorio Astronomico di Roma, Via di Frascati 33, 00078, Monte Porzio
Catone, Italy  Lorenzo Napolitano INAF-Osservatorio Astronomico di Roma, Via
di Frascati 33, 00078, Monte Porzio Catone, Italy  Dipartimento di Fisica,
Sapienza, Universit$\grave{a}$ di Roma, Piazzale Aldo Moro 5, 00185, Roma,
Italy  Luca Graziani Dipartimento di Fisica, Sapienza, Universit$\grave{a}$ di
Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy  INFN, Sezione di Roma I,
Piazzale Aldo Moro 2, 00185, Roma, Italy  INAF-Osservatorio Astronomico di
Roma, Via di Frascati 33, 00078, Monte Porzio Catone, Italy  Raffaella
Schneider Dipartimento di Fisica, Sapienza, Universit$\grave{a}$ di Roma,
Piazzale Aldo Moro 5, 00185, Roma, Italy  INFN, Sezione di Roma I, Piazzale
Aldo Moro 2, 00185, Roma, Italy  INAF-Osservatorio Astronomico di Roma, Via di
Frascati 33, 00078, Monte Porzio Catone, Italy

###### Abstract

The gas surrounding first-generation (Pop III) stars is expected to emit a
distinct signature in the form of the HeII recombination line at 1640
Å(HeII$\lambda$1640). Here we explore the challenges and opportunities in
identifying this elusive stellar population via the HeII$\lambda$1640 in
$M_{\star}>10^{7.5}~{}\mathrm{M_{\odot}}$ galaxies during the Epoch of
Reionization (EoR, $z\simeq 6-10$), using JWST/NIRSpec. With this aim in mind,
we combine cosmological dustyGadget simulations with analytical modeling of
the intrinsic HeII emission. While tentative candidates with bright HeII
emission like GN-z11 have been proposed in the literature, the prevalence of
such bright systems remains unclear due to significant uncertainties involved
in the prediction of the HeII luminosity. In fact, similar Pop III clumps
might be almost two orders of magnitude fainter depending on the underlying
Pop III model (e.g., with/without stellar mass loss and/or assuming different
Pop III-formation efficiencies in star-forming clouds). Moreover, up to $\sim
90\%$ of these clumps might be missed with NIRSpec/MOS due to the limited FoV,
while this problem appears to be less severe with NIRSpec/IFU. We propose deep
spectroscopy targeting potential peripheral Pop III clumps around bright,
massive galaxies as a promising approach to achieve a clear detection of the
first stars.

stars: Population III — galaxies: star formation –– galaxies: high-redshift ––
dark ages, reionization, first stars –– cosmology: theory

††facilities:  JWST(NIRSpec) ††software:  numpy https://numpy.org, matplotlib
https://matplotlib.org, scipy https://scipy.org, astropy
http://www.astropy.org, JWST Exposure Time Calculator
https://jwst.etc.stsci.edu

##  1 Introduction

The deployment of JWST has opened new frontiers for modern astrophysics,
enabling us to explore the depth of the high-$z$ Universe with unprecedented
sensitivity and resolution. Notably, it paves the way for the exciting
possibility of directly detecting the first generation of stars, known as
Population III (Pop III) stars.

The HeII recombination line at 1640 $\mathrm{\SIUnitSymbolAngstrom}$
(HeII$\lambda$1640) has been indicated as a potential tracer of Pop IIIs
(Tumlinson & Shull, 2000; Tumlinson et al., 2001; Bromm et al., 2001b;
Schaerer, 2002, 2003; Raiter et al., 2010). Due to their pristine chemical
composition, Pop III stars are expected in fact to be predominantly massive
(Abel et al., 2002; Bromm et al., 2002), up to $\sim$100s $\mathrm{M}_{\odot}$
(Hosokawa et al., 2011; Hirano et al., 2014; Stacy et al., 2016; Chon et al.,
2023), or even $\sim$1000s $\mathrm{M}_{\odot}$ (Hirano et al., 2015a, b; Susa
et al., 2014; Hosokawa et al., 2016; Sugimura et al., 2020; Latif et al.,
2022). This massive component should power a very hard radiation (> 54.4 eV),
able to doubly ionise He in the nearby gas, and therefore trigger the
HeII$\lambda$1640 line emission through the cascading recombination of HeIII.

While Pop III stars are predicted to start forming at Cosmic Dawn, around
$z\sim 20-30$ (Bromm, 2013; Klessen & Glover, 2023), cosmological simulations
(Xu et al., 2016; Jaacks et al., 2019; Liu & Bromm, 2020; Sarmento et al.,
2018; Sarmento & Scannapieco, 2022; Skinner & Wise, 2020; Venditti et al.,
2023) and semi-analytical models (Visbal et al., 2020; Trinca et al., 2023)
suggest that pristine gas reservoirs hosting Pop IIIs might persist down to
the Epoch of Reionization (EoR, $z\sim 6-10$). There are already a few
candidates with tentative HeII detection, possibly indicative of Pop IIIs at
these epochs (Wang et al., 2022; Maiolino et al., 2023; Vanzella et al.,
2023); however, their confirmation is still pending. Hence, the following
questions arise: how many more sources similar to present candidates can we
discover? If the plethora of models predicting Pop III star formation at later
cosmic times is accurate, why aren’t more of these candidates observed?

Before considering a radical overhaul of our star formation and feedback
recipes, we need to account for all the systematics involved in the detection
of Pop IIIs. Perhaps the sensitivity of our instruments is insufficient,
either because the intrinsic HeII flux arising from Pop IIIs is too faint, or
because it is heavily absorbed by dust (Venditti et al., 2023; Curtis-Lake et
al., 2023; Roberts-Borsani et al., 2023). Moreover, a significant number of
Pop III systems at these redshifts might fall outside the field-of-view (FoV)
of our instruments, if they reside at the periphery of their hosting dark-
matter haloes (Venditti et al., 2023). Finally, the HeII recombination
signature is expected to be short-lived, due to the brief lifetime of the most
massive stars ($\sim$ few Myr, Schaerer, 2002, 2003; Katz et al., 2023).
Understanding all these challenges is crucial to design robust strategies for
a systematic search of Pop IIIs during the EoR, with the goal of expanding our
pool of available candidates.

This letter aims to explore all these aspects, by combining the statistics of
late Pop III clumps inferred from the cosmological simulations introduced in
Di Cesare et al. (2023) and Venditti et al. (2023) with an analytical
modelling of the HeII emission arising from Pop III stars (Schaerer, 2002). In
Section 2, we describe our cosmological simulations (Section 2.1) and the
adopted procedure to estimate the HeII luminosity from Pop III stellar
populations (Section 2.2). In Section 3 we presents our results, i.e.: (i) our
predictions of the HeII luminosity, compared with the sensitivity of
JWST/NIRSpec in different configurations (Section 3.1); (ii) the expected bias
due to the limited FoV (Section 3.2); (iii) the expected number of HeII-
emitting Pop III systems in existing JWST surveys (Section 3.3). In Section 4
we critically discuss our findings, with particular reference to the effect of
dust absorption/scattering (Section 4.1) and to the potential of using HeII
for the identification of Pop III stars compared to other indicators, e.g.,
pair-instability supernovae (PISNe, Section 4.2). Finally, Section 5 presents
our conclusions.

##  2 Methodology

###  2.1 Simulating the cosmological environment

The cosmological simulations employed in the present work have been performed
with the hydrodynamical code dustyGadget (Graziani et al., 2020), and they are
described in Di Cesare et al. (2023). They consist of eight simulated volumes
(U6 - U13111Data from U9 and U11 are not included in the present work as these
simulations present different snapshot dumpings with respect to the others; in
fact, these cubes are less star-forming and hence of lower interest for the
present study.), with a comoving side of
$50h^{-1}~{}$\mathrm{c}\mathrm{M}\mathrm{p}\mathrm{c}$$, a total number of
$2\times 672^{3}$ particles and a mass resolution for dark matter/gas
particles of $3.53\times 10^{7}h^{-1}~{}$\mathrm{M}_{\odot}$$/$5.56\times
10^{6}h^{-1}~{}$\mathrm{M}_{\odot}$$ each, evolved from $z\simeq 100$ down to
$z\simeq 4$. A $\mathrm{\Lambda}$CDM cosmology consistent with Planck
Collaboration et al. (2016) is assumed ($\Omega_{\mathrm{m,0}}=0.3089$,
$\Omega_{\mathrm{b,0}}=0.0486$, $\Omega_{\mathrm{\Lambda},0}=0.6911$,
$h=0.6774$).

Detailed information on the dustyGadget code, and particularly on its
innovative self-consistent modelling for dust production and evolution, can be
found in Graziani et al. (2020). We here briefly summarize the main features
of the adopted prescriptions for star formation and feedback, of particular
interest for the present work.

Stellar particles with a mass of $\sim 2\times 10^{6}~{}$\mathrm{M}_{\odot}$$
are generated from gas particles with a cold phase number density
$n_{\mathrm{cold}}\gtrsim 300~{}$\mathrm{c}\mathrm{m}^{-3}$$. They represent
stellar populations born in an instantaneous burst with an assigned initial
mass function (IMF). Depending on the gas metallicity, below or above a
critical metallicity222Here we assume $Z_{\mathrm{\odot}}=0.02$ (Anders &
Grevesse, 1989). ($Z_{\mathrm{crit}}=10^{-4}~{}$\mathrm{Z}_{\odot}$$, Bromm et
al. 2001a; Maio et al. 2010; Graziani et al. 2020) we define a stellar
population to be Pop III or Pop II/I, respectively. We assume a Salpeter-
shaped IMF (Salpeter, 1955) with a mass range of [0.1, 100] ([100, 500])
$\mathrm{M}_{\odot}$ for Pop II/I (Pop III). The impact of the contribution of
low-mass Pop III stars, ($\sim 1-40~{}$\mathrm{M}_{\odot}$$, the mass range
inferred from stellar archaeology, Iwamoto et al. 2005; Keller et al. 2014;
Ishigaki et al. 2014; de Bennassuti et al. 2014, 2017; Hartwig et al. 2015;
Fraser et al. 2017; Rossi et al. 2021; Magg et al. 2022; Aguado et al. 2023)
is extensively discussed in Venditti et al. (2023) and Venditti et al. (2024).
In fact, although the aforementioned studies show that a precise modelling of
the low-mass end of the IMF is required to reconstruct the detailed
nucleosynthetic pattern of old, metal-poor stars, here we are mostly
interested in the high-mass tail that is mainly responsible for He ionisation,
because of its hard UV photon budget.

