# Universal criteria for single femtosecond pulse ultrafast magnetization
switching in ferrimagnets

F. Jakobs  Dahlem Center for Complex Quantum Systems and Fachbereich Physik,
Freie Universität Berlin, 14195 Berlin, Germany  U. Atxitia  unai.atxitia@fu-
berlin.de. Dahlem Center for Complex Quantum Systems and Fachbereich Physik,
Freie Universität Berlin, 14195 Berlin, Germany

###### Abstract

Single-pulse switching has been experimentally demonstrated in ferrimagnetic
GdFeCo and Mn2RuxGa alloys. Complete understanding of single-pulse switching
is missing due to the lack of an established theory accurately describing the
transition to the non-equilibrium reversal path induced by femtosecond laser
photo-excitation. In this work we present general macroscopic theory for the
magnetization dynamics of ferrimagnetic materials upon femtosecond laser
excitation. Our theory reproduces quantitatively all stages of the switching
process observed in experiments. We directly compare our theory to computer
simulations using atomistic spin dynamics methods for both GdFeCo and Mn2RuxGa
alloys. We provide explicit expressions for the magnetization relaxation rates
in terms of microscopic parameters which allows us to propose universal
criteria for switching in ferrimagnets.

Ultrafast magnetization switching induced by a single femtosecond laser pulse
has attracted a lot of attention as a promising solution for low energy,
faster memory applications. Stanciu  _et al._ (2007); Radu  _et al._ (2011);
Ostler  _et al._ (2012); Le Guyader  _et al._ (2012); Graves  _et al._ (2013);
Liu  _et al._ (2015); El Hadri  _et al._ (2016); Lalieu  _et al._ (2017); Chen
_et al._ (2017); Lalieu  _et al._ (2019); Iacocca  _et al._ (2019); Kimel and
Li (2019); Ignatyeva  _et al._ (2019); van Hees  _et al._ (2020); Davies  _et
al._ (2020a). Until recently only GdFeCo alloys and synthetic ferrimagnets
Lalieu  _et al._ (2017), presented the ability to switch under either optical
femtosecond laser- Ostler  _et al._ (2012); Le Guyader  _et al._ (2012);
Lalieu  _et al._ (2017) or electric picosecond current-pulses Yang  _et al._
(2017); El-Ghazaly  _et al._ (2020). Although several micro- and macroscopic
models have reproduced single-pulse switching in GdFeCo ferrimagnets Vahaplar
_et al._ (2009); Mentink  _et al._ (2012); Barker  _et al._ (2013); Wienholdt
_et al._ (2013); Schellekens and Koopmans (2013); Baryakhtar  _et al._ (2013);
Baral and Schneider (2015); Gridnev (2016); Kalashnikova and Kozub (2016);
Zhang  _et al._ (2016); Krivoruchko (2016); Gerlach  _et al._ (2017); Gridnev
(2018); Beens  _et al._ (2019); Vogler  _et al._ (2019); Davies  _et al._
(2020b), complete understanding of the role of electrons, lattice and spin
sublattices and their mutual interactions remains a challenge Carva  _et al._
(2017). The existing criteria for switching rely on the existence of two
antiferromagnetically coupled magnetic sublattices showing distinct dynamical
response to femtosecond laser photo-excitation. While in single species
ferromagnets such as 3d transition metals, relaxation of angular momentum
occurs via dissipation into other degrees of freedom – relativistic relaxation
– in two-sublattice magnets, additionally, relaxation can occur via angular
momentum exchange between sublattices – exchange relaxation. By driving the
spin system into a non-equilibrium state where exchange relaxation dominates,
a non-equilibrium ultrafast reversal path opens. One of the most outstanding,
open questions is about the conditions or criteria for the onset of the
exchange dominated relaxation regime. Crucially, it is unclear how previous
understanding gained from observation in GdFeCo translates into the recent
discovery of single-pulse switching in the ferrimagnetic Heusler alloy
Mn2RuxGa Banerjee  _et al._ (2020), where the two antiferromagnetically
coupled Mn atoms are of the same kind in comparison to Gd and FeCo.

In this work we present a general theoretical framework for the description of
single pulse switching of ferrimagnets. We provide explicit expressions for
the relativistic and exchange relaxation parameters as a function of
microscopic material parameters, including their dependence on temperature and
non-equilibrium sublattice magnetization. This allows us to uncover the
criteria for the onset of the exchange-dominated relaxation regime and
switching. We verify the validity of the model by direct comparison to
atomistic spin model simulations of both GdFeCo and Mn2RuxGa alloys.

We shall describe each magnetic atom at site $i$ as a classical spin vector
$\mathbf{s}$ of a unit length. The magnetic and mechanical moments of each
atom/element are given by $\bm{\mu}=\mu_{s}\mathbf{s}$ and
$\mathbf{S}=\mu_{s}\mathbf{s}/\gamma$ where $\mu_{s}$ is the magnetic moment
and $\gamma$ is the gyromagnetic ratio. We consider a classical Heisenberg
spin HamiltonianNowak (2007):

$\mathcal{H}=-\sum_{i\neq j\\\ \langle
ij\rangle}J_{ij}\mathbf{s}_{i}\cdot\mathbf{s}_{j}-\sum_{i}d^{z}_{i}(s^{z}_{i})^{2}.$
(1)

To model a ferrimagnet, one needs to consider two sublattices with different
and antiparallel magnetic moments $\mu_{a}$ and $\mu_{b}$, with three
different exchange coupling constants. Two ferromagnetic couplings for each
sublattice coupling with itself ($J_{a}$ and $J_{b}$ >0) and a third for the
antiferromagnetic interaction between them, $J_{ab}<0$ Ostler  _et al._ . The
second term in Eq. (1) describes the contribution to the energy of on-site
uniaxial anisotropy with easy-axis in $z$-direction and anisotropy constant,
$d^{z}_{i}$.

The macroscopic model we propose is derived from a microscopic spin model,
where the equation of motion for the spin dynamics of each atomic spin is the
stochastic Landau-Lifshitz-Gilbert (LLG) equation Nowak (2007):

$\frac{\partial\mathbf{s}_{i}}{\partial
t}=-\frac{|\gamma|}{(1+\lambda^{2})\mu_{i}}\left[\left(\mathbf{s}_{i}\times\mathbf{H}_{i}\right)-\lambda\left(\mathbf{s}_{i}\times\left(\mathbf{s}_{i}\times\mathbf{H}_{i}\right)\right)\right].$
(2)

$\lambda$ is the local intrinsic atomic damping, the effective field
$\mathbf{H}_{i}=\bm{\zeta}_{i}-\frac{\partial\mathcal{H}}{\partial\mathbf{s}_{i}}$,
where thermal fluctuations are represented by the stochastic field
$\bm{\zeta}_{i}$.

Figure 1: Degrees of freedom involved in single-pulse switching; electrons,
lattice and element-specific spin systems. Electrons act as heat-bath to the
spin system with an element-specific coupling strength given by
$\alpha_{a(b)}$. Exchange relaxation rate
$\alpha_{\rm{ex}}\sim\alpha_{a}/m_{a}+\alpha_{b}/m_{b}$, where $m_{a(b)}$ are
the normalized sublattice magnetization magnitude.

The non-equilibrium macroscopic magnetization dynamics of the element-specific
angular momentum $S_{a}=\mu_{a}\langle s_{a}\rangle/\gamma$, where
$m_{a}=\langle s_{a}\rangle$ is the first moment of the non-equilibrium
distribution function, can be described by Mentink  _et al._ (2012):

$\frac{dS_{a}}{dt}=\alpha_{a}\mu_{a}H_{a}+\alpha_{\rm{ex}}(\mu_{a}H_{a}-\mu_{b}H_{b})$
(3)

where $a\neq b$. The macroscopic relativistic damping parameter in Eq. (3)
reads Atxitia  _et al._ (2012)

$\alpha_{a}=2\lambda_{a}\frac{L(\xi_{a})}{\xi_{a}}.$ (4)

Here, $L(\xi)$ stands for the Langevin function. The relaxation parameter
strongly depends on temperature and non-equilibrium magnetization state
through the thermal field $\xi_{a}=\beta\mu_{a}H_{a}^{\rm{MFA}}$. In the
exchange approximation, the MFA field acting on sublattice $a$ is:

$\mu_{a}H_{a}^{\rm{MFA}}=z_{a}J_{aa}m_{a}+z_{ab}J_{ab}m_{b}.$ (5)

Here, $z_{a}$ is the number of nearest neighbours (n.n.) of spins of type $a$,
and $z_{ab}$ is the amount of n.n. of type $b$. The macroscopic damping
increases with temperature up to a value $\alpha_{a}=2/3$ at the critical
temperature Atxitia  _et al._ (2012). The exchange relaxation parameter
$\alpha_{\rm{ex}}$ is given by

$\alpha_{\rm{ex}}=\frac{1}{2}\left(\frac{\alpha_{a}}{z_{ab}m_{a}}+\frac{\alpha_{b}}{z_{ab}m_{b}}\right).$
(6)

This expression is the extension of the non-local exchange relaxation in
ferromagnets to local exchange relaxation in ferrimagnets. The role of
$\alpha_{\rm{ex}}$, $\alpha_{a}$ and $\alpha_{b}$ as the coupling between the
sublattices and heat baths is visualized in Fig. 1. In single species
ferromagnets, sublattices $a$ and $b$ represent the same spin lattice, hence
$\alpha_{a}=\alpha_{b}$. Therefore, $\alpha_{\rm{ex}}=\alpha_{a}/(zm_{a})$,
and $\mu_{a}H_{a}-\mu_{b}H_{b}=\mu_{a}H_{\rm{ex}}a_{0}^{2}\Delta m_{a}$, with
$a_{0}$ representing the lattice constant. Hence, $\Gamma_{\rm{ex}}^{\rm{non-
loc.}}=\alpha_{\rm{ex}}(\mu_{a}H_{a}-\mu_{b}H_{b})=\alpha_{a}(A/M_{a}(T))\Delta
m_{a}$, where $A$ is the so-called micromagnetic exchange stiffness Atxitia
_et al._ (2010). $M_{a}(T)=(\mu_{a}/\upsilon_{a})m_{a}$ is the magnetization
density at temperature $T$, where $\upsilon_{a}$ is the unit cell volume. Non-
local exchange relaxation plays a minimal role in the field of ultrafast
magnetization dynamics since $\Delta m_{a}\ll 1$.

The non-equilibrium fields ($\mu_{a}H_{a}=0$ at equilibrium) are given by

$\mu_{a}H_{a}=\frac{(m_{a}-L(\xi_{a}))}{\beta L^{\prime}(\xi_{a})}.$ (7)

where, $L^{\prime}(\xi)=dL/d\xi$. Note that they are different to the MFA
fields in Eq. (5), and have been derived previously Garanin (1997); Atxitia
_et al._ (2012). As the magnetic system approaches thermal equilibrium, the
non-equilibrium fields can be cast into Landau-like expressions Mentink  _et
al._ (2012); Atxitia  _et al._ (2012).

Figure 2: Normalized exchange relaxation parameter $\alpha_{\rm{ex}}/\lambda$
as function of the sublattice magnetization $m_{a}$ and $m_{b}$, at $T=600$ K.
System parameters correspond to GdFeCo.
$\lambda_{a}=\lambda_{b}=\lambda=0.01$. The dotted black line corresponds to
$\alpha_{\rm{ex}}=\alpha_{a}$. The white dot represents the starting
sublattice magnetization ($m_{a},m_{b}$). The closed dashed orange line
describes a trajectory meeting a no-switching criteria. The open solid white
line describes a trajectory meeting a switching criteria.

Equation (3) has been proposed before based on symmetry arguments Mentink  _et
al._ (2012); Baryakhtar  _et al._ (2013) as a direct generalization of the
Landau-Lifshitz equation with longitudinal relaxation terms. These models have
introduced the relaxation parameters at a purely phenomenological level and to
some extent their values are arbitrary. Moreover, since they were taken as
constant values, most of the non-equilibrium spin physics was not taken into
account. Our model overcomes these assumptions by providing expressions for
the relativistic and exchange relaxation parameters as a function of the
sublattice specific atomic relaxation parameter, $\lambda_{a(b)}$, and
normalized magnetization $m_{a(b)}$.

Figure 3: Element-specific magnetization dynamics of single-pulse switching in
ferrimagnets using an atomistic spin dynamics model (symbols) and a
macroscopic model (solid lines) for Mn2Ru0.86Ga (a) and Gd25FeCo (b). The red
area corresponds to the critical values ($m_{a}^{c},m_{b}^{c}$) to enter
exchange relaxation regime. The blue area corresponds to the time lapse where
relativistic relaxation dominates. Different contributions to the element-
specific magnetization relaxation for the single-pulse switching dynamics in
Mn2Ru0.86Ga (c) and Gd25FeCo (d). $\Gamma^{r}_{a(b)}$ stands for the
relativistic relaxation rate of sublattice $a=4a$ (c) and $a=$Fe (d), and
$b=4c$ (c) and $b=$Gd (d). $\Gamma^{\rm{ex}}$ is the exchange relaxation rate,
which is the same for both sublattices.
$\lambda_{a}=\lambda_{b}=\lambda=0.01$.

This insight is paramount to find the criteria for the onset of a exchange
relaxation dominated state. In Fig. 2 we present a diagram of the normalized
exchange relaxation parameter $\alpha_{\rm{ex}}/\lambda$ for GdFeCo alloy
parameters as a function of $m_{a}$ and $m_{b}$ at a fixed temperature,
$T=600$ K, which corresponds to the Curie temperature of the alloy. We observe
that the exchange relaxation parameter strongly depends on the magnitude of
sub-lattice magnetization and its value ranges over three orders of magnitude,
$\alpha_{\rm{ex}}/\lambda=0.1-10$. Importantly, only when magnetic states
reduce significantly, does the exchange relaxation dominates over relativistic
relaxation.

So far only two classes of ferrimagnets have shown single-pulse switching,
Gdx(FeCo)1-x alloys and Mn2RuxGa. Switching in GdFeCo has been thoroughly
studied both experimentally Stanciu  _et al._ (2007); Radu  _et al._ (2011);
Ostler  _et al._ (2012); Le Guyader  _et al._ (2012); Graves  _et al._ (2013)
and theoretically Ostler  _et al._ (2012); Barker  _et al._ (2013);
Schellekens and Koopmans (2013); Wienholdt  _et al._ (2013); Baryakhtar  _et
al._ (2013); Beens  _et al._ (2019); Iacocca  _et al._ (2019); Vogler  _et
al._ (2019), whereas in Mn2RuxGa has been only recently demonstrated for a
range of Ru concentrations ($x\geq 0.9$) Banerjee  _et al._ (2020). In GdFeCo
alloy, switching is characterised by a fast response of the Fe sublattice and
slower response of the Gd sublattice to femtosecond laser photo-excitation.
This difference roots to their distinct magnetic moment,
$\mu_{\rm{Gd}}=7.6\mu_{\rm{B}}$ and $\mu_{\rm{Fe}}=1.92\mu_{\rm{B}}$.
Differently to GdFeCo, antiferromagnetically coupled Mn spins in Mn2RuxGa have
similar atomic magnetic moments. We demonstrate the universality of our theory
by direct comparison of the photo-induced magnetization switching to computer
simulations based on atomistic spin dynamics (Eq. (2)) for GdFeCo (disordered
spin structure) and Mn2RuxGa (ordered spin structure). Their magnetic
properties such as magnetic moments and exchange interactions differ
substantially. We consider two typical compositions, that show switching,
Gd25(FeCo)75 alloys and Mn2Ru0.86Ga. It is noteworthy; that while for GdFeCo
atomistic spin models have been used thoroughly, in Mn2Ru0.86Ga similar
simulations are missing. We derive the necessary material parameters based on
experimental measurements Fowley  _et al._ (2018); Betto  _et al._ (2015);
Thiyagarajah  _et al._ (2015); Kurt  _et al._ (2014); Bonfiglio  _et al._
(2021). We find that for Mn2Ru0.86Ga the atomic magnetic moments, $\mu_{4a}$ =
2.88 $\mu_{\rm{B}}$ and $\mu_{4c}=$ 4.05 $\mu_{\rm{B}}$, and the exchange
parameters, $J_{a}=1.28\times 10^{21}$ J, $J_{b}=4.0\times 10^{22}$ J and
$J_{ab}=-4.85\times 10^{22}$ J describe the equilibrium magnetization well as
function of temperature. Using these parameters the Curie temperature becomes
$T_{c}=600$ K and the compensation temperature $T_{\rm{M}}\approx 300$ K. We
note that in the experimental samples, those temperatures are sensitive to the
growth conditions as well as material composition. However our temperatures
are within the reported range of experimentally found temperatures Fowley  _et
al._ (2018); Betto  _et al._ (2015); Thiyagarajah  _et al._ (2015); Kurt  _et
al._ (2014); Bonfiglio  _et al._ (2021). We use the so-called two-temperature
model (TTM) to describe the dynamics of the electron and phonon systemsKaganov
_et al._ (1957); Chen  _et al._ (2006). For both materials we use the same
parameters in the TTM and thus the electron and phonon temperatures are the
same for the same fluence in both materials Bonfiglio  _et al._ (2021). The
heat-bath to which the spins are coupled is represented by the electron
system.

Figures 3 (a) and (b) show excellent agreement between our macroscopic model
and atomistic spin dynamics simulations for both alloys for all stages of the
magnetization dynamics leading to switching, from fast demagnetization,
transient ferromagnetic-like state, to magnetization recovery. Interestingly,
the switching dynamics in Mn2Ru0.86Ga (Fig. 3 (a)) differs to GdFeCo (Fig. 3
(b)), both Mn sublattices demagnetize at similar rate and switch almost
simultaneously. Although demagnetization timescales are similar in Mn2Ru0.86Ga
and GdFeCo, the recovery of the magnetization in Mn2Ru0.86Ga is significantly
slower. The relaxation of the sublattice magnetization (Eq. 3) can be split
into two contributions, the relativistic relaxation:
$\Gamma^{r}_{a}=\alpha_{a}\mu_{a}H_{a}$, and exchange relaxation
$\Gamma^{\rm{ex}}=\alpha_{\rm{ex}}(\mu_{a}H_{a}-\mu_{b}H_{b})$ (see Fig. 3(c)
and (d)). For both ferrimagnetic alloys, $\Gamma_{a}^{r}$ drives the dynamics
in the first hundreds of femtoseconds, until sublattice magnetization reduces
sufficiently to enter the exchange relaxation dominated region
$\Gamma^{\rm{ex}}>\Gamma_{a}^{r}$ (Fig. 2). In this regime, the exchange
relaxation steers the systems towards an intermediate metastable state defined
by the condition $\mu_{a}H_{a}=\mu_{b}H_{b}$. Under some conditions this
intermediate state precede switching. It is cumbersome however to directly
analyse Eqs. (7) due to its highly non-linear character. Therefore in the
following we investigate some limiting scenarios.

In the high temperature limit,
$\xi_{a}=\beta\mu_{a}H_{a}^{\rm{MFA}}\rightarrow 0$,
$\Gamma^{r}_{a}=2\lambda_{a}k_{B}Tm_{a}$, which is the so-called thermal
fluctuation field. The corresponding relaxation time,
$\tau_{a}=\mu_{a}/(\gamma 2\lambda_{a}k_{B}T)$, associated to relativistic
relaxation, has been discussed before Mentink  _et al._ (2012); Radu  _et al._
(2015). Similarly, we can estimate the high-temperature limit of the exchange
relaxation rate

$\Gamma^{\rm{ex}}_{\infty}(m_{a},m_{b})=\lambda\frac{k_{B}T}{z}\frac{(m_{a}+m_{b})(m^{z}_{a}-m_{b}^{z})}{m_{a}m_{b}}.$
(8)

High temperature limits are valid when the temperature is larger than the
exchange energy acting on the spins. Otherwise, intermediate-to-high
temperature limit are necessary. This limit adds corrections to previous
estimations. For instance,
$\mu_{a}H_{a}-\mu_{b}H_{b}=3k_{B}((T-T_{c}^{a})m_{a}+(T-T_{c}^{b})m_{b})$,
where $J_{0,a}+J_{0,ab}=3k_{B}T^{a}_{c}$. The exchange relaxation rate is

$\Gamma^{\rm{ex}}_{T}(m_{a},m_{b})=\Gamma^{\rm{ex}}_{\infty}\left(1-\frac{1}{T}\frac{T_{c}^{a}m_{a}+T_{c}^{b}m_{b}}{m_{a}+m_{b}}\right)$
(9)

For element-specific critical temperature $T_{c}^{a}\approx T_{c}^{b}$, the
correction can be cast as
$\Gamma^{\rm{ex}}_{T}=\Gamma^{\rm{ex}}_{\infty}(T-T^{a}_{c})/T$.Since the
relativistic relaxation rate is also modified as
$\Gamma^{r}_{a}=\Gamma^{r}_{a,\infty}(T-T^{a}_{c})/T$, we can fairly
investigate the crossover from relativistic to exchange dominated regime by
comparing $\Gamma^{\rm{ex}}_{\infty}$ and $\Gamma^{r}_{a,\infty}$. Two cases
of interest exist, i) one sublattice is faster than the other, and ii) both
sublattices demagnetize at the same rate.

In the first case, $\tau_{a}\ll\tau_{b}$, sublattice $a$ demagnetizes faster
than sublattice $b$. This corresponds to GdFeCo (Fig. (3)(b)). Soon after the
application of a fs laser pulse, $m_{a}\ll m_{b}$, and consequently,
$\Gamma^{\rm{ex}}\approx\lambda k_{B}Tm_{b}/(zm_{a})$ (cf. Eq.(8)). We
estimate the conditions for the transition from relativistic to exchange-
dominated regimes (see Fig. (3)(d)): From $\Gamma^{\rm{ex}}>\Gamma^{r}_{a}$,
$m_{a}<\sqrt{m_{b}/2z_{ab}}\approx 0.288\sqrt{m_{b}}$, and from
$\Gamma^{\rm{ex}}>\Gamma^{r}_{b}$ and $m_{a}\leq 1/2z=0.0833$ (red colored
area in Fig. 3 (b)). A second case, $\tau_{a}\approx\tau_{b}$, might also
arise when demagnetization times of both sublattices are similar. This is the
case of Mn2Ru0.86Ga alloys (Fig. (3)(c)). One can estimate the conditions for
the transition $\Gamma^{\rm{ex}}=\Gamma_{a}^{r}$. This happens for both
sublattices when $m_{a,b}=(m_{a}(0)+m_{b}(0))^{2}/(m_{a}(0)m_{b}(0))/2z_{ab}$.
Assuming a realistic value of $m_{b}(0)/m_{a}(0)=0.9$, the exchange-dominate
regime is reached when $m_{a,b}=0.334$ (red colored area in Fig. 3 (c)).
Notably, this condition only depends on the initial values of the
magnetization, $m_{a(b)}(0)$. These results are simple and general, and one of
the main result of the present work. One can interpret the transition from
relativistic to exchange-dominated regime as follows: as the magnetization of
one sublattice decreases, the phase space of available states for the spins of
the other sublattice to switch by exchange of angular momentum dramatically
increases. While switching spin via coupling to an external bath becomes
increasingly difficult as the magnetization reduces.

Once the system has entered the exchange dominated regime the dynamics follows
a path where total angular momentum is conserved towards a magnetic state
where $\mu_{a}H_{a}=\mu_{b}H_{b}$. In the high temperature limit, this
condition reduces to $m^{z}_{a}=m^{z}_{b}$, meaning that the exchange
relaxation drives the magnetization of both sublattices into the same
polarity, i.e. a ferromagnetic-like state Radu  _et al._ (2011). From this
condition arises, that in order to have a final $m_{a}^{z}<0$ the following
condition is necessary: $S^{\rm{ex}}_{a}+S_{b}^{\rm{ex}}<0$. Here
$S_{a(b)}^{\rm{ex}}$ stands for the angular magnetic moment when the system
enters the exchange dominated regime. For example, in Mn2RuxGa, since Mn spins
are assumed to demagnetize at the same rate, this condition reduces to
$S^{\rm{ex}}_{a}+S_{b}^{\rm{ex}}\approx(S_{a}(0)+S_{b}(0))\exp(-t/\tau_{a})<0$.
Namely, only for a starting temperature below the compensation temperature,
$S_{a}(0)+S_{b}(0)<0$, conditions for switching are fulfilled, in complete
agreement to experimental observations Banerjee  _et al._ (2020). Yet, the
exchange relaxation regime needs to be active for a significant time in order
for the magnetization to switch. We compare the time scales associated to both
the relativistic and exchange relaxation. Relativistic relaxation rate follows
$\Gamma_{a}^{r}=2\lambda_{a}k_{B}(T-T_{c}^{a})m^{z}_{a}$, which is strongly
reduced by the ultrafast dynamics of $m^{z}_{a}$, in only a few hundred of
femtoseconds $\Gamma_{a}^{r}\rightarrow 0$. Whereas $\Gamma^{\rm{ex}}$ rather
follows the dynamics of the temperature $T$, and therefore decays slower than
$\Gamma_{a}^{r}$ (Fig. (3)(c) and (d)).

The characteristic time scale of the electron and lattice temperature is
described by the TTM and for common parameter values in the range of 2 $-$ 3
ps. As the temperature reduces, the exchange relaxation drives the system
towards $(T-T_{c}^{a})m^{z}_{a}=(T-T_{c}^{b})m^{z}_{b}$. For
$T_{c}^{b}<T<T_{c}^{a}$, the exchange relaxation drives the system back into
an antiferromagnetic, but switched sate. As the magnetization builds up in the
opposite direction, $\Gamma^{\rm{ex}}$, decreases and the relativistic
relaxation takes over. Interestingly, our theory predicts that for systems
where $T_{c}^{a}\approx T_{c}^{b}$ switching would be unlikely, which has been
recently demonstrated in rare-earth free synthetic ferrimagnets Liao  _et al._
(2019). Further, one can accelerate the transition from exchange relaxation
dominated to the relativistic regime, and speed up complete switching by
increasing difference between $T_{c}^{a}$ and $T_{c}^{b}$, an effect which has
been observed by the substitution of Fe by Co, namely, GdFe by GdCo Ceballos
_et al._ (2021). While GdFe recovers in tens of picoseconds, GdCo alloys only
need of a couple of picoseconds.