The gas chemical evolution model is adopted from Tornatore et al. (2007); Maio
et al. (2010, 2011). We include mass-dependent yields from Pop III stars in
the range [140, 260] $\mathrm{M}_{\odot}$, ending their life as PISNe (Heger &
Woosley, 2002), and mass and metallicity-dependent yields from Pop II/I stars
with low-intermediate mass (long-lived, van den Hoek & Groenewegen 1997) and
high mass (> 8 $\mathrm{M}_{\odot}$, dying as core-collapse supernovae,
Woosley & Weaver 1995), also considering type Ia supernovae (Thielemann et
al., 2003). For simplicity, we assume that Pop II/I stars more massive than 40
$\mathrm{M}_{\odot}$ and Pop III stars outside the PISN range directly
collapse into black holes and do not participate in the metal enrichment
process. This clearly is an idealization, and effects such as rapid rotation
could contribute to enrichment across a broader range of stellar masses (e.g.,
Liu et al., 2021). Dust and metals are spread in the inter-stellar medium
(ISM) through a spline kernel. Galactic winds are also modelled following
Springel & Hernquist (2003) with a constant velocity of
$500~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$ (Ginolfi et al., 2020).

The simulations have demonstrated good agreement with available model
predictions and observations of the cosmic star-formation-rate/stellar-mass
density evolution and with important scaling relations (i.e., the main
sequence of star-forming galaxies, the stellar-to-halo mass relation and the
dust-to-stellar mass relation), including early JWST data (Di Cesare et al.,
2023). They have also been employed to investigate C envelopes around merging
galaxies as a possible origin of the [CII]158 $\mu$m emission in the circum-
galactic medium surrounding individual, resolved galaxies, observed by the
ALPINE333ALMA Large Program to Investigate [CII] at Early Times Survey (Le
Fèvre et al., 2020; Faisst et al., 2020; Béthermin et al., 2020,
http://alpine.ipac.caltech.edu/). survey at $z\sim 4.5$ (Di Cesare et al.,
2024). Most notably, they are the largest simulated volumes currently
available that include a model for Pop III stars, making them a powerful tool
to understand the statistics of Pop III star formation across cosmic time
(Venditti et al., 2023, 2024). However, we emphasize that the limited mass
resolution, together with the lack of a proper treatment of radiative
feedback444The simulations model a homogeneous UV background as in Haardt &
Madau (1996). However, this does not produce feedback on cosmic star formation
at $z>6$., allows us to provide reliable results only for halos with a stellar
mass555Corresponding to a number of stellar particles $\gtrsim 20$. of
$\mathrm{log}(M_{\star}/$\mathrm{M}_{\odot}$)\gtrsim 7.5$ at $6\lesssim
z\lesssim 10$; note that all Pop III stars in this mass regime at the
considered redshifts are found to be coexisting with Pop II stellar components
in our simulations (Venditti et al., 2023). We also currently do not include a
model for metal mixing and turbulent metal diffusion below our gas mass
resolution (as e.g. in Sarmento et al., 2016, 2017, 2018; Sarmento &
Scannapieco, 2022). We refer the reader to Venditti et al. (2023, 2024) for a
thorough discussion of these limitations for Pop III studies.

###  2.2 Computing the HeII luminosity of Pop III clumps

The intrinsic luminosity of the HeII$\lambda$1640 line ($L_{\mathrm{HeII}}$)
emitted from a Pop III clump can be inferred from the mass of the clump
($M_{\mathrm{III}}$) as follows:

$L_{\mathrm{HeII}}=\overline{\varepsilon}_{\mathrm{HeII}}E_{\mathrm{HeII}}\times
M_{\mathrm{III}},$ (1)

with $E_{\mathrm{HeII}}\simeq 1.21\times 10^{-11}$ erg the energy of a
HeII$\lambda$1640 photon and $\overline{\varepsilon}_{\mathrm{HeII}}$ the
average HeII photon emissivity per unit stellar mass of a Pop III stellar
population. We do not take into account dust attenuation, whose expected
impact will be further discussed in Section 4.1.

Figure 1: Time-averaged HeII photon emissivity per unit stellar mass
$\varepsilon_{\mathrm{HeII}}$ as a function of the initial Pop III mass $m$,
from Schaerer (2002), in the mass range of our assumed Salpeter-like IMF
([100, 500] $\mathrm{M}_{\odot}$). The black, solid line refers to the model
assuming no mass loss (no ML, see table 4 of the original paper), while the
black, dotted line refers to the model assuming strong mass loss (strong ML,
see their table 5). The average emissivity
$\overline{\varepsilon}_{\mathrm{HeII}}$ of Pop III stellar populations for
the two models with our assumed IMF – i.e., the values adopted in Equation 2
for the present work – are also indicated on top of the horizontal,
solid/dotted, black lines. The red, dashed-dotted line further shows the cases
of 300 $\mathrm{M}_{\odot}$ and 500 $\mathrm{M}_{\odot}$ stars evolving on the
ZAMS along the entire stellar lifetime as a reference (Bromm et al., 2001b),
to exemplify how mass loss keeps stellar evolution closer to the ZAMS,
enhancing the time-averaged photon emissivity (as in the black, dotted line).

Figure 1 shows the time-averaged photon production rate
$\varepsilon_{\mathrm{HeII}}$ of individual Pop III stars of various masses
$m$, assuming either strong mass loss (ML) arising from high-mass stars
(dotted line) or no mass loss at all (solid line), from tables 5 and 4 of
Schaerer (2002) respectively. We compute
$\overline{\varepsilon}_{\mathrm{HeII}}$ by integrating over the IMF
$\phi(m)$:

$\overline{\varepsilon}_{\mathrm{HeII}}=\frac{\int_{m_{\mathrm{low}}}^{m_{\mathrm{up}}}\varepsilon_{\mathrm{HeII}}(m)\phi(m)\differential{m}}{\int_{m_{\mathrm{low}}}^{m_{\mathrm{up}}}\phi(m)\differential{m}}.$
(2)

For our assumed Salpeter-like IMF (whose lower and upper limits are
$m_{\mathrm{low}}=100~{}$\mathrm{M}_{\odot}$$ and
$m_{\mathrm{up}}=500~{}$\mathrm{M}_{\odot}$$, Section 2.1), this results in
$\overline{\varepsilon}_{\mathrm{HeII}}\simeq 1.53\times 10^{47}/6.48\times
10^{46}~{}${phot.}\,\mathrm{s}^{-1}\,\mathrm{M}_{\odot}^{-1}$$ for the case
with strong/no mass loss respectively. Note that mass loss causes the stars to
evolve close to the Zero-Age-Main-Sequence (ZAMS) for longer times, resulting
in higher time-averaged photon emissivities. In fact, the model666Shown for
the cases of 300 $\mathrm{M}_{\odot}$ and 500 $\mathrm{M}_{\odot}$ Pop III
stars, from private communication. See table 1 of Bromm et al. (2001b) for the
case of a 1000 $\mathrm{M}_{\odot}$ Pop III star. of Bromm et al. (2001b) –
assuming Pop III stars always evolving along the ZAMS – lies closer to the
strong-mass-loss case (red, dashed-dotted line).

As in Venditti et al. (2024), we consider the possibility of a Pop III mass
$M_{\mathrm{III}}$ in our Pop III clumps (in Equation 1) that is lower than
our resolution element $M_{\mathrm{III,res}}\sim 2\times
10^{6}~{}$\mathrm{M}_{\odot}$$ (i.e. the mass of a Pop III stellar particle in
our simulations), by introducing an efficiency factor $\eta_{\mathrm{III}}<1$:

$M_{\mathrm{III}}=\eta_{\mathrm{III}}M_{\mathrm{III,res}}.$ (3)

By interpreting $M_{\mathrm{III,res}}$ as the amount of extremely metal-poor
gas above our density threshold that is available for star formation, and
$M_{\mathrm{III}}$ as the amount of stellar mass actually produced in a single
star-formation event, we can place a lower limit on $\eta_{\mathrm{III}}\sim
0.01$ from simulations describing Pop III star formation in the first mini-
haloes (see e.g. Bromm 2013 and references therein). However, higher values
might be found in more massive haloes at later times, which are expected to
host more efficient star-formation sites. In fact, a past simulation (Greif et
al., 2008) of primordial gas collapsing into an atomic-cooling halo at $z\sim
10$ shows that the gas experiences a boost in ionization (e.g. through shocks)
resulting in more efficient cooling through the HD channel (Bromm et al.,
2009); this intermediate regime – previously referred to as Pop III.2 or Pop
II.5 – between the very first episodes of star formation and later Pop II/I
star formation has been predicted to yield higher star-formation efficiencies,
even a factor $\sim$10 higher than star formation in minihalos at Cosmic Dawn
(Greif & Bromm, 2006). In the absence of tight constraints in the mass regime
we are currently probing, we explore values up to $\eta_{\mathrm{III}}\sim
0.1$. We note that a similar value would also be implied considering the ratio
of the mass777The mass estimate from Maiolino et al. (2023) has been updated
with respect to Venditti et al. (2024) to match the accepted version of the
paper. inferred for the supposed Pop III clusters in GN-z11 at $z=10.6$ ($\sim
2-2.5\times 10^{5}~{}$\mathrm{M}_{\odot}$$, Maiolino et al., 2023) and
RXJ2129-z8HeII at $z\simeq 8.2$ ($7.8\pm 1.4\times
10^{5}~{}$\mathrm{M}_{\odot}$$, Wang et al., 2022) with respect to our
resolution element.