To summarize, in this work we have proposed a general macroscopic theory for
the magnetization dynamics of ferrimagnetic materials driven by femtosecond
laser photo-excitation. Our theory reproduces quantitatively all stages of the
switching process observed in experiments. Notably, we have directly compared
our theory to computer simulations using atomistic spin dynamics methods for
both GdFeCo and Mn2RuxGa alloys. The magnetization dynamics transits from a
relativistic relaxation path to an exchange dominated regime due to the strong
enhancement of the exchange relaxation. We demonstrate that switching occurs
when the sublattice magnetization reaches a threshold value. These criteria
substitute previous ones and pave the way for the discovery of new class of
ferrimagnets showing switching.

_Acknowledgement._ The authors thank Ilie Radu and Jon Gorchon for useful
discussions and critical reading of the manuscript. We gratefully acknowledge
support by the Deutsche Forschungsgemeinschaft through SFB/TRR 227 "Ultrafast
Spin Dynamics", Project A08.

## References

  * Stanciu  _et al._ (2007) C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Phys Rev Lett 99, 047601 (2007).
  * Radu  _et al._ (2011) I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V. Kimel, Nature 472, 205 (2011).
  * Ostler  _et al._ (2012) T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L. J. J. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov, A. M. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk, T. Rasing, and A. V. Kimel, Nature Communications 3, 666 (2012).
  * Le Guyader  _et al._ (2012) L. Le Guyader, S. El Moussaoui, M. Buzzi, R. V. Chopdekar, L. J. Heyderman, a. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, A. V. Kimel, and F. Nolting, Applied Physics Letters 101, 022410 (2012).
  * Graves  _et al._ (2013) C. E. Graves, A. H. Reid, T. Wang, B. Wu, S. de Jong, K. Vahaplar, I. Radu, D. P. Bernstein, M. Messerschmidt, L. Müller, R. Coffee, M. Bionta, S. W. Epp, R. Hartmann, N. Kimmel, G. Hauser, A. Hartmann, P. Holl, H. Gorke, J. H. Mentink, A. Tsukamoto, A. Fognini, J. J. Turner, W. F. Schlotter, D. Rolles, H. Soltau, L. Strüder, Y. Acremann, A. V. Kimel, A. Kirilyuk, T. Rasing, J. Stöhr, A. O. Scherz, and H. A. Dürr, Nature Materials 12, 293 (2013).
  * Liu  _et al._ (2015) T.-M. M. Liu, T. Wang, A. H. Reid, M. Savoini, X. Wu, B. Koene, P. Granitzka, C. E. Graves, D. J. Higley, Z. Chen, G. Razinskas, M. Hantschmann, A. Scherz, J. Stoehr, A. Tsukamoto, B. Hecht, A. V. Kimel, A. Kirilyuk, T. Rasing, and H. A. Duerr, Nano Letters 15, 6862 (2015).
  * El Hadri  _et al._ (2016) M. S. El Hadri, P. Pirro, C.-H. Lambert, S. Petit-Watelot, Y. Quessab, M. Hehn, F. Montaigne, G. Malinowski, and S. Mangin, Phys. Rev. B 94, 064412 (2016).
  * Lalieu  _et al._ (2017) M. L. M. Lalieu, M. J. G. Peeters, S. R. R. Haenen, R. Lavrijsen, and B. Koopmans, Phys. Rev. B 96, 220411 (2017).
  * Chen  _et al._ (2017) J.-Y. Chen, L. He, J.-P. Wang, and M. Li, Physical Review Applied 7, 21001 (2017).
  * Lalieu  _et al._ (2019) M. L. M. Lalieu, R. Lavrijsen, and B. Koopmans, Nat Commun 10 (2019), https://doi.org/10.1038/s41467-018-08062-4.
  * Iacocca  _et al._ (2019) E. Iacocca, T.-M. M. Liu, A. H. Reid, Z. Fu, S. Ruta, P. W. Granitzka, E. Jal, S. Bonetti, A. X. Gray, C. E. Graves, R. Kukreja, Z. Chen, D. J. Higley, T. Chase, L. Le Guyader, K. Hirsch, H. Ohldag, W. F. Schlotter, G. L. Dakovski, G. Coslovich, M. C. Hoffmann, S. Carron, A. Tsukamoto, A. Kirilyuk, A. V. Kimel, T. Rasing, J. Stöhr, R. F. L. Evans, T. Ostler, R. W. Chantrell, M. A. Hoefer, T. J. Silva, and H. A. Dürr, Nat Commun 10, 1756 (2019).
  * Kimel and Li (2019) A. V. Kimel and M. Li, Nature Reviews Materials 4, 189 (2019).
  * Ignatyeva  _et al._ (2019) D. O. Ignatyeva, C. S. Davies, D. A. Sylgacheva, A. Tsukamoto, H. Yoshikawa, P. O. Kapralov, A. Kirilyuk, V. I. Belotelov, and A. V. Kimel, Nature Communications 10, 4786 (2019).
  * van Hees  _et al._ (2020) Y. L. W. van Hees, P. van de Meugheuvel, B. Koopmans, and R. Lavrijsen, Nature Communications 11, 3835 (2020).
  * Davies  _et al._ (2020a) C. S. Davies, G. Bonfiglio, K. Rode, J. Besbas, C. Banerjee, P. Stamenov, J. M. D. Coey, A. V. Kimel, and A. Kirilyuk, Physical Review Research 2, 32044 (2020a).
  * Yang  _et al._ (2017) Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Science Advances 3, e1603117 (2017).
  * El-Ghazaly  _et al._ (2020) A. El-Ghazaly, J. Gorchon, R. B. Wilson, A. Pattabi, and J. Bokor, Journal of Magnetism and Magnetic Materials 502, 166478 (2020).
  * Vahaplar  _et al._ (2009) K. Vahaplar, A. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Physical Review Letters 103, 117201 (2009).
  * Mentink  _et al._ (2012) J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov, A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson, and T. Rasing, Physical Review Letters 108, 057202 (2012).
  * Barker  _et al._ (2013) J. Barker, U. Atxitia, T. A. Ostler, O. Hovorka, R. W. Chantrell, O. Chubykalo-Fesenko, and R. W. Chantrell, Scientific reports 3, 3262 (2013).
  * Wienholdt  _et al._ (2013) S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and U. Nowak, Physical Review B 88, 020406 (2013).
  * Schellekens and Koopmans (2013) A. J. Schellekens and B. Koopmans, Phys. Rev. B 87, 020407 (2013).
  * Baryakhtar  _et al._ (2013) V. G. Baryakhtar, V. I. Butrim, and B. A. Ivanov, JETP Letters 98, 289 (2013).
  * Baral and Schneider (2015) A. Baral and H. C. Schneider, Physical Review B 91, 100402 (2015).
  * Gridnev (2016) V. N. Gridnev, Journal of Physics: Condensed Matter 28, 476007 (2016).
  * Kalashnikova and Kozub (2016) A. M. Kalashnikova and V. I. Kozub, Physical Review B - Condensed Matter and Materials Physics 93, 1 (2016), arXiv:arXiv:1506.06585v2 .
  * Zhang  _et al._ (2016) G. P. Zhang, T. Latta, Z. Babyak, Y. H. Bai, and T. F. George, Modern Physics Letters B 30, 16300052 (2016).
  * Krivoruchko (2016) V. N. Krivoruchko, Physical Review B 94, 54434 (2016).
  * Gerlach  _et al._ (2017) S. Gerlach, L. Oroszlany, D. Hinzke, S. Sievering, S. Wienholdt, L. Szunyogh, and U. Nowak, Physical Review B 95, 224435 (2017).
  * Gridnev (2018) V. N. Gridnev, Phys. Rev. B 98, 014427 (2018).
  * Beens  _et al._ (2019) M. Beens, M. L. M. Lalieu, A. J. M. Deenen, R. A. Duine, and B. Koopmans, Physical Review B 100, 220409 (2019).
  * Vogler  _et al._ (2019) C. Vogler, C. Abert, F. Bruckner, and D. Suess, Physical Review B 100, 54401 (2019).
  * Davies  _et al._ (2020b) C. S. Davies, T. Janssen, J. H. Mentink, A. Tsukamoto, A. V. Kimel, A. F. G. van der Meer, A. Stupakiewicz, and A. Kirilyuk, Phys. Rev. Applied 13, 24064 (2020b).
  * Carva  _et al._ (2017) K. Carva, P. Baláž, and I. Radu (Elsevier, 2017) pp. 291–463.
  * Banerjee  _et al._ (2020) C. Banerjee, N. Teichert, K. E. Siewierska, Z. Gercsi, G. Y. P. Atcheson, P. Stamenov, K. Rode, J. M. D. Coey, and J. Besbas, Nature Communications 11, 4444 (2020).
  * Nowak (2007) U. Nowak, Classical Spin Models in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley and Sons, Ltd 2007) (2007).
  * (37) T. A. Ostler, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, I. Radu, R. Abrudan, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. Kimel, Phys. Rev. B 84, 024407.
  * Atxitia  _et al._ (2012) U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Physical Review B 86, 104414 (2012).
  * Atxitia  _et al._ (2010) U. Atxitia, D. Hinzke, O. Chubykalo-Fesenko, U. Nowak, H. Kachkachi, O. Mryasov, R. Evans, and R. Chantrell, Physical Review B - Condensed Matter and Materials Physics 82 (2010), 10.1103/PhysRevB.82.134440.
  * Garanin (1997) D. A. Garanin, Physical Review B 55, 3050 (1997).
  * Fowley  _et al._ (2018) C. Fowley, K. Rode, Y.-C. Lau, N. Thiyagarajah, D. Betto, K. Borisov, G. Atcheson, E. Kampert, Z. Wang, Y. Yuan, S. Zhou, J. Lindner, P. Stamenov, J. M. D. Coey, and A. M. Deac, Phys. Rev. B 98, 220406 (2018).
  * Betto  _et al._ (2015) D. Betto, N. Thiyagarajah, Y.-C. Lau, C. Piamonteze, M.-A. Arrio, P. Stamenov, J. M. D. Coey, and K. Rode, Phys. Rev. B 91, 094410 (2015).
  * Thiyagarajah  _et al._ (2015) N. Thiyagarajah, Y.-C. Lau, D. Betto, K. Borisov, J. M. D. Coey, P. Stamenov, and K. Rode, Applied Physics Letters 106, 122402 (2015).
  * Kurt  _et al._ (2014) H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Y.-C. Lau, E. Fonda, and J. M. D. Coey, Phys. Rev. Lett. 112, 027201 (2014).
  * Bonfiglio  _et al._ (2021) G. Bonfiglio, K. Rode, G. Y. P. Atcheson, P. Stamenov, J. M. D. Coey, A. V. Kimel, T. Rasing, and A. Kirilyuk, Journal of Physics: Condensed Matter 33, 135804 (2021).
  * Kaganov  _et al._ (1957) M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, JETP 173 (1957).
  * Chen  _et al._ (2006) J. K. Chen, D. Y. Tzou, and J. E. Beraun, International Journal of Heat and Mass Transfer 49, 307 (2006).
  * Radu  _et al._ (2015) I. Radu, C. Stamm, A. Eschenlohr, F. Radu, R. Abrudan, K. Vahaplar, T. Kachel, N. Pontius, R. Mitzner, K. Holldack, A. Föhlisch, T. a. Ostler, J. H. Mentink, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, a. V. Kimel, and T. Rasing, SPIN 5, 1550004 (2015).
  * Liao  _et al._ (2019) J.-W. Liao, P. Vallobra, L. O’Brien, U. Atxitia, V. Raposo, D. Petit, T. Vemulkar, G. Malinowski, M. Hehn, E. Martínez, S. Mangin, and R. Cowburn, Advanced Science 6 (2019), 10.1002/advs.201901876.
  * Ceballos  _et al._ (2021) A. Ceballos, A. Pattabi, A. El-Ghazaly, S. Ruta, C. P. Simon, R. F. L. Evans, T. Ostler, R. W. Chantrell, E. Kennedy, M. Scott, J. Bokor, and F. Hellman, Physical Review B 103, 24438 (2021).


# Counting the number of trigonal curves of genus five over finite fields

Tom Wennink

###### Abstract

The trigonal curves form a subvariety of $M_{5}$, the moduli space of smooth
curves of genus five. The cohomological data of these spaces can be found by
counting their numbers of points over finite fields. The trigonal curves of
genus five can be represented by projective plane quintics that have one
singularity, which is of delta-invariant one. We use a partial sieve method
for plane curves to count the number of trigonal curves over any finite field.
The result agrees with the findings of a computer program we have written that
counts the number of trigonal curves over the finite fields of two and three
elements.

##  1 Introduction.

Inside the moduli space $M_{5}$ of smooth curves of genus five there is the
subvariety $\mathcal{T}_{5}$ of trigonal curves. In this article we count
$\\#\mathcal{T}_{5}(k)$ for all finite fields $k$.

######  Theorem 7.2.

The number of smooth trigonal curves of genus five over a finite field
$\mathbb{F}_{q}$ is given by

$\\#\mathcal{T}_{5}(\mathbb{F}_{q})=q^{11}+q^{10}-q^{8}+1.$

We can find cohomological data of a space from knowing its numbers of points
over finite fields. Let $\mathcal{X}$ be a Deligne-Mumford stack over
$\mathbb{Z}$ of dimension $d$. Bogaart and Edixhoven have shown in [1, Theorem
2.1] that under certain conditions we have
$\\#\mathcal{X}(\mathbb{F}_{q})=P(q)$, where $P(t)=\sum P_{i}t^{i}$ is a
polynomial over $\mathbb{Q}$ of degree $d$ such that $P_{i}=P_{d-i}$.
Furthermore the cohomology of $\mathcal{X}$ is determined by $P$.

From Theorem $F$ in [2, p.15] and the fact that $\overline{M}_{5}$ is
unirational it follows that $\overline{M}_{5}$ satisfies the required
conditions. By purity, knowing the cohomology of $\overline{M}_{5}$ is
equivalent to knowing the Hodge Euler characteristic of $\overline{M}_{5}$.
This can be computed from the Hodge Euler characteristic of $M_{5}$ and the
$\mathbb{S}_{n}$-equivariant Hodge Euler characteristics of the $M_{g,n}$
occurring in the description of the boundary $\Delta_{5}$. Not all the
$\\#M_{4,n}(\mathbb{F}_{q})$ are known yet but again with the help of Theorem
$F$ from [2, p.15] we find that all $\\#M_{g,n}(\mathbb{F}_{q})$ are
polynomial. (We say a function of $\mathbb{F}_{q}$ is polynomial if it is an
element of $\mathbb{Q}[q]$.) From this it follows that
$\\#M_{5}(\mathbb{F}_{q})$ is polynomial. We have counted the trigonal curves
in this article and for hyperelliptic curves we know
$\\#\mathcal{H}_{5}(\mathbb{F}_{q})=q^{9}$. The curves in
$M_{5}(\mathbb{F}_{q})$ that are neither hyperelliptic nor trigonal have a
canonical embedding as a complete intersection of three quadrics in
$\mathbb{P}^{4}$. The number of these curves over finite fields
$\mathbb{F}_{q}$ is given by a polynomial $P(q)=\sum P_{i}q^{i}$. The
dimension of $M_{5}$ is $12$ so if we are able to compute $P_{i}$ for $6\leq
i\leq 12$ then we know $P(q)$ (using Poincaré Duality for $\overline{M}_{5}$).

There is a bijection between smooth trigonal curves of genus five and plane
quintics that have one singularity of delta-invariant one and no other
singularities. We fix a point $P$ and then aim to count quintics that have a
singularity at $P$ of delta-invariant one and no other singularities, see
Section 3. First we take all quintics that have a singularity at $P$ of delta-
invariant one and then apply a sieve method, see Section 4. This is done by
adding and subtracting various loci of quintics with singular points besides
$P$. The computation for these loci is found in Section 5. With each step of
the sieve method more and more curves that have singular points besides $P$
will have been removed exactly once. This process does not terminate so we
stop at some point. We then have to correct the count so that all curves that
have singularities besides $P$ have been removed exactly once. The necessary
computations for this are found in Section 6. Section 7 combines the results
and contains a weblink to a computer program we have written that counts
$\\#\mathcal{T}_{5}(\mathbb{F}_{2})$ and $\\#\mathcal{T}_{5}(\mathbb{F}_{3})$.
The results of this program confirm Theorem 7.2.

During our counting we come across a lot of cases. We have computed all the
cases we need and have listed the results. The methods we use are illustrated
by a number of examples in which we describe how we compute a single case.

### Acknowledgements

I would like to thank Carel Faber for his help and support. He was the advisor
for my master thesis and afterwards helped me transform it into this article.

##  2 Preliminaries and tools.

Our notation for partitions and tuples of points is in the style of [3].

######  Definition 2.1.

We write $\lambda=[1^{\lambda_{1}},\ldots,v^{\lambda_{v}}]$ for the partition
where $i$ appears $\lambda_{i}$ times. This partition has weight
$|\lambda|:=\sum_{i=1}^{v}i\cdot\lambda_{i}$. We consider the empty partition
$[]$ to have weight $0$. For the sake of notation we leave out the zero
powers, e.g. $[1^{2},2^{0},3^{0},4^{1}]$ is the same as $[1^{2},4^{1}]$.

######  Notation 2.2.

With $k$ we denote a finite field with $q$ elements. We define $k_{i}$ to be
the finite field extension of $k$ that has $q^{i}$ elements.

######  Notation 2.3.

When we talk about the Frobenius map $\mathcal{F}$ we mean the geometric
Frobenius.

Let $\mathbb{P}^{2}$ be the projective plane over $k$. The geometric Frobenius
on $\mathbb{P}^{2}$ is the endomorphism
$\mathcal{F}:\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ defined by
$(x:y:z)\mapsto(x^{q}:y^{q}:z^{q})$.

######  Definition 2.4.

Let $X$ be a scheme defined over $k$. An $n$-tuple $(x_{1},\ldots,x_{n})$ of
distinct subschemes of $X_{\bar{k}}$ is called a conjugate $n$-tuple if
$\mathcal{F}(x_{i})=x_{i+1}$ for $1\leq i<n$ and $\mathcal{F}(x_{n})=x_{1}$,
where $\mathcal{F}$ is the Frobenius map.

A $|\lambda|$-tuple $(x_{1},\ldots,x_{|\lambda|})$ of distinct subschemes of
$X_{\bar{k}}$ is called a $\lambda$-tuple if it consists of $\lambda_{1}$
conjugate $1$-tuples, followed by $\lambda_{2}$ conjugate $2$-tuples, etc.

For a scheme $X$ defined over $k$ we write $X(\lambda)$ for the set of
$\lambda$-tuples of points of $X$.

Let $X$ be a $k$-scheme, $N_{r}:=|X(k_{r})|$ and let $(\mu\ast
N)(n)=\sum_{d|n}\mu(\frac{n}{d})N_{d}$ denote the convolution of $N_{-}$ with
the Möbius function.

######  Lemma 2.5.

For any scheme $X$ defined over $k$, the number of $\lambda$-tuples of points
of $X$ is equal to

$|X(\lambda)|=\prod_{i=1}^{v}\binom{\frac{(\mu\ast N)(i)}{i}}{\lambda_{i}}.$

######  Definition 2.6.

For any scheme $X$ defined over $k$ and $w\in\mathbb{Z}_{\geq 0}$ we define

$\pi_{w}(X):=\sum_{|\lambda|=w}(-1)^{\sum_{i}\lambda_{i}}\cdot|X(\lambda)|.$

######  Lemma 2.7.

The inverse Hasse-Weil zeta function generates $\pi$, that is if $X$ is a
scheme of finite type over $k$ we have

$\frac{1}{Z(X;t)}=\sum_{w=0}^{\infty}\pi_{w}(X)t^{w}.$

###### Proof.

We have

$\displaystyle\sum_{w=0}^{\infty}\pi_{w}(X)t^{w}$
$\displaystyle=\sum_{w=0}^{\infty}\sum_{|\lambda|=w}\prod_{i=1}^{v}\binom{\frac{(\mu\ast
N)(i)}{i}}{\lambda_{i}}(-1)^{\lambda_{i}}t^{i\cdot\lambda_{i}}$
$\displaystyle=\prod_{i=1}^{\infty}\sum_{k=0}^{\infty}\binom{\frac{(\mu\ast
N)(i)}{i}}{k}(-t^{i})^{k}$
$\displaystyle=\prod_{i=1}^{\infty}(1-t^{i})^{\frac{(\mu\ast N)(i)}{i}}$
$\displaystyle=\prod_{i=1}^{\infty}\text{exp}\left(\frac{(\mu\ast
N)(i)}{i}\cdot\text{log}(1-t^{i})\right)$
$\displaystyle=\text{exp}\left(\sum_{i=1}^{\infty}\frac{(\mu\ast
N)(i)}{i}\sum_{k=1}^{\infty}\frac{-t^{ik}}{k}\right)$
$\displaystyle=\text{exp}\left(-\sum_{i=1}^{\infty}\sum_{k=1}^{\infty}\frac{(\mu\ast
N)(i)\cdot t^{ik}}{ik}\right)$
$\displaystyle=\text{exp}\left(-\sum_{i=1}^{\infty}\sum_{d|i}\frac{(\mu\ast
N)(d)\cdot t^{i}}{i}\right)$
$\displaystyle=\text{exp}\left(-\sum_{i=1}^{\infty}N_{i}\frac{t^{i}}{i}\right)$
$\displaystyle=\frac{1}{Z(X;t)}$

∎

Using the fact that
$Z(\mathbb{P}^{n};t)=\frac{1}{(1-t)(1-qt)\cdots(1-q^{n}t)}$ and
$Z(\mathbb{P}^{n}-\\{\mathbf{P}\\};t)=\frac{1}{(1-qt)\cdots(1-q^{n}t)}$ we get
the following corollaries.

######  Corollary 2.8.

$\pi_{w}(\mathbb{P}^{n})=0$ for $w\geq n+2$ and
$\pi_{w}(\mathbb{P}^{n}-\\{\mathbf{P}\\})=0$ for $w\geq n+1$.

######  Corollary 2.9.

$\sum_{w=0}^{\infty}\pi_{w}(\mathbb{P}^{n})=0.$

##  3 Plane curves.

Let a smooth trigonal curve of genus five be given. The dual system of the
$g^{1}_{3}$ is a $g^{2}_{5}$. This gives the following.

######  Proposition 3.1.

There is a bijection between the smooth trigonal curves of genus five and
projective plane quintics that have precisely one singularity which is of
delta-invariant one.

We want to count

$\\#\mathcal{T}_{5}(k)=\sum_{C/k}\frac{1}{|\text{Aut}_{k}(C)|}$

where the sum is over representatives of $k$-isomorphism classes of smooth
trigonal curves of genus five over $k$. By the above proposition it is
equivalent to sum over representatives of $k$-isomorphism classes of plane
quintics that have precisely one singularity which is of delta-invariant one.

Let $C$ be a smooth trigonal curve of genus five. Because the $g^{1}_{3}$ and
the $g^{2}_{5}$ are unique the automorphism group of $C$ as an abstract curve
is isomorphic to the automorphism group of $C$ as a plane curve. Plane curve
automorphisms are elements of $\text{PGL}_{3}$. So the automorphism group of
$C$ is the stabilizer of $C$ for the action of $\text{PGL}_{3}$ on
$\mathbb{P}^{2}$. We get

$\sum_{C/k}\frac{1}{|\text{Aut}_{k}(C)|}=\sum_{C/k}\frac{1}{|\text{Stab}_{k}(C)|}=\sum_{C/k}\frac{|\text{Orb}_{k}(C)|}{|\text{PGL}_{3}(k)|}=\frac{1}{|\text{PGL}_{3}(k)|}\sum_{C\in
T(k)}1.$

Here $T$ is the set of plane quintics with exactly one singularity which has
delta-invariant one. This enables us to simply count plane curves rather than
plane curves up to $k$-isomorphism.

######  Definition 3.2.

Let $C$ be a plane quintic with polynomial equation
$\sum_{i=0}^{5}F_{i}(x,y)z^{5-i}$ and let $L,L^{\prime}$ be lines. We say $C$
has tangents $L,L^{\prime}$ at $(0:0:1)$ if $C$ has a singularity of
multiplicity $2$ at $(0:0:1)$ and ${L\cdot L^{\prime}}=F_{2}(x,y)$. We say $C$
has tangents $L,L^{\prime}$ at a point $P$ if there exists a linear
transformation $\phi$ such that $\phi(P)=(0:0:1)$ and $\phi(C)$ has tangents
$\phi(L),\phi(L^{\prime})$ at $(0:0:1)$.

The curves in $T(k)$ can be separated into those which have an ordinary cusp
and those which have an ordinary node. The curves that have an ordinary node
can then be further separated into those that have both tangents defined over
$k$, in which case it is a split node and those for which the tangents are a
conjugate $2$-tuple, a non-split node. We will count these three types of
curves separately.

Let $C$ be a curve in $T(k)$ and let $P$ be its singular point. Since $P$ is
the only singular point of $C$ it is defined over $k$. If $P$ is a split node
then we can apply a $k$-linear coordinate change such that $P$ gets mapped to
$(0:0:1)$ and the tangents at $P$ get mapped to the lines $x$ and $y$. We can
count the number of curves in $T(k)$ that have a split node by first counting
the curves that have a singularity at $(0:0:1)$ with tangents $x,y$ and then
dividing by a suitable subset of $\text{PGL}_{3}(k)$. We develop some notation
for this.

######  Notation 3.3.

We write $\mathbf{P}$ for the fixed point $(0:0:1)$.

######  Notation 3.4.

We write $\mathbf{C}$ for the space of plane $k$-quintics, which is a
$\mathbb{P}^{20}(k)$.

######  Definition 3.5.

We define $\mathbf{C}_{\text{split}}$ to be the subset of $\mathbf{C}$
consisting of curves that have a singularity at $\mathbf{P}$ of multiplicity
$2$ with tangents $x$ and $y$.