##  3 Results

###  3.1 HeII luminosity vs. JWST/NIRSpec sensitivity

Figure 2: Average HeII line luminosity ($L_{\mathrm{HeII}}$, shaded regions)
arising from Pop III-hosting dustyGadget galaxies at $z\sim 6.7-10$, assuming
a Salpeter-like IMF in the range [100, 500] $\mathrm{M}_{\odot}$, compared
with available observational candidates, i.e., LAP1 at $z\simeq 6.6$ (Vanzella
et al., 2023, purple star), RXJ2129-z8HeII at $z\simeq 8.2$ (Wang et al.,
2022, green star), and GN-z11 at $z\simeq 10.6$ (Maiolino et al., 2023, blue
star). The red/dark-red shaded regions refer to the models assuming no/strong
mass loss (ML) respectively, with variable Pop III formation efficiency
$\eta_{\mathrm{III}}$ from 0.01 (lower-end of the regions) to 0.3 (upper-end),
and considering the average Pop III mass in dustyGadget galaxies at these
redshifts in Equation 1. The oblique lines show the sensitivity of
JWST/NIRSpec corresponding to an integrated $\mathrm{S/N}\sim 3$ line
detection in both the IFU (left) and MOS (right) observing modes at $z\simeq
6.7$ and $z\simeq 10$ for different configurations, i.e., different resolving
powers ($R\sim 1000/2700/100$, with different thickness and markers), exposure
times ($\sim 10/50$ h, grey/golden) and line widths (500/50
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$, dashed/dotted). The corresponding
lines for an integrated $\mathrm{S/N}\sim 5$ line detection with medium
resolving power and a $\sim$ 50 h exposure are also shown in red, with the
same linestyles as the $\mathrm{S/N}\sim 3$ case for the two assumed line
widths.

Figure 2 provides predictions of the HeII luminosity ($L_{\mathrm{HeII}}$)
arising from dustyGadget galaxies hosting Pop IIIs as a function of redshift
$z$, considering different assumptions on the Pop III formation efficiency
$\eta_{\mathrm{III}}$ and mass loss. We estimate the average Pop III mass
($M_{\mathrm{III}}$) in Pop III-hosting galaxies between redshifts $\sim 6$
and $\sim 10$ and compute the resulting $L_{\mathrm{HeII}}$ from Equation 1.
The upper end of the red shaded regions in the plots is associated with the
models assuming strong mass loss and high $\eta_{\mathrm{III}}$ (up to a
value888This upper limit is derived by considering the ratio of the Pop III
mass inferred for RXJ2129-z8HeII ($\sim 7.8\times
10^{5}~{}$\mathrm{M}_{\odot}$$) to our typical Pop III stellar particle
masses. $\eta_{\mathrm{III}}=0.3$), while the lower end is associated with no
mass loss and low $\eta_{\mathrm{III}}$ (down to $\eta_{\mathrm{III}}=0.01$).
The purple, green and blue stars provide a comparison with the luminosity of
available observational candidates, i.e., LAP1999Note that, while the authors
provide constraints on the expected HeII flux in LAP1, the identification of
the observed feature as a HeII$\lambda$1640 line is weakened by the presence
of a small blueshift relative to the Balmer lines. Moreover, the measured flux
would require a quite extreme Pop III scenario. Hence, the authors
conservatively consider the line non-detected and derive an upper limit on the
HeII flux, also shown in the plot. at $z\simeq 6.6$ (Vanzella et al., 2023),
RXJ2129-z8HeII at $z\simeq 8.2$ (Wang et al., 2022) and GN-z11101010The two
points for GN-z11 in the left panel indicate two different NIRSpec/IFU
measures, considering a small aperture around the HeII clump (bottom point),
and a larger aperture aimed at also capturing an additional, more extended
emission, possibly coming from a fainter, less significant clump (top point).
The point in the right panel refers instead to the NIRSpec/MOS measure.
Although the two measures are consistent in terms of wavelength, comparing the
fluxes is non-trivial due to the uncertainty on the exact location of the MSA
shutter, and hence on the covered fraction of the putative HeII clump
(Maiolino et al., 2023). at $z\simeq 10.6$ (Maiolino et al., 2023).

The oblique lines indicate sensitivity limits for JWST/NIRSpec (Jakobsen et
al., 2022) in both the Integral Field Unit (IFU) and Multi-Object Spectroscopy
(MOS) modes at $z\simeq 6.7$ and $z\simeq 10$ for different configurations.
The limits are computed using the JWST Exposure Time
Calculator111111https://jwst.etc.stsci.edu, assuming a point source with no
continuum and a line centered at $\lambda\simeq
1.64\times[(1+z)/10]~{}\mu$\mathrm{m}$$, plus medium background. We adopt an
operational integrated signal-to-noise (S/N) threshold of $\sim 3$ to
determine the minimum observable line flux, which depends on the specific
observational setup as well as the chosen line width. We explore two possible
values for the line width, $\Delta
v=500/50~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$ (dashed/dotted lines),
corresponding respectively to the typical width of emission lines in high-$z$
galaxies ($\Delta v\simeq 50~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$) and
to a more extreme scenario121212A similar value of $\Delta v\simeq
428~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$ has been found for the
HeII$\lambda$1640 line detected in the lensed galaxy RXCJ2248-ID at $z=6.1$
(Topping et al., 2024), whose spectrum broadly resembles that of GN-z11
(Maiolino et al., 2023), minus the AGN signatures. ($\Delta v\simeq
500~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$). We further consider
observations with two exposure times, $t\simeq 10/50$ h (grey/golden lines),
with the appropriate grating/filter pair depending on the redshifted
wavelength of the line at medium/high resolution (i.e., resolving power
$R=1000/2700$) and with the Prism/CLEAR at low resolution131313For the
PRISM/CLEAR, only the 500 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ case is
shown, as the instrument cannot discriminate emission lines with $\Delta
v\lesssim 5000~{}$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$$ at these
wavelengths, and hence using a lower $\Delta v$ does not change the results.
($R=100$).

In the IFU mode, the instrument is centered on the source with an
aperture141414As we aim for the smallest possible aperture in order to
optimise the sampled flux at the center, we consider a value of the order of
the Point Spread Function (PSF) Full Width Half Maximum (FWHM) at $\lambda\sim
1.5~{}\mu$\mathrm{m}$$ (as a reference, note that the line is redshifted at
$\lambda\simeq 1.3/1.8~{}\mu$\mathrm{m}$$ at $z\simeq 6.7/10$ respectively).
of 0.05”, while the sky annulus151515The choice of the sky annulus used for
background subtraction does not change our results appreciably, as we are
considering a uniform background. spans a range between 0.3” and 0.9”. In the
MOS mode, we select a three-shutters (-1,0,1) slitlet shape, with the source
placed in shutter 0 and the Micro-Shutter Assembly (MSA) located in quadrant 3
center; we apply the MSA full shutter extraction strategy for background
subtraction. The Improved Reference Sampling and Subtraction (IRS2, Rauscher
et al. 2012) readout pattern is employed for both cases.

It is evident that even with $\sim 10$ h observations and considering the
narrow-line, best-case scenario at all the available spectral resolutions, we
would only be able to capture some of the most extreme Pop III systems
($\gtrsim 10^{41}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$$). All
the observed candidates lie in fact in the upper part of this plot. Very low-
luminosity systems ($\lesssim 4\times
10^{40}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$$) – having low
$\eta_{\mathrm{III}}$ and/or no mass loss – might be missed even in the
deepest exposures ($\sim 50$ h). However, it is to be noted that more
favourable conditions are possible. For example, while here we conservatively
assumed a medium background, the very low background during the NIRSpec/IFU
observation of GN-z11 allowed the detection of a fainter HeII line than
expected (with a $\simeq 10.6$ h exposure, in the small aperture). Moreover,
the line appears unresolved in the G235M/F170LP grating/filter pair ($R\sim
1000$), meaning even narrower lines (with higher S/N) may be found.

###  3.2 Pop IIIs outside JWST/NIRSpec field-of-view

Another factor contributing to the potential oversight of Pop III systems is
their placement outside the FoV of our instruments. As star formation is
typically more efficient in the dense, central regions of galaxies (e.g,
Carrasco et al., 2010; van Dokkum et al., 2014), peripheral areas tend to
evolve at a slower pace, preserving their chemically pristine state for
extended periods of time. Additionally, these regions may experience gas
infall from the external environment. As a result, Pop III stars might be
found as far as $\sim 20$ kpc from the galactic centre (see figure 10 of
Venditti et al. 2023), especially in massive, evolved galaxies, which have
undergone prolonged periods of star formation. Maiolino et al. (2023), for
example, find a potential Pop III clump at a distance $\sim$ 2 kpc from the
host galaxy of GN-z11 ($M_{\star}\sim 8\times 10^{8}~{}$\mathrm{M}_{\odot}$$).

Figure 3: Illustration of a single JWST/NIRSpec pointing centered on Pop III-
hosting galaxies (blue clouds) in the IFU/MOS modes, respectively (left/right;
in the MOS mode, three consecutive shutters are considered). The distances
$d_{\mathrm{proj}}$ of the peripheral Pop III clusters (small blue clouds +
red stars) from the galactic center in the displayed projection are shown as
blue arrows, while the largest distances such that the clusters would
certainly fall within the FoV independent of the instrument’s orientation
($d_{\mathrm{min}}$), or that conversely they would fall within for at least
one possible orientation ($d_{\mathrm{max}}$), are shown as dashed/solid
arrows, spanning the whole dashed/solid circumferences. The dimensions of
NIRSpec/IFU FoV and of the three shutters of NIRSpec/MOS are also indicated in
the figure (not to scale). Depending on the orientation of the galaxies with
respect to our instruments, peripheral Pop III systems might fall out of the
FoV and would thus be missed in observations.

We estimate the number of Pop III systems that might be missed in a single
pointing of JWST/NIRSpec centered on dustyGadget haloes. Note that two degrees
of freedom need to be taken into account in this calculation:

  1. 1.

the orientation of the galaxy with respect to our instruments. In fact, when
projecting a three-dimensional galaxy onto a two-dimensional image on the sky
along an arbitrary direction, the hosted Pop III cluster will be observed at a
distance $d_{\mathrm{proj}}$ from the galactic center which is at most equal
to the three-dimensional distance $d_{\mathrm{3D}}$. Particularly, there is
always a direction along which the Pop III cluster is exactly aligned with the
center along our line-of-sight to the source. For each simulated galaxy, we
consider the worst possible projection in which
$d_{\mathrm{proj}}=d_{\mathrm{3D}}$, hence providing an upper limit on the
number of Pop III systems that we expect to be missing when pointing towards
the galactic center;

  2. 2.

the orientation of the instrument. As shown in Figure 3, we consider two
cases: (i) a Pop III system is “missed” when we are never able to see it
however we rotate the FoV, i.e., when $d_{\mathrm{proj}}>d_{\mathrm{max}}$
(falling out of the solid circumferences encompassing all the possible
orientations of the instrument); (ii) a Pop III system is “potentially missed”
when it might be missed depending on the particular orientation of the FoV,
i.e., $d_{\mathrm{proj}}>d_{\mathrm{min}}$ (falling out of the dashed
circumference enclosing the area always covered with a random, fixed
orientation of the instrument). For the IFU/MOS NIRSpec modes (the latter
assumed in a three-shutters configuration), we have $d_{\mathrm{max}}\simeq
2.1/1.0^{\prime\prime}$ and $d_{\mathrm{min}}\simeq 1.5/0.1^{\prime\prime}$.

Figure 4: Top panels: Number density of Pop III particles ($n_{\mathrm{III}}$)
that we expect to find at a given redshift $z$ in haloes within a given range
of stellar mass $M_{\star}$ as a function of $M_{\star}$, computed in six bins
of $M_{\star}$ (with a spacing of 0.5 dex in the range
$7.5\leq\mathrm{Log}M_{\star}/$\mathrm{M}_{\odot}$<10.5$). The total
$n_{\mathrm{III}}$ is shown as a grey, solid line, while the
$n_{\mathrm{III}}$ found considering a single pointing of JWST/NIRSpec
centered on the host galaxy in the IFU/MOS mode, respectively, are shown as
golden/brown lines. The solid/dashed linestyle refer to the best/worst-case
scenario for the orientation for the instrument, considering the worst line-
of-sight to the source (see text and Figure 3 for details). The total number
of Pop III particles found in each bin is also indicated on top of the bins.
Bottom panels: Fraction of Pop IIIs missed in JWST/NIRSpec pointings, with
same colour/linestyle convention. Results are shown for the combined simulated
volumes U6, U7, U8, U10, U12, and U13 at redshifts $z=8.1$ (left panels),
$z=7.3$ (middle panels) and $z=6.7$ (right panels). We find that a significant
number of Pop III systems can be overlooked in these configurations,
especially in high-mass galaxies, although this problem appears to be less
severe with NIRSpec/IFU thanks to its larger FoV.

The top panels of Figure 4 show the number density of Pop III particles that
would be found within the NIRSpec FoV in all the considered configurations,
compared to the total. Results are shown at $z=8.1$, 7.3 and 6.7 in
bins161616The total number of Pop III particles found in each bin among all
the simulated cubes is indicated in the plots, serving as a cautionary note
regarding the limited statistics at the highest-stellar-mass bins. More
reliable results would necessitate even larger simulated boxes or more
simulated volumes. of stellar mass171717For reference, the relation between
stellar mass and dark matter mass in dustyGadget galaxies is shown in figure 7
of Di Cesare et al. (2023). $M_{\star}$. The bottom panels illustrate,
conversely, the fraction of Pop IIIs missed in all the aforementioned
scenarios. We find that a significant number of Pop III systems can be
overlooked in these configurations, especially in high-mass galaxies, although
this problem appears to be less severe with NIRSpec/IFU thanks to its larger
FoV.

Note that, in the case of GN-z11, the HeII$\lambda$1640 emission found to peak
at about $d_{\mathrm{proj}}=0.5^{\prime\prime}$ from the center (i.e.
$d_{\mathrm{min}}<d_{\mathrm{proj}}<d_{\mathrm{max}}$ for the MOS mode), would
have been missed with a different orientation of the MSA, while it is always
included in the FoV of NIRSpec/IFU ($d_{\mathrm{proj}}<d_{\mathrm{min}}$).
Interestingly, the small size of the MSA slits have also been found to lead to
a possible underestimation of the Ly$\alpha$ flux in Ly$\alpha$ emitters in
the presence of a spatial offset between the UV and Ly$\alpha$ emission, or of
an extended diffuse Ly$\alpha$ emission (Nakane et al., 2023; Napolitano et
al., 2024).

###  3.3 Expected Pop III systems in JWST surveys

Figure 5: Number of potentially observable HeII-emitting Pop III systems
$N_{\mathrm{HeII}}$ as a function of the effective survey volume
$V_{\mathrm{eff}}$ at $z=8.1$ (see text for details). Vertical, dashed lines
indicate the effective volume of selected JWST surveys and their cumulative
volume at $z\simeq 8$, with $\Delta z=1$ (filled circles, see text for
details); the number of Pop III systems found in our six dustyGadget cubes is
also indicated in the plot (DG, empty squares). A horizontal, dashed line
further indicates the reference value of 1 system per volume. The black line
refers to the number found in all haloes with
$7.5<\mathrm{Log}(M_{\star}/$\mathrm{M}_{\odot}$)\leq 9.5$, while the coloured
lines refer to the number found in haloes of different stellar mass bins (see
Figure 4).

Figure 5 shows the predicted number $N_{\mathrm{III}}$ of HeII-emitting Pop
III systems in galaxies of various masses
($7.5<\mathrm{Log}(M_{\star}/$\mathrm{M}_{\odot}$)\leq 9.5$) at $z=8.1$ that
are potentially observable with a single pointing of JWST/NIRSpec in its IFU
mode, as a function of effective survey volume $V_{\mathrm{eff}}$. We consider
the systems that would fall within the FoV in the worst possible projection
($d_{\mathrm{proj}}=d_{\mathrm{3D}}$) with any orientation of the instrument
($d_{\mathrm{proj}}<d_{\mathrm{min}}$; see Section 3.2), hence this should be
considered as a lower limit. Specifically, we multiply the values of
$n_{\mathrm{III}}$ from the simulations (first panel of Figure 4, golden,
dashed line) by the comoving volume of JWST surveys at $z\simeq 8$, with
$\Delta z=1$:

  1. 1.

the Next Generation Deep Extragalactic Exploratory Public (NGDEEP) Survey
(Finkelstein et al., 2021; Pirzkal et al., 2023; Bagley et al., 2023);

  2. 2.

the Grism Lens-Amplified Survey from Space
(GLASS181818https://glass.astro.ucla.edu/ers/, Treu et al. 2017, 2022;
Castellano et al. 2022);

  3. 3.

the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization
(UNCOVER191919https://jwst-uncover.github.io/, Bezanson et al. 2022; Furtak et
al. 2023; Weaver et al. 2023);

  4. 4.

the Cosmic Evolution Early Release Science Survey
(CEERS202020https://ceers.github.io/ceersi-first-images-release, Finkelstein
et al. 2017, 2022, 2023);

  5. 5.

the Public Release IMaging for Extragalactic Research
(PRIMER212121https://primer-jwst.github.io/) survey (Dunlop et al., 2021);

  6. 6.

the PANORAMIC survey (Williams et al., 2021);

  7. 7.

the Cosmic Evolution Survey (COSMOS-
Web222222https://cosmos.astro.caltech.edu/, Casey et al. 2023).

Our simulations suggest that more than 400 Pop III systems could be discovered
in galaxies with $M_{\star}\gtrsim 10^{7.5}~{}$\mathrm{M}_{\odot}$$ around
$z\simeq 8$, within all these combined JWST fields, and more than one system
is supposed to be found within each individual survey. However, costly
spectroscopic follow-up on these sources is required for their identification.
Moreover, the number of HeII emitters above the sensitivity limits for a given
instrumental setup strongly depends on the assumed model for Pop III star
formation (see Section 3.1). As an example, only $\sim 75/85\%$ of the whole
range of luminosities spanned by the models in Figure 2 would be covered with
a $\sim 10$ h exposure at medium spectral resolution, assuming a line width of
500/50 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ respectively, while up to $\sim
90/95\%$ would be covered with a $\sim 50$ h exposure.

These Pop III models are likely not equiprobable in reality. Depending on the
actual nature of Pop III stars, fainter Pop III models might be favoured
(especially in low-mass galaxies, which are more numerous), causing a large
fraction of these systems to only be accessible through very deep exposures.
As discussed in Section 2.2, both the Pop III-formation efficiency and IMF
could in fact vary depending on environmental conditions, such as the mass of
the dark matter host where the Pop IIIs are formed. We emphasize the need for
more in-depth studies of Pop III star formation in mini-halos versus
Ly-$\alpha$ cooling halos, that would allow us to infer a probability
distribution function for the values of the two parameters
$\overline{\varepsilon}_{\mathrm{HeII}}$ (Equation 2) and
$\eta_{\mathrm{III}}$ (Equation 3) – and hence for the HeII luminosity
$L_{\mathrm{HeII}}$ (Equation 1) – as a function of host mass.

##  4 Discussion

###  4.1 Dust absorption/scattering

In Figure 2 we considered the intrinsic HeII emission arising from Pop III
star-forming regions. However, light can be removed from our lines-of-sight to
the sources because of both absorption and scattering by interstellar dust,
which should be accounted for via an additional factor $\sim\exp{-\tau}$ in
Equation 1, with $\tau$ the dust optical depth at
$\lambda=1640~{}$\mathrm{\SIUnitSymbolAngstrom}$$. Models of dust mixtures
reproducing the observed extinction in the Milky
Way232323https://www.astro.princeton.edu/~draine/dust/dust.html (Weingartner &
Draine, 2001; Li & Draine, 2001; Draine, 2003a, b, c; Glatzle et al., 2019)
predict in fact non-negligible values of the dust absorption cross section per
unit dust mass ($\sim 30\%$ of the peak value242424A maximum dust absorption
cross section per unit dust mass of $\simeq 1.5\times
10^{5}~{}$\mathrm{c}\mathrm{m}^{2}\,\mathrm{g}^{-1}$$ is found assuming an
extinction ratio $R_{\mathrm{V}}=3.1$ (see e.g. Figure 1 of Glatzle et al.
2019).) and of the albedo/scattering asymmetry parameter ($\sim 0.4/0.6$) at
$\lambda=1640~{}$\mathrm{\SIUnitSymbolAngstrom}$$.