######  Definition 3.6.

Let $\mathbf{C}^{\prime}$ be a subset of $\mathbf{C}$ and let $\lambda$ be a
partition. Here we allow for $\lambda$ to be an infinite partition
$[1^{\lambda_{1}},\ldots,i^{\lambda_{i}},\ldots]$. We define
$\mathbf{C}^{\prime}(\lambda)$ to be the subset of $\mathbf{C}^{\prime}$
consisting of those curves whose singularities form precisely a
$\lambda$-tuple of points. We also define
$\mathbf{C}^{\prime}(\mathbf{P},\lambda)$ to be the subset of
$\mathbf{C}^{\prime}$ consisting of those curves whose singularities besides
$\mathbf{P}$ form precisely a $\lambda$-tuple of points.

######  Remark 3.7.

Note that $\mathbf{C}(\lambda)$ as defined above is different from the set of
$\lambda$-tuples of points $\mathbb{P}^{20}(\lambda)$, we trust that this
difference will be clear from the context.

With this notation $\mathbf{C}_{\text{split}}(\mathbf{P},[])$ consists of the
curves that have no singularities besides $\mathbf{P}$. Let
$\text{Stab}_{k}(\mathbf{P},\\{x,y\\})$ be the subgroup of $\text{PGL}_{3}(k)$
that fixes $\mathbf{P}$ and fixes the set $\\{x,y\\}$. It is the group that
fixes $\mathbf{C}_{\text{split}}(\mathbf{P},[])$ so we get

$\frac{|\\{C\in T(k)\;|\;\text{$C$ has a split
node}\\}|}{|\text{PGL}_{3}(k)|}=\frac{|\mathbf{C}_{\text{split}}(\mathbf{P},[])|}{|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|}.$

We do the corresponding thing for non-split nodes and cusps.

######  Definition 3.8.

Let $\alpha$ be a fixed element of $k_{2}$ that is not in $k$. The set
$\mathbf{C}_{\text{non-split}}\subset\mathbf{C}$ is given by those curves that
have a singularity at $\mathbf{P}$ of multiplicity $2$ with tangents $x+\alpha
y$ and $x+\mathcal{F}(\alpha)y$.

######  Definition 3.9.

The set $\mathbf{C}_{\text{cusp}}\subset\mathbf{C}$ is given by those curves
that have a singularity at $\mathbf{P}$ of multiplicity $2$ with double
tangent $y$ and nonzero coefficient for $x^{3}z^{2}$.

We have

$\frac{|T(k)|}{|\text{PGL}_{3}(k)|}=\frac{|\mathbf{C}_{\text{split}}(\mathbf{P},[])|}{|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|}+\frac{|\mathbf{C}_{\text{non-
split}}(\mathbf{P},[])|}{|\text{Stab}_{k}(\mathbf{P},\\{x+\alpha
y,x+\mathcal{F}(\alpha)y\\})|}+\frac{|\mathbf{C}_{\text{cusp}}(\mathbf{P},[])|}{|\text{Stab}_{k}(\mathbf{P},\\{y\\})|}.$

We compute the sizes of the three stabilizer groups. The matrices that fix
$P$, $x$ and $y$ have the form

$\begin{pmatrix}a&0&0\\\ 0&b&0\\\ c&d&1\end{pmatrix}$

where $ab\neq 0$. There are $q^{2}(q-1)^{2}$ such matrices. We can also
permute $x$ and $y$ which adds a factor two so
$|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|=2q^{2}(q-1)^{2}$.

If $\text{char}(k)\neq 2$ then the field $k$ has a quadratic nonresidue $r$
and we can take $\alpha$ such that $\alpha^{2}=r$. The matrices that fix $P$,
$x+\alpha y$ and $x+\mathcal{F}(\alpha)y$ have the form

$\begin{pmatrix}a&rb&0\\\ b&a&0\\\ c&d&1\end{pmatrix}\text{ if
$\text{char}(k)\neq
2$},\quad\begin{pmatrix}a&\frac{\alpha\cdot\mathcal{F}(\alpha)\cdot(a+b)}{\alpha+\mathcal{F}(\alpha)}&0\\\
\frac{a+b}{\alpha+\mathcal{F}(\alpha)}&b&0\\\ c&d&1\end{pmatrix}\text{ if
$\text{char}(k)=2$}.$

In either case we get $q^{2}\cdot(q^{2}-1)$ matrices so with the tangent
permutation we have $|\text{Stab}_{k}(\mathbf{P},\\{x+\alpha
y,x+\mathcal{F}(\alpha)y\\})|=2(q^{4}-q^{2})$.

The matrices that fix $P$ and $y$ have the form

$\begin{pmatrix}a&b&0\\\ 0&c&0\\\ d&e&1\end{pmatrix}$

where $ac\neq 0$. Since we have no permutation of tangents here we get
$|\text{Stab}_{k}(\mathbf{P},\\{y\\})|=q^{3}(q-1)^{2}$.

##  4 The partial sieve method.

We want to compute $|\mathbf{C}_{\text{split}}(\mathbf{P},[])|$,
$|\mathbf{C}_{\text{non-split}}(\mathbf{P},[])|$ and
$|\mathbf{C}_{\text{cusp}}(\mathbf{P},[])|$ using the sieve principle. In this
section we focus on $|\mathbf{C}_{\text{split}}(\mathbf{P},[])|$, the other
two cases are similar. We take $|\mathbf{C}_{\text{split}}|$ and then sieve by
adding and subtracting various loci of curves with singular points besides
$\mathbf{P}$.

######  Definition 4.1.

Let $\mathbf{C}^{\prime}$ be a subset of $\mathbf{C}$ and let
$S\subset\mathbb{P}^{2}$ be a set of points. We define
$\mathbf{C}^{\prime}(S)$ to be the subset of $\mathbf{C}^{\prime}$ consisting
of those curves that have a singularity at each point in $S$.

The locus of singular curves in $\mathbf{C}_{\text{split}}$ is the union of
all $\mathbf{C}_{\text{split}}(S)$ where
$S\subset\mathbb{P}^{2}-\\{\mathbf{P}\\}$ is a conjugate $n$-tuple of points
for some $n\geq 1$. We can use the sieve principle to compute the elements of
this union. That is, we sum
$(-1)^{m+1}|\mathbf{C}_{\text{split}}(S_{1})\cap\ldots\cap\mathbf{C}_{\text{split}}(S_{m})|$
over all $m\geq 1$ and all unordered choices of distinct
$S_{1},\ldots,S_{m}\subset\mathbb{P}^{2}-\\{\mathbf{P}\\}$ where every $S_{i}$
is a conjugate $n_{i}$-tuple of points.

Since
$\mathbf{C}_{\text{split}}(S_{1})\cap\mathbf{C}_{\text{split}}(S_{2})=\mathbf{C}_{\text{split}}(S_{1}\cup
S_{2})$ we find that when we subtract this sum from
$|\mathbf{C}_{\text{split}}|$ we get

$\sum_{\lambda}\left((-1)^{\sum_{i}\lambda_{i}}\cdot\sum_{S\in(\mathbb{P}^{2}-\\{\mathbf{P}\\})(\lambda)}|\mathbf{C}_{\text{split}}(S)|\right).$

This sum won’t terminate because there are curves that have infinitely many
singularities. Another problem with computing this sum is that
$|\mathbf{C}_{\text{split}}(S)|$ becomes increasingly hard to compute as the
number of points in $S$ grows. So instead we apply a partial sieve principle
where we pick a number $N$ and only sieve for all $\lambda$ with
$|\lambda|\leq N$.

######  Definition 4.2.

We define

$\mathcal{S}_{\text{split},N}:=\sum_{|\lambda|\leq
N}\left((-1)^{\sum_{i}\lambda_{i}}\cdot\sum_{S\in(\mathbb{P}^{2}-\\{\mathbf{P}\\})(\lambda)}|\mathbf{C}_{\text{split}}(S)|\right).$

We also have to account for the curves that have more than $N$ singularities
besides $\mathbf{P}$. This is done by explicitly counting the number of curves
that have precisely a $\lambda$-tuple of singlarities besides $\mathbf{P}$ for
all $\lambda$ such that $|\lambda|>N$. For each curve we have to subtract it
as often as it has been counted in the computation of
$\mathcal{S}_{\text{split}}$.

######  Definition 4.3.

We define

$\sigma_{N}(\lambda):=-\sum_{\begin{subarray}{c}\mu\subset\lambda\\\ |\mu|\leq
N\end{subarray}}(-1)^{\sum_{i}\mu_{i}}\prod_{i=1}^{v}\binom{\lambda_{i}}{\mu_{i}},$

where $\mu\subset\lambda$ means that $\mu_{i}\leq\lambda_{i}$ for all $i$.

We have

$|\mathbf{C}_{\text{split}}(\mathbf{P},[])|=\mathcal{S}_{\text{split},N}+\sum_{|\lambda|>N}|\mathbf{C}_{\text{split}}(\mathbf{P},\lambda)|\cdot\sigma_{N}(\lambda).$

######  Definition 4.4.

Let $C$ be a projective plane curve, we define $\\#\delta_{k}^{1}(C)$ to be
the number of (isolated) singularities over $k$ on $C$ of delta-invariant one.

Let $[\lambda,1^{1}]$ be the partition $\mu$ where $\mu_{1}=\lambda_{1}+1$ and
$\mu_{i}=\lambda_{i}$ for $i>1$. Since $\text{Stab}_{k}(\mathbf{P},\\{x,y\\})$
is the group that fixes $\mathbf{C}_{\text{split}}(\mathbf{P},\lambda)$ we
have

$\frac{|\mathbf{C}_{\text{split}}(\mathbf{P},\lambda)|}{|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|}=\frac{1}{|\text{PGL}_{3}(k)|}\cdot\sum_{C\in\mathbf{C}([\lambda,1^{1}])}\\#\\{\text{ordinary
split nodes over $k$ on $C$}\\}.$

This means we get

$\displaystyle\frac{|T(k)|}{|\text{PGL}_{3}(k)|}=$
$\displaystyle\frac{\mathcal{S}_{\text{split},N}}{|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|}+\frac{\mathcal{S}_{\text{non-
split},N}}{|\text{Stab}_{k}(\mathbf{P},\\{x+\alpha
y,x+\mathcal{F}(\alpha)y\\})|}+\frac{\mathcal{S}_{\text{cusp},N}}{|\text{Stab}_{k}(\mathbf{P},\\{y\\})|}$
$\displaystyle+\frac{1}{|\text{PGL}_{3}(k)|}\sum_{|\lambda|>N}\sum_{C\in\mathbf{C}([\lambda,1^{1}])}\\#\delta_{k}^{1}(C)\cdot\sigma_{N}(\lambda).$

The choice of $N$ is made by considering the difficulty of computing
$\mathcal{S}_{\text{split},N}$ versus the above sum over $|\lambda|>N$. We
choose $N$ to be five.

##  5 The sieving part.

In this section we will compute $\mathcal{S}_{\text{split},5}$,
$\mathcal{S}_{\text{non-split},5}$ and $\mathcal{S}_{\text{cusp},5}$.

###  5.1 A split node.

To know $\mathcal{S}_{\text{split},5}$ we have to compute
$|\mathbf{C}_{\text{split}}(S)|$ for all $S$ such that $|S|\leq 5$. The space
$\mathbf{C}_{\text{split}}(S)$ is an $\mathbb{A}^{15}$ with $3\cdot|S|$ linear
conditions on it. For different choices of $S$ we will need to prove the
independence of a subset of these conditions. To do this look at the matrix
that has the monomial coefficients of degree five as columns and the linear
conditions as rows. To make this explicit we need to decide what the three
conditions are for each point. We have the condition that the point is on the
curve and we pick two from the three conditions that we get by demanding the
derivatives with respect to $x,y,z$ to be zero at the point. When we have
constructed this matrix we take a subset of the columns such that we get a
square matrix. If the determinant is nonzero then the conditions are
independent.

We also want the conditions to be independent from the condition that there
must be a split node at $\mathbf{P}$ with tangents $x,y$. This means that in
our choice of a subset of columns we must avoid the columns corresponding to
the coefficients of $x^{2}z^{3},xyz^{3},y^{2}z^{3},xz^{4},yz^{4},z^{5}$.

Because $S$ is a $\lambda$-tuple of points for some partition $\lambda$, the
Frobenius morphism maps $S$ to itself and all linear conditions can be
expressed using coefficients over $k$. The $k$-dimension of
$\mathbf{C}_{\text{split}}(S)$ as a vector space over $k$ is therefore the
same as the $\bar{k}$-dimension of $\mathbf{C}_{\text{split}}(S)$ as a vector
space over $\bar{k}$. So the dimension of $\mathbf{C}_{\text{split}}(S)$ over
$k$ does not change when we apply a $\bar{k}$-change of coordinates, that
fixes $\mathbf{P}$, to the points in $S$.

Note that after applying such a transformation the conditions that
$x^{2}z^{3}=y^{2}z^{3}=0$ and $xyz^{3}\neq 0$ might change, but they will
still be conditions on the coefficients of $x^{2}z^{3},xyz^{3},y^{2}z^{3}$.

######  Example 5.1.

Let $|S|=5$ so that together with $\mathbf{P}$ we have six points such that
there are no four points on a line, no six on an irreducible conic and no
three on a line through $\mathbf{P}$. We apply a $\bar{k}$-linear
transformation that fixes $\mathbf{P}$ and maps the points in $S$ to
$(1:0:0),(0:1:0),(1:1:1),(1:\alpha:\beta),(1:\gamma:\delta)$ for some
$\alpha,\beta,\gamma,\delta$ where $\alpha,\gamma\neq 0,1$ and
$\alpha\neq\gamma$. We also have
$\alpha\beta\gamma-\alpha\beta\delta-\alpha\gamma\delta+\beta\gamma\delta-\beta\gamma+\alpha\delta\neq
0$ for otherwise there would be six points on a conic.