Candidate HeII emitters such as GN-z11 (Jiang et al., 2021; Tacchella et al.,
2023), RXJ2129-z8HeII (Wang et al., 2022), LAP1 (Vanzella et al., 2023) and
RXCJ2248-ID (Topping et al., 2024) are consistent with essentially no dust
attenuation. However, recent ALMA programs, such as ALPINE and
REBELS252525Reionization Era Bright Emission Line Survey (Bouwens et al.,
2022)., have unveiled a population of dusty, obscured star-forming galaxies at
$4\lesssim z\lesssim 9$, which is estimated to contribute $\sim 10-25\%$ to
the $z>6$ cosmic star formation rate density (Fudamoto et al., 2021). The
presence of a significant population of red, optically-faint galaxies at these
redshifts, especially at the high-mass end of the stellar mass
function262626Gottumukkala et al. (2024) find that the obscured galaxy SMF at
$6<z<8$ overtakes the pre-JWST SMF around
log$(M_{\star}/$\mathrm{M}_{\odot}$)\sim 10.375$. By integrating the SMF at
log$(M_{\star}/$\mathrm{M}_{\odot}$)>9.25$, they estimate that the stellar
mass density might even double with respect to pre-JWST studies. (SMF), is
further confirmed by JWST (e.g. Xiao et al. 2023; Gottumukkala et al. 2024). A
lack of prominent emission lines272727Particularly, Curtis-Lake et al. (2023)
reported $2\sigma$ upper limits of $\simeq
6-15.4~{}$\mathrm{\SIUnitSymbolAngstrom}$$ on the HeII equivalent widths. Two
out of the four analysed objects also indicate moderate levels of dust (V-band
optical depth, $\tau_{\mathrm{V}}\sim 0.2$), albeit with large uncertainties.
– possibly ascribed to high levels of dust absorption, and/or a combination of
low overall star formation rate and intrinsic faintness – has also been
reported in the spectra of metal-poor galaxies at $z\gtrsim 10$ observed with
JWST (Curtis-Lake et al., 2023; Roberts-Borsani et al., 2023).

A detailed estimate of the dampening of the HeII line caused by interstellar
dust would require full radiative-transfer calculations, that are beyond the
goals of the present work. However, here we remark that the results shown in
Figure 2 should be interpreted as an upper limit, while the actual HeII
luminosity will depend on the global dust content of the galaxies282828See
e.g. Di Cesare et al. (2023) for the dust-to-stellar mass scaling relations of
dustyGadget galaxies. and on their viewing angle, due to their very
inhomogeneous dust distribution (Venditti et al. 2023; see also Smith et al.
2019). Focusing on an individual Pop III-hosting galaxy at $z=7.3$, we found
that the optical depth ($\tau$) resulting from dust absorption only (i.e.
neglecting the contribution of scattering) can vary from $\sim 10^{-8}$ up to
$\sim 10$ depending on the specific line-of-sight to the sources (Venditti et
al., 2023). A strong viewing angle dependence of dust attenuation is further
demonstrated by Cochrane et al. (2024), focusing on HST-dark galaxies at
$4<z<7$.

###  4.2 Detecting Pop III through HeII vs. PISNe

In Venditti et al. (2024), we discussed an alternative channel to identify Pop
III-hosting galaxies, looking for massive Pop III stars
($140~{}$\mathrm{M}_{\odot}$<m<260~{}$\mathrm{M}_{\odot}$$, Heger & Woosley
2002) at the moment of their death as PISNe. These supernovae are expected to
be extremely bright, reaching bolometric luminosities higher than $\sim
10^{45}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$$ during the short
shock-breakout phase and $\sim
10^{44}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-1}$$ during their
long-term light-curve evolution (Kasen et al., 2011), i.e. $\sim 2-3$ orders
of magnitude brighter than our most optimistic scenario for the HeII line.
Moreover, they could be more straightforwardly identified even without
requiring costly spectroscopic analysis292929Hartwig et al. (2018) and Moriya
et al. (2022) discuss for example the optimal filter combinations to detect
PISNe at $z\gtrsim 6$ with JWST and with the Nancy Grace Roman Space
Telescope, and to discriminate between different types of supernovae (see also
Wang et al., 2012). In Venditti et al. (2024) we further discussed how the
peak emission from PISNe can easily outshine the stellar emission of their
hosting galaxies, especially in spatially resolved observations.. In fact, the
high temperatures required to power HeII line emission can be achieved through
a number of other confusing mechanisms/sources, including X-ray binaries
(Schaerer et al., 2019; Saxena et al., 2020a, b; Senchyna et al., 2020;
Cameron et al., 2023; Lecroq et al., 2024), Wolf-Rayet stars (Kehrig et al.,
2018; Saxena et al., 2020a; Shirazi & Brinchmann, 2012; Senchyna et al., 2021;
Cameron et al., 2023; Martins et al., 2023; Tozzi et al., 2023; Gómez-González
et al., 2024), AGNs (Saxena et al., 2020a, b; Shirazi & Brinchmann, 2012;
Tozzi et al., 2023; Liu et al., 2024; Topping et al., 2024), shocks (Kehrig et
al., 2018; Lecroq et al., 2024), and stellar winds (Upadhyaya et al., 2024).

However, the signal from PISNe is very short-lived, $\sim 1$ yr in the source
frame (Kasen et al., 2011), compared to $\sim 1$ Myr for the HeII line
(Schaerer, 2002, 2003; Katz et al., 2023). The combination of the short
lifetime and the limited mass range of PISNe progenitors makes PISNe extremely
rare phaenomena. In Venditti et al. (2024, figure 4) we found, at best $\sim
0.4$ PISNe on average among galaxies with
$7.5<\mathrm{log}(M_{\star}/$\mathrm{M}_{\odot}$)\leq 9.5$ at $z\simeq 8$,
within the effective volume of all the combined JWST surveys considered in
Figure 5, i.e. more than three orders of magnitude lower than the predicted
number of Pop III HeII emitters. Hence, a trade-off between the limitation in
statistics for PISNe and the limitation in brightness for the HeII signature
has to be taken into account when designing our strategies for Pop III
detection.

##  5 Conclusions

A systematic search for Pop III stars during the EoR through the
HeII$\lambda$1640 line poses several challenges. We predict more than 400 Pop
III systems could be discovered in $M_{\star}\gtrsim
10^{7.5}~{}$\mathrm{M}_{\odot}$$ galaxies within existing/ongoing JWST surveys
at these redshifts. However, considerable uncertainty surrounds the luminosity
of their intrinsic HeII emission, which might vary from $\sim 2\times
10^{40}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-2}$$ up to $\sim
10^{42}~{}$\mathrm{e}\mathrm{r}\mathrm{g}\,\mathrm{s}^{-2}$$ depending on the
adopted Pop III model (e.g. with/without mass loss, and for different
assumptions on the star-formation–efficiency parameter $\eta_{\mathrm{III}}$).
Dust absorption can also further dampen this emission along unfavourable
lines-of-sight. While promising candidates such as GN-z11 exist (with an
inferred high $\eta_{\mathrm{III}}\sim 0.1$, and essentially dust-free), it
remains unclear how representative such bright targets are. Moreover, many
similar targets might fall outside our FoVs, even more than 90% when
considering small FoVs as for NIRSpec/MOS.

In principle, a large portion of these Pop III systems (if not all of them)
could be too faint to be detected in wide – but shallow – blind surveys. A
more effective strategy might involve follow-up spectroscopy in the regions
surrounding bright, massive galaxies: although rarer, these are in fact more
likely to host peripheral Pop III stars (Yajima et al., 2023; Venditti et al.,
2023). Focusing on a limited number of promising targets would truly allow us
to push our instrument capabilities, particularly:

  * •

conducting very deep observations, to confirm/exclude the presence of even the
faintest Pop III systems with high confidence;

  * •

comprehensively sampling the external regions via multiple pointings, to hunt
for Pop III star-forming clumps in the outskirts.

In Venditti et al. (2023) we found indications that strong accretion of
pristine gas from the IGM at the knots of the cosmic web might favour Pop III
star formation. On the other hand, under-dense regions with a less progressed
history of star formation are also of interest. Correa Magnus et al. (2024)
suggested a novel formation pathway for Pop IIIs with major mergers as a
primary source of gas. However, the role of mergers in the global Pop III
star-formation budget – and hence whether isolated/interacting galaxies are a
better observational target – needs to be confirmed through a more thorough
statistical analysis. In future works we plan to delve into all these aspects,
to help us identify the most favourable candidates/environments for follow-up.

We thank Roberto Maiolino for valuable comments. AV acknowledges support from
Sapienza University of Rome program "Bando mobilità internazionale PhD 2022
(II edizione)" (Decreto N. 3147/2022 Prot. n. 0102185 del 15/11/2022) during
the visiting period (March-September 2023) at UT Austin, Texas (USA). LG and
RS acknowledge support from the PRIN 2022 MUR project 2022CB3PJ3 - First Light
And Galaxy aSsembly (FLAGS) funded by the European Union – Next Generation EU,
and from the Amaldi Research Center funded by the MIUR program "Dipartimento
di Eccellenza" (CUP:B81I18001170001).

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# Plane-filling curves of small degree over finite fields

Shamil Asgarli  Department of Mathematics and Computer Science
Santa Clara University
500 El Camino Real
USA 95053<EMAIL_ADDRESS>and  Dragos Ghioca  Department of Mathematics
University of British Columbia
1984 Mathematics Road
Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

A plane curve $C$ in $\mathbb{P}^{2}$ defined over $\mathbb{F}_{q}$ is called
plane-filling if $C$ contains every $\mathbb{F}_{q}$-point of
$\mathbb{P}^{2}$. Homma and Kim, building on the work of Tallini, proved that
the minimum degree of a smooth plane-filling curve is $q+2$. We study smooth
plane-filling curves of degree $q+3$ and higher.

###### Key words and phrases:

Plane curve, space-filling curve, smooth curve, finite field

###### 2020 Mathematics Subject Classification:

Primary: 14G15, 14H50; Secondary: 11G20, 14G05

##  1\. Introduction

The study of space-filling curves in $\mathbb{R}^{2}$ starts with the work of
Peano [Pea90] in the 19th century. About 100 years later, Nick Katz [Kat99]
studied space-filling curves over finite fields and raised open questions
about their existence. One version of Katz’s question was the following. Given
a smooth algebraic variety $X$ over a finite field $\mathbb{F}_{q}$, does
there always exist a _smooth_ curve $C\subset X$ such that
$C(\mathbb{F}_{q})=X(\mathbb{F}_{q})$? In other words, is it possible to pass
through all of the (finitely many) $\mathbb{F}_{q}$-points of $X$ using a
smooth curve? Gabber [Gab01] and Poonen [Poo04] independently answered this
question in the affirmative.

We will consider the special case when $X=\mathbb{P}^{2}$. We say that a curve
$C\subset\mathbb{P}^{2}$ is _plane-filling_ if
$C(\mathbb{F}_{q})=\mathbb{P}^{2}(\mathbb{F}_{q})$. Equivalently, $C$ is a
plane-filling curve $C$ if $\\#C(\mathbb{F}_{q})=q^{2}+q+1$. In a natural
sense, plane-filling curves are extremal. There are other classes of extremal
curves with respect to the set of $\mathbb{F}_{q}$-points, including blocking
curves [AGY23] and tangent-filling curves [AG23].

From Poonen’s work [Poo04], we know that there exist smooth plane-filling
curves of degree $d$ over $\mathbb{F}_{q}$ whenever $d$ is sufficiently large
with respect to $q$. It is natural to ask for the minimum degree of a smooth
plane-filling curve over $\mathbb{F}_{q}$. Homma and Kim [HK13] proved that
the minimum degree is $q+2$. More precisely, by building on the work of
Tallini [Tal61a, Tal61b], they showed that a plane-filling curve of the form

$(ax+by+cz)(x^{q}y-xy^{q})+y(y^{q}z-yz^{q})+z(z^{q}x-zx^{q})=0$

is smooth if and only if the polynomial
$t^{3}-(ct^{2}+bt+a)\in\mathbb{F}_{q}[t]$ has no $\mathbb{F}_{q}$-roots. In a
sequel paper [Hom20], Homma investigated further properties of plane-filling
curves of degree $q+2$. The automorphism group of these special curves was
studied by Duran Cunha [Dur18]. As another direction, Homma and Kim [HK23]
investigated space-filling curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

In this paper, we investigate the existence of smooth plane-filling curves of
degree $q+3$ and higher. The guiding question for our paper is the following.

######  Question 1.1.

Let $q$ be a prime power. Does there exist a smooth plane-filling curve of
degree $q+3$ defined over $\mathbb{F}_{q}$?

The three binomials $x^{q}y-xy^{q},y^{q}z-yz^{q}$, and $z^{q}x-zx^{q}$
generate the ideal of polynomials defining plane-filling curves; see
[HK13]*Proposition 2.1 for proof of this assertion. Thus, any plane-filling
curve of degree $q+3$ must necessarily be defined by

$Q_{1}(x,y,z)\cdot(x^{q}y-xy^{q})+Q_{2}(x,y,z)\cdot(y^{q}z-yz^{q})+Q_{3}(x,y,z)\cdot(z^{q}x-zx^{q})=0$

for some homogeneous quadratic polynomials
$Q_{1},Q_{2},Q_{3}\in\mathbb{F}_{q}[x,y,z]$. The difficulty is finding
suitable $Q_{1},Q_{2},Q_{3}$ for which the corresponding curve is smooth.

Our first result gives a necessary and sufficient condition for the plane-
filling curve $C_{k}$ to be smooth at all the $\mathbb{F}_{q}$-points.

######  Theorem 1.2.

For each $k\in\mathbb{F}_{q}$, consider the plane-filling curve $C_{k}$
defined by

$x^{2}(x^{q}y-xy^{q})+y^{2}(y^{q}z-yz^{q})+(z^{2}+kx^{2})(z^{q}x-zx^{q})=0.$
(1)

Then $C_{k}$ is smooth at every $\mathbb{F}_{q}$-point of $\mathbb{P}^{2}$ if
and only if the polynomial $x^{7}+kx^{3}-1$ has no zeros in $\mathbb{F}_{q}$.

To ensure that the previous theorem is not vacuous, we need to show that there
exists some $k\in\mathbb{F}_{q}$ such that $x^{7}+kx^{3}-1$ has no zeros in
$\mathbb{F}_{q}$.

######  Proposition 1.3.

There exists a value $k\in\mathbb{F}_{q}$ such that
$x^{7}+kx^{3}-1\in\mathbb{F}_{q}[x]$ has no zeros in $\mathbb{F}_{q}$.

###### Proof.

When $x=0$, there is no $k\in\mathbb{F}_{q}$ such that $x^{7}+kx^{3}-1=0$. For
each $x\in\mathbb{F}_{q}^{\ast}$, there is a _unique_ value of
$k\in\mathbb{F}_{q}$ such that $x^{7}+kx^{3}-1=0$. Thus, there are at most
$q-1$ values of $k\in\mathbb{F}_{q}$ such that the polynomial $x^{7}+kx^{3}-1$
has a zero in $\mathbb{F}_{q}$. ∎

The next result improves Proposition 1.3.

######  Theorem 1.4.

There exist at least $\frac{q}{6}-1-\frac{28}{3}\sqrt{q}$ many values of
$k\in\mathbb{F}_{q}$ such that $x^{7}+kx^{3}-1\in\mathbb{F}_{q}[x]$ has no
zeros in $\mathbb{F}_{q}$.

Note that Theorem 1.2 and Proposition 1.3 together yields that for each odd
$q$, there exists at least one value $k\in\mathbb{F}_{q}$ for which the
corresponding curve $C_{k}$ has no singular $\mathbb{F}_{q}$-points.
Furthermore, we expect that the curves in Theorem 1.2 are smooth if and only
if they are smooth at all their $\mathbb{F}_{q}$-points. Our main conjecture
below restates this prediction.

######  Conjecture 1.5.

Suppose $q$ is odd. The plane-filling curve $C_{k}$ defined by (1) is smooth
if and only if the polynomial $x^{7}+kx^{3}-1$ has no zeros in
$\mathbb{F}_{q}$.

We have verified Conjecture 1.5 using Macaulay2 [M2] for all odd prime powers
$q<200$. When $q=2^{m}$ is even, the curve $C_{k}$ defined by (1) turns out to
be singular (for _every_ $k\in\mathbb{F}_{q}$). As a replacement, we consider
another curve $D_{k}$ in this case:

$x^{2}(x^{q}y-xy^{q})+y^{2}(y^{q}z-yz^{q})+(z^{2}+kxy)(z^{q}x-zx^{q})=0.$ (2)

We make a similar conjecture regarding the smoothness of the curves $D_{k}$.

######  Conjecture 1.6.

Suppose $q$ is even. The plane-filling curve $D_{k}$ defined by (2) is smooth
if and only if the polynomial $x^{7}+kx^{5}+1$ has no zeros in
$\mathbb{F}_{q}$.

The polynomial $x^{7}+kx^{5}+1$ featured above is prominent because one can
show, similar to Theorem 1.2, that a plane-filling curve $D_{k}$ is smooth at
all of its $\mathbb{F}_{q}$-points (when $q$ is even) if and only if
$x^{7}+kx^{5}+1$ has no $\mathbb{F}_{q}$-roots. We have verified Conjecture
1.6 using Macaulay2 [M2] for $q=2^{m}$ when $1\leq m\leq 9$.

We prove the following as partial progress towards Conjecture 1.5.

######  Theorem 1.7.

Suppose $q$ is odd. There exists a suitable choice of $k\in\mathbb{F}_{q}$
such that the plane-filling curve $C_{k}$ defined by by (1) is smooth at all
$\mathbb{F}_{q^{2}}$-points.

A similar argument as the one employed in Theorem 1.7 yields an analogous
result when $q$ is even, and the curve $C_{k}$ is replaced by $D_{k}$.

To prove Theorem 1.7, we will prove that any plane-filling curve of degree
$q+3$ which is smooth at $\mathbb{F}_{q}$-points and has no
$\mathbb{F}_{q}$-linear component must be smooth at each of its
$\mathbb{F}_{q^{2}}$-points.

We also investigate plane-filling curves of degree $q+r+1$ where $r\geq 2$ is
arbitrary.

######  Theorem 1.8.

For each $k\in\mathbb{F}_{q}$, consider the plane-filling curve $C_{k,r}$
defined by

$x^{r}(x^{q}y-xy^{q})+y^{r}(y^{q}z-yz^{q})+(z^{r}+kx^{r})(z^{q}x-zx^{q})=0.$

Then $C_{k,r}$ is smooth at every $\mathbb{F}_{q}$-point of $\mathbb{P}^{2}$
if and only if the polynomial $x^{r^{2}+r+1}+kx^{r+1}-1=0$ has no zeros in
$\mathbb{F}_{q}$.

### Structure of the paper

In Section 2, we prove Theorem 1.4. We devote Section 3 to Theorem 1.7, and
Section 4 to Theorem 1.8.

##  2\. Proof of Theorem 1.4

We begin this section by noting that Theorem 1.2 is a special case of Theorem
1.8 which will be proven in Section 4. Our Theorem 1.2 provides a criterion
that tests whether the plane-filling curve $C_{k}$ defined by (1) is smooth at
every $\mathbb{F}_{q}$-point.

The following technical result will be employed in our proof of Theorem 1.4.

######  Lemma 2.1.

The polynomial $x^{3}y^{3}(x+y)(x^{2}+y^{2})+(x^{2}+xy+y^{2})$ is irreducible
in $\overline{\mathbb{F}_{q}}[x,y]$.