We construct a matrix as described above where we take derivatives w.r.t. $x$
and $z$. This gives us a $15\times 21$ matrix. If we leave out the columns
corresponding to $x^{2}z^{3},xyz^{3},y^{2}z^{3},xz^{4},yz^{4},z^{5}$ we get a
square matrix with determinant

$(\alpha\beta\gamma-\alpha\beta\delta-\alpha\gamma\delta+\beta\gamma\delta-\beta\gamma+\alpha\delta)^{4}(\alpha-\gamma)(\alpha-1)\alpha^{2}(\gamma-1)\gamma^{2}.$

So the conditions are independent and
$\mathbf{C}_{\text{split}}(S)=\mathbb{A}^{0}$.

The above argument also ensures independence for the cases where we have
$|S|\leq 4$ such that together with $\mathbf{P}$ there are no four points on a
line and no three on a line through $\mathbf{P}$.

######  Example 5.2.

Let us have two lines and let $|S|=5$ be such that the intersection point of
the lines is in $S$, one of the lines has three more points from $S$ on it,
and the other line has $\mathbf{P}$ and the last point in $S$ on it.

By Bézout any quintic that has singularities at the points in $S$ has the two
lines as a component. If the line through $\mathbf{P}$ is not either $x$ or
$y$ then we get a contradiction with the tangent conditions and
$\mathbf{C}_{\text{split}}(S)=\emptyset$. Without loss of generality we take
the line to be $y$. Denote the intersection point of the two lines by $Q$. Now
$\mathbf{C}_{\text{split}}(S)$ is isomorphic to the space of cubics that pass
through the points in $S-\\{Q\\}$ and pass through $\mathbf{P}$ with tangent
$x$.

We use a $\bar{k}$-linear transformation that keeps $\mathbf{P}$ fixed and
maps the points in $S$ to $(1:0:0),(0:1:0),(1:1:0),(1:\alpha:0),(1:0:1)$ where
$\alpha\neq 0,1$. Taking the columns for $x^{3},x^{2}y,xy^{2},y^{3}$ gives
determinant $-\alpha(\alpha-1)$. So the conditions are independent and
$\mathbf{C}_{\text{split}}(S)=\mathbb{A}^{3}$.

######  Example 5.3.

Let $S$ have at least four points on a line that passes through $\mathbf{P}$.
By Bézout the line is a double component of any quintic in
$\mathbf{C}_{\text{split}}(S)$. This contradicts the fact that $\mathbf{P}$ is
an ordinary node so $\mathbf{C}_{\text{split}}(S)=\emptyset$.

We list the possible cases for $S$ where after every case we describe
$\mathbf{C}_{\text{split}}(S)$. The cases where
$\mathbf{C}_{\text{split}}(S)=\emptyset$ have been left out. Whenever we have
three points (including $\mathbf{P}$) on a line through $\mathbf{P}$, we leave
implicit that the line is either $x$ or $y$ (see Example 5.2).

  * •

No four points on a line, no six on an irreducible conic and no three on a
line through $\mathbf{P}$: $\mathbb{A}^{15-3\cdot|S|}$

  * •

$\mathbf{P}$ and two more points on a line and $0\leq n\leq 3$ points outside
the line, no four points on a line and no six on a conic: $\mathbb{A}^{10-3n}$

  * •

$\mathbf{P}$ and two more points on a line and the three other points on
another line, such that none of the points is on the intersection of the two
lines: $\mathbb{A}^{2}$

  * •

$\mathbf{P}$ and two points on one line and $\mathbf{P}$ and two points on
another line, and $0\leq n\leq 1$ points outside these lines:
$\mathbb{A}^{5-3n}$

  * •

$\mathbf{P}$ and three more points on a line and $0\leq n\leq 2$ points
outside the line such that there are no three on another line through
$\mathbf{P}$: $\mathbb{A}^{9-3n}$

  * •

$\mathbf{P}$ and three points on one line and $\mathbf{P}$ and two points on
another line: $\mathbb{A}^{4}$

  * •

Four points on a line and $\mathbf{P}$ and $0\leq n\leq 1$ points outside the
line such that there are no three on a line through $\mathbf{P}$:
$\mathbb{A}^{5-3n}$

  * •

Four points on a line and $\mathbf{P}$ and another point outside the line such
that there are three on a line through $\mathbf{P}$: $\mathbb{A}^{3}$

  * •

Five points on a line and $\mathbf{P}$ outside the line: $\mathbb{A}^{4}$

  * •

Six points on an irreducible conic: $\mathbb{A}^{3}$

Now we have to compute the number of possible $\lambda$-tuples of points
belonging to each case.

######  Example 5.4.

For each $\lambda$ of weight five we look at how many choices of
$\lambda$-tuples of points there are such that there are $5$ points on a line
and the point $\mathbf{P}$ outside the line.

Pick a line that does not pass through $\mathbf{P}$, there are $q^{2}$ such
lines. We then take a $\lambda$-tuple of points on the chosen line. Because of
Corollary 2.8 the contribution to $\mathcal{S}_{\text{split}}$ is given by

$\displaystyle\sum_{|\lambda|=5}(-1)^{\sum_{i}\lambda_{i}}\cdot
q^{2}\cdot\sum_{S\in\mathbb{P}^{1}(\lambda)}|\mathbf{C}_{\text{split}}(S)|$
$\displaystyle=q^{2}\cdot|\mathbb{A}^{4}(k)|\cdot\sum_{|\lambda|=5}(-1)^{\sum_{i}\lambda_{i}}\cdot|\mathbb{P}^{1}(\lambda)|$
$\displaystyle=q^{2}\cdot|\mathbb{A}^{4}(k)|\cdot\pi_{5}(\mathbb{P}^{1})$
$\displaystyle=0$

In the above list all cases besides the first one have four points on a line,
six on an irreducible conic or three on a line through $\mathbf{P}$. Using
Corollary 2.8 it turns out that all these cases contribute zero. This means we
only need to know for every partition $\lambda$ with $|\lambda|\leq 5$, the
number of $\lambda$-tuples such that there are no four points on a line, six
on an irreducible conic or three on a line through $\mathbf{P}$. This is given
by $|\mathbb{P}^{2}(\lambda)|$ minus the number of choices of $\lambda$-tuples
of points corresponding to all other cases. However these numbers all become
zero when we sum them with a factor $(-1)^{\sum_{i}\lambda_{i}}$. So we can
just pretend we don’t have to subtract anything at all.

$\displaystyle\frac{\mathcal{S}_{\text{split},5}}{|\text{Stab}_{k}(\mathbf{P},\\{x,y\\})|}$
$\displaystyle=\frac{1}{2q^{2}(q-1)^{2}}\sum_{|\lambda|\leq
5}(-1)^{\sum_{i}\lambda_{i}}\cdot\sum_{S\in(\mathbb{P}^{2}-\\{\mathbf{P}\\})(\lambda)}|\mathbb{A}^{15-3\cdot|S|}|$
$\displaystyle=\frac{1}{2q^{2}(q-1)^{2}}\sum_{i=1}^{5}\pi_{i}(\mathbb{P}^{2}-\\{\mathbf{P}\\})\cdot
q^{15-3i}$ $\displaystyle=\frac{q^{15}-q^{14}-q^{13}+q^{12}}{2q^{2}(q-1)^{2}}$
$\displaystyle=\frac{1}{2}(q^{11}+q^{10})$

###  5.2 A non-split node.

If $S$ is a set of points such that $\mathbf{P}$ and two points in $S$ are on
a line then by Bézout that line is part of any curve in
$\mathbf{C}_{\text{non-split}}(S)$, so it is one of the tangent lines. Because
the sets of points we consider are closed under Frobenius there have to be
just as many points on the other tangent line. So for non-split nodes we only
have the following non-empty cases:

  * •

No four points on a line, no six on an irreducible conic and no three on a
line through $\mathbf{P}$: $\mathbb{A}^{15-3\cdot|S|}$

  * •

$\mathbf{P}$ and two points on one line and $\mathbf{P}$ and two points on
another line, and $0\leq n\leq 1$ points outside these lines:
$\mathbb{A}^{5-3n}$

  * •

Four points on a line and $\mathbf{P}$ and $0\leq n\leq 1$ points outside the
line such that there are no three on a line through $\mathbf{P}$:
$\mathbb{A}^{5-3n}$

  * •

Four points on a line and $\mathbf{P}$ and another point outside the line such
that there are three on a line through $\mathbf{P}$: $\mathbb{A}^{3}$

  * •

Five points on a line and $\mathbf{P}$ outside the line: $\mathbb{A}^{4}$

  * •

Six points on an irreducible conic: $\mathbb{A}^{3}$

Because we used $\bar{k}$-linear transformations for the computations of
$\mathbf{C}_{\text{split}}(S)$ their results also apply to
$\mathbf{C}_{\text{non-split}}(S)$ for these cases.

Choosing four points (closed under Frobenius) such that we have $\mathbf{P}$
and two points on one tangent line and $\mathbf{P}$ and two points on the
other line is the same as choosing two non-$\mathbf{P}$ points on one of the
tangent lines and then taking their conjugates. So if we work over $k_{2}$
then we can use the fact that $\pi_{2}(\mathbb{P}^{1}-\\{\mathbf{P}\\})=0$.
Choosing five non-$\mathbf{P}$ points on a $k_{2}$-conic can also be done over
$k_{2}$. These two cases become zero and the other cases are exactly the same
as they are for split nodes. This means we get the result

$\frac{\mathcal{S}_{\text{non-
split},5}}{|\text{Stab}_{k}(\mathbf{P},\\{x+\alpha
y,x+\mathcal{F}(\alpha)y\\})|}=\frac{q^{15}-q^{14}-q^{13}+q^{12}}{2(q^{4}-q^{2})}=\frac{1}{2}(q^{11}-q^{10}).$

###  5.3 A cusp.

For $\mathbf{C}_{\text{cusp}}(S)$ we can not as easily reuse the previous
computations since we have the added condition that the coefficient for
$x^{3}z^{2}$ is nonzero. This means that we need to keep the line $y$ fixed
when we do a linear transformation. If $S$ is a set of points such that
$\mathbf{P}$ and two points in $S$ are on a line then by Bézout that line is
part of any curve in $\mathbf{C}_{\text{cusp}}(S)$. For any such curve the
cusp at $\mathbf{P}$ is not ordinary and so $\mathbf{C}_{\text{cusp}}(S)$ is
empty. Similarly $\mathbf{C}_{\text{cusp}}(S)$ is empty for any set of points
$S$ such that there are five points on a conic that has tangent $y$ at
$\mathbf{P}$. We have the following non-empty cases:

  * •

No four points on a line, no six on an irreducible conic, no five on a conic
that has tangent $y$ at $\mathbf{P}$, and no three on a line through
$\mathbf{P}$: $\mathbb{A}^{15-3\cdot|S|}-\mathbb{A}^{14-3\cdot|S|}$

  * •

Four points on a line and $\mathbf{P}$ and $0\leq n\leq 1$ points outside the
line such that there are no three on a line through $\mathbf{P}$:
$\mathbb{A}^{5-3n}-\mathbb{A}^{4-3n}$

  * •

Five points on a line and $\mathbf{P}$ outside the line:
$\mathbb{A}^{4}-\mathbb{A}^{3}$

######  Example 5.5.

Let $S$ have no four points on a line, no six on an irreducible conic, no five
on a conic that has tangent $y$ at $\mathbf{P}$, and no three on a line
through $\mathbf{P}$.

If $|S|=4$ then we can use a $\bar{k}$-linear transformation that fixes
$\mathbf{P}$ and $y$ and maps the points in $S$ to
$(0:1:0),(1:1:1),(1:\alpha:\beta),(1:\gamma:\delta)$ where $\alpha,\gamma\neq
1$, $\alpha\neq\gamma$ and
$\alpha\beta\gamma-\alpha\gamma\delta-\alpha\beta+\gamma\delta+\alpha-\gamma\neq
0$. Taking derivatives w.r.t. $y$ and $z$ and taking the columns for
$y^{3}z^{2},xy^{3}z,x^{2}y^{2}z,x^{3}yz,x^{4}z,x^{2}y^{3},x^{3}y^{2},$
$x^{4}y,x^{5},xy^{4},y^{4}z,y^{5}$ gives determinant

$-(\alpha\beta\gamma-\alpha\gamma\delta-\alpha\beta+\gamma\delta+\alpha-\gamma)^{3}(\alpha-\gamma)^{2}(\alpha-1)^{2}(\gamma-1)^{2}.$

If $|S|=5$ then we want to use the same arguments as in Example 5.1. But there
we don’t consider the coefficient for $x^{3}z^{2}$ so we might be counting
some non-ordinary cusps. We show that there are actually no non-ordinary cusps
on any curves that have a double tangent $y$ at $\mathbf{P}$ and singularities
at all the points in $S$. If $\mathbf{P}$ is a non-ordinary cusp then it lies
on an irreducible quartic or quintic component or on the intersection of two
irreducible components. If it lies on an irreducible quartic then having six
singularities in total implies that there will be four on a line. If there are
six singularities on an irreducible quintic then they all need to have delta-
invariant $1$. Finally if it lies on the intersection of two irreducible
components then one of those components is a line or a conic, and we will have
$\mathbf{P}$ and two more points on a line or five points on a conic that has
tangent $y$ at $\mathbf{P}$.