###### Proof.

The proof employs a technique seen in Eisenstein’s criterion. First, suppose
$p=\operatorname{char}(\mathbb{F}_{q})\neq 3$. Assume, to the contrary, that
$f(x,y)\colonequals x^{3}y^{3}(x+y)(x^{2}+y^{2})+(x^{2}+xy+y^{2})$ is
reducible over the algebraic closure $\overline{\mathbb{F}_{q}}$. Write
$f(x,y)=g(x,y)\cdot h(x,y)$, and express

$\displaystyle g(x,y)$
$\displaystyle=g_{m}(x,y)+g_{m+1}(x,y)+\cdots+g_{s}(x,y)$ $\displaystyle
h(x,y)$ $\displaystyle=h_{n}(x,y)+h_{n+1}(x,y)+\cdots+h_{t}(x,y)$

where $g_{i}(x,y)$ and $h_{j}(x,y)$ are homogeneous of degree $i$ and $j$,
respectively, for $m\leq i\leq s$ and $n\leq j\leq t$. From
$f(x,y)=g(x,y)\cdot h(x,y)$, we see that

$\begin{cases}g_{m}h_{n}=x^{2}+xy+y^{2}\\\
g_{s}h_{t}=x^{3}y^{3}(x+y)(x^{2}+y^{2})\\\ \sum_{i+j=k}h_{i}g_{j}=0\text{ for
}2<k<9\end{cases}$

Since the characteristic $p\neq 3$, the polynomial $x^{2}+xy+y^{2}$ factors
into distinct linear factors in $\overline{\mathbb{F}_{q}}[x,y]$. Let
$x+\lambda y$ be one of those linear factors with
$\lambda\in\overline{\mathbb{F}_{q}}$. Then $x^{2}+xy+y^{2}$ is divisible by
$x+\lambda y$ but not by $(x+\lambda y)^{2}$. Thus, exactly one of $g_{m}$ or
$h_{n}$ is divisible by $x+\lambda y$. Without loss of generality, assume
$x+\lambda y$ divides $g_{m}$, and not $h_{n}$. Then using
$\sum_{i+j=k}h_{i}g_{j}=0$ for $2<k<9$, we inductively see that $x+\lambda y$
divides $g_{j}$ for each $m\leq j\leq s$. In particular, $x+\lambda y$ divides
$g_{s}h_{t}$. This is a contradiction because $x+\lambda y$ does not divide
$x^{3}y^{3}(x+y)(x^{2}+y^{2})$. Indeed, $x^{2}+xy+y^{2}$ and
$x^{3}y^{3}(x+y)(x^{2}+y^{2})$ are relatively prime.

When $p=3$, a similar argument works from the other end of the polynomial: the
leading term $x^{3}y^{3}(x+y)(x^{2}+y^{2})$ is divisible by $x+y$ but not by
$(x+y)^{2}$. We deduce that $f(x,y)$ is irreducible over
$\overline{\mathbb{F}_{q}}$ for every prime power $q$. ∎

###### Proof of Theorem 1.4.

Our goal is to give a lower bound on the number of $k\in\mathbb{F}_{q}$ such
that the polynomial $x^{7}+kx^{3}-1$ has no roots in $\mathbb{F}_{q}$. As $x$
ranges in $\mathbb{F}_{q}^{\ast}$ (note that there is no $k\in\mathbb{F}_{q}$
for which $x=0$ would be a root of $x^{7}+kx^{3}-1$), the number of “bad”
choices of $k$ are parametrized by $\frac{1-x^{7}}{x^{3}}$. We will show that
there are many choices of $x$ and $y$ such that $\frac{1-x^{7}}{x^{3}}$ and
$\frac{1-y^{7}}{y^{3}}$ give rise to the same value of $k$. Setting these
expressions equal to each other, we obtain the following.

$\frac{1-x^{7}}{x^{3}}=\frac{1-y^{7}}{y^{3}}\ \ \Rightarrow\ \
x^{7}y^{3}-y^{3}=y^{7}x^{3}-x^{3}$

After rearranging and dividing both sides by $x-y$, we obtain an affine curve
$\mathcal{C}\subset\mathbb{A}^{2}$ defined by

$x^{3}y^{3}(x+y)(x^{2}+y^{2})+x^{2}+xy+y^{2}=0,$

for $x,y\in\mathbb{F}_{q}^{\ast}$ _and_ $x\neq y$. Let $G$ be a graph whose
vertex set is $\mathbb{F}_{q}^{\ast}$, and there is an edge between $x$ and
$y$ if $(x,y)$ lies on the affine curve $\mathcal{C}$. We consider undirected
edges, so the pairs $(x,y)$ and $(y,x)$ correspond to the same edge.

Claim 1. The number of edges of $G$ is at least $\frac{q}{2}-6-28\sqrt{q}$.

Let $\tilde{\mathcal{C}}\subset\mathbb{P}^{2}$ be the projectivization of
$\mathcal{C}$. By Lemma 2.1, the curve $\tilde{\mathcal{C}}$ is geometrically
irreducible. By Hasse-Weil inequality for geometrically irreducible curves
[AP96]*Corollary 2.5, $\\#\tilde{\mathcal{C}}(\mathbb{F}_{q})\geq
q+1-56\sqrt{q}$. Since the line at infinity $z=0$ can contain at most $5$
distinct $\mathbb{F}_{q}$-points, we have $\\#C(\mathbb{F}_{q})\geq
q-4-56\sqrt{q}$; furthermore, we exclude the points for which $xy=0$ and there
is only one such point $[0:0:1]\in\tilde{\mathcal{C}}$. We also need to rule
out the points on the diagonal, namely $x=y$; in this case, $4x^{9}+3x^{2}=0$
which contributes at most $7$ additional points with $x\neq 0$. Thus, the
number of $(x,y)\in C(\mathbb{F}_{q})$ with $x\neq y$ is at least
$q-12-56\sqrt{q}$. The claim follows since the edges are undirected.

Claim 2. Every connected component of $G$ is a complete graph $K_{n}$ where
$n\in\\{1,2,3,4,5,6\\}$.

If $(x,y)$ and $(x,z)$ are both edges of $G$, then
$\frac{1-x^{7}}{x^{3}}=\frac{1-y^{7}}{y^{3}}$ and
$\frac{1-x^{7}}{x^{3}}=\frac{1-z^{7}}{z^{3}}$. Consequently,
$\frac{1-y^{7}}{y^{3}}=\frac{1-z^{7}}{z^{3}}$ and $(y,z)$ lies on the curve
$\mathcal{C}$, so $(y,z)$ is an edge in $G$ too. Thus, each connected
component of $G$ is a clique. In addition, from the equation of $\mathcal{C}$,
the degree of each vertex $x\in G$ is at most $6$.

For each $1\leq i\leq 6$, let $m_{i}$ denote the number of cliques of size $i$
in $G$. Counting the number of edges in $G$ leads to the following equality.

$\\#E(G)=\sum_{i=1}^{6}\frac{i(i-1)}{2}\cdot m_{i}.$

Each clique of size $i$ in $G$ increases the number of “good” values of $k$ by
an additive factor of $i-1$ because each clique corresponds to one “bad” value
of $k$, i.e., a value $k\in\mathbb{F}_{q}$ for which the equation
$x^{7}+kx^{3}-1=0$ is solvable for some $x\in\mathbb{F}_{q}$. More precisely,

$\displaystyle\\#\\{k\in\mathbb{F}_{q}\ |\ x^{7}+kx^{3}-1\text{ has no zeros
in }\mathbb{F}_{q}\\}$ $\displaystyle=q-\sum_{i=1}^{6}m_{i}$
$\displaystyle=1+(q-1)-\sum_{i=1}^{6}m_{i}$
$\displaystyle=1+\sum_{i=1}^{6}i\cdot m_{i}-\sum_{i=1}^{6}m_{i}$
$\displaystyle=1+\sum_{i=1}^{6}(i-1)\cdot m_{i}$ $\displaystyle\geq
1+\frac{1}{3}\sum_{i=1}^{6}\frac{(i-1)i}{2}\cdot m_{i}\geq
1+\frac{1}{3}\\#E(G)\geq 1+\frac{1}{3}\left(\frac{q}{2}-6-28\sqrt{q}\right)$

as desired. ∎

##  3\. Smoothness at $\mathbb{F}_{q^{2}}$-points

In this section, we show that a plane-filling curve $C$ of degree $q+3$ has
the following special property: being smooth at $\mathbb{F}_{q}$-points
implies being smooth at $\mathbb{F}_{q^{2}}$-points under a mild condition.

######  Proposition 3.1.

Suppose $C$ is a plane-filling curve of degree $q+3$ such that

  1. (i)

The curve $C$ is smooth at all the $\mathbb{F}_{q}$-points.

  2. (ii)

The curve $C$ has no $\mathbb{F}_{q}$-linear component.

Then $C$ is smooth at each $\mathbb{F}_{q^{2}}$-point.

###### Proof.

Assume, to the contrary, that $C$ is singular at some
$\mathbb{F}_{q^{2}}$-point $Q$. Then $Q$ is not an $\mathbb{F}_{q}$-point due
to the hypothesis (i). Let $Q^{\sigma}$ denote the Galois conjugate of $Q$
under the Frobenius automorphism. More explicitly, if
$Q=[x:y:z]\in\mathbb{P}^{2}$, then $Q^{\sigma}=[x^{q}:y^{q}:z^{q}]$. Note that
$Q^{\sigma}$ is also contained in $C$ (since $C$ is defined over
$\mathbb{F}_{q}$). Moreover, $Q^{\sigma}$ is also a singular point of $C$.

Consider the line $L$ joining $Q$ and $Q^{\sigma}$, which is an
$\mathbb{F}_{q}$-line by Galois theory. By hypothesis (ii), the line $L$ must
intersect $C$ in exactly $q+3$ points (counted with multiplicity). However,
$L$ already contains $q+1$ distinct $\mathbb{F}_{q}$-points of $C$ (because
$C$ is plane-filling), and passes through the two singular points $Q$ and
$Q^{\sigma}$, each contributing intersection multiplicity at least $2$. Thus,
the total intersection multiplicity between $L$ and $C$ is at least
$(q+1)+2+2=q+5$, a contradiction. ∎

######  Remark 3.2.

We can weaken the hypothesis of Proposition 3.1 by replacing the condition
$\deg(C)=q+3$ with $\deg(C)\leq q+4$. Indeed, the same proof works verbatim.

Next, we show that the plane-filling curves $C_{k}$ of degree $q+3$ considered
in equation (1) indeed satisfy condition (ii) when $q$ is odd.

######  Proposition 3.3.

The curve $C_{k}$ defined by (1) has no $\mathbb{F}_{q}$-linear components
when $q$ is odd.

###### Proof.

There are three types of $\mathbb{F}_{q}$-lines in $\mathbb{P}^{2}$.

Type I. The line $L$ is given by $z=0$.

The curve $C_{k}$ meets the line $\\{z=0\\}$ at finitely many points
determined by $x^{2}(x^{q}y-xy^{q})=0$. In particular, $\\{z=0\\}$ is not a
component of $C$.

Type II. The line $L$ is given by $x=az$ for some $a\in\mathbb{F}_{q}$.

The curve $C_{k}$ meets the line $\\{x=az\\}$ at finitely many points
determined by

$(az)^{2}((az)^{q}y-(az)y^{q})+y^{2}(y^{q}z-yz^{q})+(z^{2}+k(az)^{2})(z^{q}(az)-z(az)^{q})=0.$

After simplifying and using $a^{q}=a$, the last term cancels and we obtain:

$a^{3}z^{q+2}y-a^{3}z^{3}y^{q}+y^{q+2}z-y^{3}z^{q}=0$

In particular, $\\{x=az\\}$ is not a component of $C$.

Type III. The line $L$ is given by $y=ax+bz$ for some $a,b\in\mathbb{F}_{q}$.

If $a=0$ or $b=0$, then $y=bz$ or $y=ax$, and the analysis is very similar to
the previous case. We will assume that $a\neq 0$ and $b\neq 0$. We substitute
$y=ax+bz$ into the equation (1) and collect terms to obtain:

$\displaystyle(b+a^{3}-k)x^{q+2}z+(2a^{2}b)x^{q+1}z^{2}+(b^{2}a-1)x^{q}z^{3}+$
$\displaystyle(-b-a^{3}+k)x^{3}z^{q}+(-2ab)x^{2}z^{q+1}+(-ab^{2}+1)xz^{q+2}=0$

The coefficient of $x^{q+1}z^{2}$ is $2a^{2}b$, which is nonzero since $q$ is
odd (so $2\neq 0$), $a\neq 0$ and $b\neq 0$. Thus, $L$ is not a component of
$C_{k}$. ∎

We are now in a position to prove Theorem 1.7 on the existence of
$k\in\mathbb{F}_{q}$ such that the plane-filling curve $C_{k}$ is smooth at
all its $\mathbb{F}_{q^{2}}$-points.

###### Proof of Theorem 1.7.

The result follows immediately from Proposition 1.3, Proposition 3.1, and
Proposition 3.3. ∎

##  4\. Higher degree plane-filling curves

We begin by establishing Theorem 1.8, which provides a necessary and
sufficient condition for the plane-filling curve $C_{k,r}$ to be smooth at all
the $\mathbb{F}_{q}$-points.

###### Proof of Theorem 1.8.

We consider the curve $C_{k,r}$ given by the equation:

$x^{r}\cdot(x^{q}y-xy^{q})+y^{r}\cdot(y^{q}z-yz^{q})+(z^{r}+kx^{r})\cdot(z^{q}x-zx^{q})=0.$
(3)

We analyze the singular locus of $C_{k,r}$ and get the equations:

$rx^{r-1}\cdot(x^{q}y-xy^{q})+x^{r}\cdot(-y^{q})+krx^{r-1}\cdot(z^{q}x-zx^{q})+(z^{r}+kx^{r})\cdot
z^{q}=0$ (4) $x^{r}\cdot
x^{q}+ry^{r-1}\cdot(y^{q}z-yz^{q})+y^{r}\cdot(-z^{q})=0$ (5) $y^{r}\cdot
y^{q}+rz^{r-1}\cdot(z^{q}x-zx^{q})+(z^{r}+kx^{r})\cdot(-x^{q})=0.$ (6)

We next analyze the possibility that we have a singular point when $xyz=0$.

If $x=0$, then equation (4) yields $z=0$, which is then employed in (6) to
derive $y=0$, contradiction.

If $y=0$, then equation (5) yields $x=0$ and then equation (4) yields $z=0$,
contradiction.

If $z=0$, then equation (5) yields $x=0$ and then equation (6) yields $y=0$,
contradiction.

So, the only possible singular points are of the form $[x:1:z]$.

We search for possible singular points
$[x:1:z]\in\mathbb{P}^{2}(\mathbb{F}_{q})$. Then equations (4), (5) and (6)
read:

$-x^{r}+z^{r+1}+kx^{r}z=0$ (7) $x^{r+1}-z=0$ (8) $1-z^{r}x-kx^{r+1}=0.$ (9)

Substituting $z=x^{r+1}$ from equation (8) into equations (7) and (9), we
obtain

$-x^{r}+x^{r^{2}+2r+1}+kx^{2r+1}=0\text{ and }1-x^{r^{2}+r+1}-kx^{r+1}=0,$

that is, there exists a singular $\mathbb{F}_{q}$-rational point on $C_{k,r}$
if and only if there exists $x\in\mathbb{F}_{q}^{\ast}$ such that

$x^{r^{2}+r+1}+kx^{r+1}-1=0,$ (10)

as desired.

We end the proof by mentioning that some care is needed to treat the case when
the characteristic $p$ of the field divides the degree of the curve (i.e., $p$
divides $r+1$ in this setting). Indeed, the singular locus of any projective
curve $\\{f=0\\}$ is defined by $\\{f=\frac{\partial f}{\partial
x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0\\}$. When $p$
divides $\deg(f)$, it is _not_ enough to consider the points in the locus
$\\{\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial
f}{\partial z}=0\\}$. Fortunately, in our case, the $\mathbb{F}_{q}$-point
$[x:1:z]$ is automatically on the curve $C_{k,r}$ because $C_{k,r}$ is plane-
filling. ∎

It may be natural to make a prediction identical to Conjecture 1.5 for higher-
degree curves. However, some care is needed, as the following two examples
show. We found these examples using Macaulay2 [M2].

######  Example 4.1.

Let $r=5$, $q=11$, and $k=9$. The plane-filling curve $C_{9,5}$ over
$\mathbb{F}_{11}$ is smooth at all the $\mathbb{F}_{11}$-points because the
polynomial $x^{31}+9x^{6}-1$ is an irreducible polynomial over
$\mathbb{F}_{11}$. However, $C_{9,5}$ is singular at two Galois-conjugate
$\mathbb{F}_{11^{2}}$-points.

In the previous example, the curve $C_{9,5}$ is irreducible over
$\mathbb{F}_{11}$. Thus, $C_{9,5}$ satisfies the two conditions of Theorem 3.1
and yet it is singular at two $\mathbb{F}_{11^{2}}$-points. Since
$\deg(C_{9,5})=q+6$, we see that Remark 3.2 is close to being sharp.

######  Example 4.2.

Let $r=7$, $q=5$. In this case, the plane-filling curve $C_{k,7}$ defined over
$\mathbb{F}_{5}$ is singular for each $k\in\mathbb{F}_{5}$. Indeed, the
associated polynomial $x^{57}+kx^{8}-1$ has an $\mathbb{F}_{5}$-root for
$k\in\\{0,2,3,4\\}$. For these values of $k$, the curve $C_{k,r}$ is singular
at an $\mathbb{F}_{5}$-point. For $k=1$, the curve $C_{1,7}$ is singular at
four points, namely, two pairs of Galois-conjugate
$\mathbb{F}_{5^{2}}$-points.

The two examples above illustrate that Conjecture 1.5 needs to be modified for
plane-filling curves of degree $q+r+1$ when $r$ is arbitrary. We propose two
related conjectures on the smoothness of the curve $C_{k,r}$ from Theorem 1.8.
Recall that $C_{k,r}\subset\mathbb{P}^{2}$ is defined by

$x^{r}(x^{q}y-xy^{q})+y^{r}(y^{q}z-yz^{q})+(z^{r}+kx^{r})(z^{q}x-zx^{q})=0$

where $r\geq 2$ is a positive integer and $k\in\mathbb{F}_{q}$.

######  Conjecture 4.3.

Let $r\geq 2$. There exists an integer $m\colonequals m(r)$ with the following
property. For all finite fields $\mathbb{F}_{q}$ with cardinality $q>m$ and
characteristic not dividing $r$, there exists some $k\in\mathbb{F}_{q}$ such
that the curve $C_{k,r}$ is smooth.

Using Macaulay2 [M2], we enumerated through values of $r$ in the range
$[2,17]$ and $q$ in the range $[2,100]$ with $\gcd(r,q)=1$. We found only the
following pairs $(r,q)$ for which $C_{k,r}$ is singular for _every_
$k\in\mathbb{F}_{q}$: $(r,q)=(7,5)$, $(13,3)$, $(16,9)$, and $(17,7)$.

######  Conjecture 4.4.

Let $r\geq 2$. There exists an integer $s\colonequals s(r)$ with the following
property. For all finite fields $\mathbb{F}_{q}$ with characteristic not
dividing $r$, and for all $k\in\mathbb{F}_{q}$, if $C_{k,r}$ is smooth at all
of its $\mathbb{F}_{q^{s}}$-points, then $C_{k,r}$ is smooth.

As a motivation for Conjecture 4.4, we mention the following general fact
about pencils of plane curves. The family of plane curves $C_{k}$ forms a
_pencil_ of plane curves since the parameter $k\in\mathbb{F}_{q}$ appears
linearly in the defining equation. If $\mathcal{L}$ is a pencil of plane
curves in $\mathbb{P}^{2}$ parametrized by $\mathbb{A}^{1}$, then
$\mathbb{F}_{q}$-members of $\mathcal{L}$ are defined by
$f(x,y,z)+kg(x,y,z)=0$ where $k\in\mathbb{F}_{q}$ is arbitrary. We will use
$X_{k}$ to denote this plane curve in the following proposition.

######  Proposition 4.5.

Let $\mathcal{L}$ be a pencil of plane curves
$\\{X_{k}\\}_{k\in\mathbb{F}_{q}}$ of degree $d$ defined over a finite field
$\mathbb{F}_{q}$. Suppose that for every $s\geq 1$, there exists some
$k\in\mathbb{F}_{q}$ such that $X_{k}$ is smooth at all of its
$\mathbb{F}_{q^{s}}$-points. Then there exists some $\ell\in\mathbb{F}_{q}$
such that $X_{\ell}$ is smooth.

###### Proof.

Assume, to the contrary, that $X_{k}$ is singular for each
$k\in\mathbb{F}_{q}$. For each $k\in\mathbb{F}_{q}$, let $n_{k}\in\mathbb{N}$
such that the curve $X_{k}$ is singular at some
$\mathbb{F}_{q^{n_{k}}}$-point. Let
$N\colonequals\prod_{k\in\mathbb{F}_{q}}n_{k}$. By construction, no $X_{k}$ is
smooth at all of its $\mathbb{F}_{q^{N}}$-points, contradicting the
hypothesis. ∎

Proposition 4.5 asserts that to find a smooth member of any pencil
$\mathcal{L}$ defined over $\mathbb{F}_{q}$, it is sufficient to find a member
which is smooth at all points of an (arbitrary) finite degree. Conjecture 4.4
strengthens the conclusion by predicting that for a pencil of plane-filling
curves, one finds a smooth member by only checking smoothness at all points of
_fixed_ finite degree.

## References