As with nodes all cases besides the first one listed become zero and we have

$\displaystyle\frac{\mathcal{S}_{\text{cusp},5}}{|\text{Stab}_{k}(\mathbf{P},\\{y\\})|}$
$\displaystyle=\frac{1}{q^{3}(q-1)^{2}}\sum_{i=1}^{5}\pi_{i}(\mathbb{P}^{2}-\\{\mathbf{P}\\})\cdot(q^{15-3i}-q^{14-3i})$
$\displaystyle=\frac{q^{15}-2q^{14}+2q^{12}-q^{11}}{q^{3}(q-1)^{2}}$
$\displaystyle=q^{10}-q^{8}.$

##  6 The explicit part.

In this section we will compute

$-\frac{1}{|\text{PGL}_{3}(k)|}\sum_{|\lambda|>5}\sum_{C\in\mathbf{C}([\lambda,1^{1}])}\\#\delta_{k}^{1}(C)\cdot\sigma_{5}(\lambda).$
(1)

######  Definition 6.1.

We define the type of a projective plane quintic over $k$ to be given by the
following information:

  * •

the degree and multiplicity of the irreducible components

  * •

the fields over which the irreducible components are defined

  * •

the number of points in which every pair of irreducible components intersects

  * •

the number of singularities of each irreducible component

  * •

the delta-invariants of singularities of each irreducible components

  * •

the fields over which the singularities of each irreducible component are
defined

######  Remark 6.2.

This definition is naturally rather analogous to [3, Definition 7.2].

By knowing the type of a curve we know over what fields its singularities are
defined and the delta-invariants of its singularities. If we can count how
many curves there are of each type then that gives us enough information to
compute (1). It is sufficient to compute the number of curves for those types
where the singularities form a $[\lambda,1^{1}]$-tuple such that
$|\lambda|>5$, $\sigma_{5}(\lambda)\neq 0$ and there is at least one
(isolated) singularity of delta-invariant one.

We first separate the types by the degree and multiplicity of their
irreducible components. Section 6.1 considers types where there is at least
one component of higher multiplicity. In Sections 6.2 to 6.7 all components
will have multiplicity one.

The results will be listed in tables where rows correspond to the various
types. The column marked $\\#\delta_{k}^{1}$ contains the number of (isolated)
singularities of delta-invariant one. The column marked $\\#\\{C\\}$ contains
the number of curves of each type divided by $|\text{PGL}_{3}(k)|$. Note that
our tables will not contain the types for which there are no singularities of
delta-invariant one or for which $\sigma_{5}(\lambda)=0$.

######  Remark 6.3.

It is not necessary for our purpose but in general it might be interesting to
separately count split nodes, non-split nodes and cusps. The main reason we do
not do this is that for the types of curves we have to count it is hard to
determine whether a singularity on an irreducible component is a split node, a
non-split node or a cusp.

###  6.1 A component of higher multiplicity.

Let $\lambda$ be an infinite partition such that there is a curve
$C\in\mathbf{C}([\lambda,1^{1}])$ with $\\#\delta_{k}^{1}(C)\neq 0$. We need
an isolated singularity of delta-invariant one so the double component $C$ can
only be one double (or triple) line. Let $\eta$ be the partition such that the
singularities of $C$ that are not on the double line form a $\eta$-tuple of
points. Let $\rho$ be the infinite partition such that $\rho_{i}$ is the
number of conjugate $i$-tuples in $\mathbb{P}^{1}$.

$\displaystyle\sigma_{5}(\lambda)$
$\displaystyle=-\sum_{\begin{subarray}{c}\mu\subset\lambda\\\ |\mu|\leq
5\end{subarray}}(-1)^{\sum_{i}\mu_{i}}\prod_{i=1}^{v}\binom{\lambda_{i}}{\mu_{i}}$
$\displaystyle=-\sum_{\begin{subarray}{c}\nu\subset\eta\\\ |\nu|\leq
5\end{subarray}}\sum_{\begin{subarray}{c}\mu\subset\rho\\\ |\mu|\leq
5-|\nu|\end{subarray}}(-1)^{\sum_{i}\mu_{i}+\nu_{i}}\prod_{i=1}^{v}\binom{\rho_{i}}{\mu_{i}}\binom{\eta_{i}}{\nu_{i}}$
$\displaystyle=-\sum_{\begin{subarray}{c}\nu\subset\eta\\\ |\nu|\leq
5\end{subarray}}(-1)^{\sum_{i}\nu_{i}}\prod_{i=1}^{v}\binom{\eta_{i}}{\nu_{i}}\sum_{|\mu|\leq
5-|\nu|}(-1)^{\sum_{i}\mu_{i}}|\mathbb{P}^{1}(\mu)|$

Since $|\nu|\leq|\eta|\leq 3$ we can use Corollaries 2.8 and 2.9 to get

$\sum_{|\mu|\leq
5-|\nu|}(-1)^{\sum_{i}\mu_{i}}|\mathbb{P}^{1}(\mu)|=\sum_{i=0}^{5-|\nu|}\pi_{i}(\mathbb{P}^{1})=\sum_{i=0}^{\infty}\pi_{i}(\mathbb{P}^{1})=0.$

So the types of curves that have a multiple component contribute zero.

###  6.2 Five lines.

The five lines can be in general position and intersect in ten points or there
can be three lines through one point so that we get eight points. Any other
type of curve consisting of five distinct lines has fever than seven
intersection points.

####  6.2.1 General position.

######  Example 6.4.

We count the number of curves of the type where the lines are all defined over
$k$. First we take two $k$-points and then through each of these points a pair
of $k$-lines such that none of the lines passes through both points. This can
be done in $\binom{q^{2}+q+1}{2}\binom{q}{2}^{2}$ ways. The four lines have
six intersection points. Take as the fifth line any line over $k$ that does
not go through any of these points. There are $(q-2)(q-3)$ such lines. We can
end up with the same five lines if we start with another choice of two of the
intersection points such that none of the five lines passes through both
points. There are $15$ such choices so we get

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot\frac{1}{15}\binom{q^{2}+q+1}{2}\binom{q}{2}^{2}(q-2)(q-3)=\frac{1}{120}(q-2)(q-3).$

Whenever $\lambda$ is a finite partition we can use

$\displaystyle\sigma_{N}(\lambda)$
$\displaystyle=-\sum_{\begin{subarray}{c}\mu\subset\lambda\\\ |\mu|\leq
N\end{subarray}}(-1)^{\sum_{i}\mu_{i}}\prod_{i=1}^{v}\binom{\lambda_{i}}{\mu_{i}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mu\subset\lambda\\\
|\mu|>N\end{subarray}}(-1)^{\sum_{i}\mu_{i}}\prod_{i=1}^{v}\binom{\lambda_{i}}{\mu_{i}}$
$\displaystyle=\sum_{\begin{subarray}{c}\mu\subset\lambda\\\
|\mu|<|\lambda|-N\end{subarray}}(-1)^{\sum_{i}(\lambda_{i}-\mu_{i})}\prod_{i=1}^{v}\binom{\lambda_{i}}{\lambda_{i}-\mu_{i}}$
$\displaystyle=(-1)^{\sum_{i}\lambda_{i}}\sum_{\begin{subarray}{c}\mu\subset\lambda\\\
|\mu|<|\lambda|-N\end{subarray}}(-1)^{\sum_{i}\mu_{i}}\prod_{i=1}^{v}\binom{\lambda_{i}}{\mu_{i}}.$

So here we get

$\sigma_{5}([1^{9}])=(-1)\cdot(1-\binom{9}{1}+\binom{9}{2}-\binom{9}{3})=56.$

lines |  $[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---|---
$[1^{5}]$ | $[1^{10}]$ | $56$ | $10$ | $\frac{1}{120}(q-2)(q-3)$
$[1^{3},2^{1}]$ | $[1^{4},2^{3}]$ | $6$ | $4$ | $\frac{1}{12}q(q-1)$
$[1^{2},3^{1}]$ | $[1^{1},3^{3}]$ | $2$ | $1$ | $\frac{1}{6}(q+1)q$

####  6.2.2 Three of the lines intersect in one point $P$.

lines |  $[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---|---
$[1^{5}]$ | $[1^{8}]$ | $6$ | $7$ | $\frac{1}{12}(q-2)$
$[1^{3},2^{1}]$, both $k_{2}$-lines through $P$ | $[1^{4},2^{2}]$ | $2$ | $3$ | $\frac{1}{4}q$

###  6.3 One conic and three lines.

One or two of the lines can be tangent to the conic, two lines can intersect
on the conic, or three lines can go through one point. We only need to
consider these as separate cases because combining them will result in fever
than seven singularities. We say the components are in general position when
none of the above cases applies.

####  6.3.1 Two of the lines are tangent to the conic.

######  Example 6.5.

We count the number of curves of the type where all components and
singularities are defined over $k$. The plane conics form a $\mathbb{P}^{5}$
and every reducible $k$-conic is either a pair of different $k$-lines, a
conjugate pair of lines, or a double $k$-line. So the number of irreducible
plane conics over $k$ is given by

$\frac{q^{6}-1}{q-1}-\binom{q^{2}+q+1}{2}-\frac{1}{2}(q^{4}-q)-(q^{2}+q+1)=(q^{2}+q+1)q^{2}(q-1).$

Pick two $k$-points $A,B$ on the conic and take the tangent lines at these
points, they meet in a point $P$. Choose two more $k$-points on the conic and
take the line through them. We need to subtract the case where $P$ is on all
three lines. If $\text{char}(k)\neq 2$ then the lines through $A$ and $B$ are
the only lines through $P$ that are tangent to the conic. There are $q-1$
other $k$-points on the conic that each determine a line through $P$. Each
line gets determined twice this way so we get

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot(q^{2}+q+1)q^{2}(q-1)\binom{q+1}{2}\left(\binom{q-1}{2}-\frac{q-1}{2}\right)=\frac{1}{4}(q-3).$

If $\text{char}(k)=2$ then all lines through $P$ are tangent to the conic so
we get

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot(q^{2}+q+1)q^{2}(q-1)\binom{q+1}{2}\binom{q-1}{2}=\frac{1}{4}(q-2).$

lines |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{3}]$ | $[1^{7}]$ | $1$ | $5$ | $\frac{1}{4}(q-3)$ | $\frac{1}{4}(q-2)$
| $[1^{5},2^{1}]$ | $-1$ | $3$ | $\frac{1}{4}(q-1)$ | $\frac{1}{4}q$
$[1^{1},2^{1}]$ | $[1^{3},2^{2}]$ | $1$ | $3$ | $\frac{1}{4}(q-1)$ | $\frac{1}{4}q$
| $[1^{1},2^{3}]$ | $-1$ | $1$ | $\frac{1}{4}(q-3)$ | $\frac{1}{4}(q-2)$

####  6.3.2 One of the lines is tangent to the conic.

lines |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{3}]$ | $[1^{8}]$ | $6$ | $7$ | $\frac{1}{8}(q-3)^{2}$ | $\frac{1}{8}(q-2)(q-4)$
| $[1^{6},2^{1}]$ | $-4$ | $5$ | $\frac{1}{4}(q-1)^{2}$ | $\frac{1}{4}q(q-2)$
| $[1^{4},2^{2}]$ | $2$ | $3$ | $\frac{1}{8}(q-1)^{2}$ | $\frac{1}{8}q(q-2)$

####  6.3.3 Two of the lines intersect on the conic.

lines |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---|---
$[1^{3}]$ | $[1^{7}]$ | $1$ | $6$ | $\frac{1}{4}(q-2)(q-3)$
| $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{4}q(q-1)$
$[1^{1},2^{1}]$ | $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{4}q(q-1)$

####  6.3.4 Three lines through one point.

######  Example 6.6.

We count the number of curves of the type where all components and
singularities are defined over $k$. We first consider the case where
$\text{char}(k)\neq 2$. Choose an irreducible conic and two $k$-points on the
conic. Take the tangent lines to the conic at the two points and denote their
intersection by $P$. There are $\frac{q-1}{2}$ lines through $P$ that
intersect the conic in two $k$-points. We can also choose an irreducible conic
and a conjugate pair of points on the conic. Let $P$ be the intersection of
the corresponding tangent lines. There are now $\frac{q+1}{2}$ lines through
$P$ that intersect the conic in two $k$-points. Putting both cases together we
get

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot(q^{2}+q+1)q^{2}(q-1)\left(\binom{q+1}{2}\binom{\frac{q-1}{2}}{3}+\frac{q^{2}-q}{2}\binom{\frac{q+1}{2}}{3}\right)=\frac{1}{48}(q-3)^{2}.$

If $\text{char}(k)=2$ then there is one $k$-point outside the conic that is
the intersection of all tangent lines to the conic. Through any other
$k$-point outside the conic there are $\frac{q}{2}$ lines that intersect the
conic in two $k$-points. So we have

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot(q^{2}+q+1)q^{2}(q-1)(q^{2}-1)\binom{\frac{q}{2}}{3}=\frac{1}{48}(q-2)(q-4).$

lines |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{3}]$ | $[1^{7}]$ | $1$ | $6$ | $\frac{1}{48}(q-3)^{2}$ | $\frac{1}{48}(q-2)(q-4)$
| $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{16}(q-1)^{2}$ | $\frac{1}{16}q(q-2)$
| $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{16}(q-1)^{2}$ | $\frac{1}{16}q(q-2)$
$[1^{1},2^{1}]$ | $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{8}(q^{2}-2q-1)$ | $\frac{1}{8}q(q-2)$
| $[1^{3},4^{1}]$ | $-1$ | $2$ | $\frac{1}{8}(q+1)(q-1)$ | $\frac{1}{8}q^{2}$

####  6.3.5 General position.

######  Example 6.7.

We count the number of curves of the type where all components and
singularities are defined over $k$ and $\text{char}(k)\neq 2$. We choose an
irreducible $k$-conic and six $k$-points on the conic. There are $15$ ways to
choose three lines connecting these six points. The lines should not intersect
in one point so we subtract the results from Example 6.6 to get

$\frac{1}{|\text{PGL}_{3}(k)|}\cdot(q^{2}+q+1)q^{2}(q-1)\binom{q+1}{6}\cdot
15-\frac{1}{48}(q-3)^{2}=\frac{1}{48}(q^{2}-7q+11)(q-3).$

lines |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{3}]$ | $[1^{9}]$ | $21$ | $9$ | $\frac{1}{48}(q^{2}-7q+11)(q-3)$ | $\frac{1}{48}(q-2)(q-4)^{2}$
| $[1^{7},2^{1}]$ | $-9$ | $7$ | $\frac{1}{16}(q^{2}-3q+1)(q-1)$ | $\frac{1}{16}q(q-2)^{2}$
| $[1^{5},2^{2}]$ | $1$ | $5$ | $\frac{1}{16}(q^{3}-2q^{2}-1)$ | $\frac{1}{16}q^{2}(q-2)$
| $[1^{3},2^{3}]$ | $3$ | $3$ | $\frac{1}{48}(q^{2}-3q+1)(q-1)$ | $\frac{1}{48}q(q-2)^{2}$
$[1^{1},2^{1}]$ | $[1^{3},2^{3}]$ | $3$ | $3$ | $\frac{1}{8}(q^{2}-q-1)(q-1)$ | $\frac{1}{8}q^{2}(q-2)$
| $[1^{1},2^{4}]$ | $-3$ | $1$ | $\frac{1}{8}(q^{2}-q-3)(q-3)$ | $\frac{1}{8}(q^{2}-2q-4)(q-2)$
| $[1^{3},2^{1},4^{1}]$ | $-1$ | $3$ | $\frac{1}{8}(q^{2}-q+1)(q+1)$ | $\frac{1}{8}q^{3}$
| $[1^{1},2^{2},4^{1}]$ | $1$ | $1$ | $\frac{1}{8}(q^{2}-q-1)(q-1)$ | $\frac{1}{8}q^{2}(q-2)$

###  6.4 Two conics and one line.

The conics can intersect in three or in four points. If they intersect in four
points then the line can intersect the conics in three or in four points. Any
other type will result in fever than seven singularities. We say the
components are in general position if the conics intersect in four points and
the line intersects the conics in four points.

In the tables the first column will denote the fields over which the
intersection points of the conics are defined.

####  6.4.1 General position and the conics are defined over $k$.

######  Example 6.8.

We count the number of curves of the type where all singularities are defined
over $k$ and $\text{char}(k)\neq 2$. There are
$\frac{1}{6}\binom{q^{2}+q+1}{2}(q^{4}-2q^{3}+q^{2})$ ways to pick four
$k$-points $P_{1},P_{2},P_{3},P_{4}$ such that there are no three on a line.
We write $\mathcal{P}$ for the pencil of conics through
$P_{1},P_{2},P_{3},P_{4}$. There are six lines through
$P_{1},P_{2},P_{3},P_{4}$ and these lines intersect in seven $k$-points. Let
$Q_{1},Q_{2},Q_{3}$ denote the other three intersection points.

Let $L$ be any line such that none of $P_{1},P_{2},P_{3},P_{4}$ is on $L$. We
have a degree two map $\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ that sends a
point $R$ on $L$ to the conic in $\mathcal{P}$ through $R$. By Hurwitz’s
theorem there are two branch points, we call these two conics the tangent
conics for $L$.

There are $3$ $k$-lines through two of $Q_{1},Q_{2},Q_{3}$. Let $L$ be such a
line. Both tangent conics for $L$ are reducible and each one intersects $L$ in
one of the $Q_{i}$. There are $q-1$ other $k$-points on $L$ so there are
$\frac{q-1}{2}$ conics in $\mathcal{P}(k)$ that intersect $L$ in two
$k$-points. One of these conics is reducible so we have $\frac{q-3}{2}$
irreducible conics.

There are $3(q-3)$ $k$-lines through precisely one of $Q_{1},Q_{2},Q_{3}$ and
not through any of $P_{1},P_{2},P_{3},P_{4}$. Let $L$ be such a line. One of
the tangent lines for $L$ intersects it in the $Q_{i}$ point on $L$ so the
other tangent conic for $L$ is also defined over $k$. There are
$\frac{q-5}{2}$ irreducible conics in $\mathcal{P}(k)$ that intersect $L$ in
two $k$-points. (This is not well defined when $q=3$ but that is not a problem
since this case occurs $3(q-3)$ times. The same idea applies to the next two
cases.)

There are $q-2$ irreducible conics in $\mathcal{P}(k)$ that each have $q-3$
$k$-points besides $P_{1},P_{2},P_{3},P_{4}$. This means we have
$\frac{1}{2}((q-2)(q-3)-3(q-3))=\frac{1}{2}(q-5)(q-3)$ $k$-lines that have two
irreducible tangent conics over $k$. For every such line there are
$\frac{q-7}{2}$ irreducible conics in $\mathcal{P}(k)$ that intersect it in
two $k$-points.

There are $(q-3)^{2}$ $k$-lines not through any of
$P_{1},P_{2},P_{3},P_{4},Q_{1},Q_{2},Q_{3}$. So there are
$(q-3)^{2}-\frac{1}{2}(q-5)(q-3)=\frac{1}{2}(q-3)(q-1)$ $k$-lines that have a
conjugate pair of irreducible tangent conics. For every such line there are
$\frac{q-5}{2}$ irreducible conics in $\mathcal{P}(k)$ that intersect it in
two $k$-points.

Putting everything together gives us

$\frac{1}{|\text{PGL}_{3}(k)|}\frac{1}{6}\binom{q^{2}+q+1}{2}(q^{4}-2q^{3}+q^{2})\Bigg{(}3\binom{\frac{q-3}{2}}{2}+3(q-3)\binom{\frac{q-5}{2}}{2}\\\
+\frac{1}{2}(q-3)(q-5)\binom{\frac{q-7}{2}}{2}+\frac{1}{2}(q-1)(q-3)\binom{\frac{q-5}{2}}{2}\Bigg{)}=\frac{1}{192}(q^{2}-9q+17)(q-3)(q-5).$

######  Example 6.9.

We count the number of curves of the type where all singularities are defined
over $k$ for $\text{char}(k)=2$. Let us choose $P_{1},P_{2},P_{3},P_{4}$ and
define $\mathcal{P}$ and $Q_{1},Q_{2},Q_{3}$ as in Example 6.8. There is one
line through $Q_{1},Q_{2},Q_{3}$ and it intersects every conic in
$\mathcal{P}$ in one point. For any other line not through any of
$P_{1},P_{2},P_{3},P_{4}$ there is precisely one conic in $\mathcal{P}$ that
intersects it in one point.

There are $3(q-2)$ $k$-lines through one of $Q_{1},Q_{2},Q_{3}$ and not
through any of $P_{1},P_{2},P_{3},P_{4}$. For any such line there are
$\frac{q-4}{2}$ irreducible conics in $\mathcal{P}(k)$ that intersect it in
two $k$-points.

There are $(q-2)(q-4)$ $k$-lines not through any of
$P_{1},P_{2},P_{3},P_{4},Q_{1},Q_{2},Q_{3}$. For any such line there are
$\frac{q-6}{2}$ irreducible conics in $\mathcal{P}(k)$ that intersect it in
two $k$-points.

Putting these two cases together we get

$\frac{1}{|\text{PGL}_{3}(k)|}\frac{1}{6}\binom{q^{2}+q+1}{2}(q^{4}-2q^{3}+q^{2})\Bigg{(}3(q-2)\binom{\frac{q-4}{2}}{2}+(q-2)(q-4)\binom{\frac{q-6}{2}}{2}\Bigg{)}\\\
=\frac{1}{192}(q-2)(q-4)(q-5)(q-6).$

con pts |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{4}]$ | $[1^{8}]$ | $6$ | $8$ | $\frac{1}{192}(q^{2}-9q+17)(q-3)(q-5)$ | $\frac{1}{192}(q-2)(q-4)(q-5)(q-6)$
| $[1^{6},2^{1}]$ | $-4$ | $6$ | $\frac{1}{96}(q^{2}-5q+3)(q-1)(q-3)$ | $\frac{1}{96}q(q-2)(q-3)(q-4)$
| $[1^{4},2^{2}]$ | $2$ | $4$ | $\frac{1}{192}(q^{2}-q+1)(q-1)(q-3)$ | $\frac{1}{192}q(q-1)(q-2)^{2}$
$[1^{2},2^{1}]$ | $[1^{6},2^{1}]$ | $-4$ | $6$ | $\frac{1}{32}(q^{2}-3q+1)(q-1)(q-3)$ | $\frac{1}{32}q(q-2)^{2}(q-3)$
| $[1^{4},2^{2}]$ | $2$ | $4$ | $\frac{1}{16}(q^{3}-2q^{2}-1)(q-1)$ | $\frac{1}{16}q^{2}(q-1)(q-2)$
$[1^{1},3^{1}]$ | $[1^{5},3^{1}]$ | $3$ | $5$ | $\frac{1}{24}(q+1)q(q-1)(q-2)$ | $\frac{1}{24}q(q+1)(q-1)(q-2)$
| $[1^{3},2^{1},3^{1}]$ | $-1$ | $3$ | $\frac{1}{12}q^{2}(q+1)(q-1)$ | $\frac{1}{12}q^{2}(q+1)(q-1)$
| $[1^{1},2^{2},3^{1}]$ | $-1$ | $1$ | $\frac{1}{24}(q+1)q(q-1)(q-2)$ | $\frac{1}{24}q(q+1)(q-1)(q-2)$
$[2^{2}]$ | $[1^{4},2^{2}]$ | $2$ | $4$ | $\frac{1}{64}(q^{3}-2q^{2}-2q-1)(q-3)$ | $\frac{1}{64}q(q^{2}-3q-2)(q-2)$
$[4^{1}]$ | $[1^{4},4^{1}]$ | $-2$ | $4$ | $\frac{1}{32}(q^{2}-q-1)(q+1)(q-1)$ | $\frac{1}{32}q^{2}(q+1)(q-2)$

####  6.4.2 General position and the conics form a conjugate pair.

con pts |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{4}]$ | $[1^{4},2^{2}]$ | $2$ | $4$ | $\frac{1}{96}(q^{3}-4q^{2}+4q+3)(q-1)$ | $\frac{1}{96}q(q-1)(q-2)^{2}$
| $[1^{4},4^{1}]$ | $-2$ | $4$ | $\frac{1}{96}(q^{2}-3q+3)(q+1)(q-1)$ | $\frac{1}{96}q^{2}(q-1)(q-2)$
$[1^{1},3^{1}]$ | $[1^{1},2^{2},3^{1}]$ | $-1$ | $1$ | $\frac{1}{12}(q+1)q(q-1)(q-2)$ | $\frac{1}{12}(q+1)q(q-1)(q-2)$
| $[1^{1},3^{1},4^{1}]$ | $1$ | $1$ | $\frac{1}{12}(q+1)q^{2}(q-1)$ | $\frac{1}{12}(q+1)q^{2}(q-1)$

####  6.4.3 The line is tangent to one of the conics.

con pts |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{4}]$ | $[1^{7}]$ | $1$ | $6$ | $\frac{1}{48}(q-3)(q-4)(q-5)$ | $\frac{1}{48}(q-2)(q-4)(q-6)$
| $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{48}(q-1)(q-2)(q-3)$ | $\frac{1}{48}q(q-2)(q-4)$
$[1^{2},2^{1}]$ | $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{8}(q-1)^{2}(q-2)$ | $\frac{1}{8}q(q-2)^{2}$
| $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{8}q(q-1)^{2}$ | $\frac{1}{8}q^{2}(q-2)$
$[1^{1},3^{1}]$ | $[1^{4},3^{1}]$ | $1$ | $3$ | $\frac{1}{6}(q+1)q(q-1)$ | $\frac{1}{6}(q+1)q(q-1)$
| $[1^{2},2^{1},3^{1}]$ | $-1$ | $1$ | $\frac{1}{6}(q+1)q(q-1)$ | $\frac{1}{6}(q+1)q(q-1)$
$[2^{2}]$ | $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{16}q(q-1)(q-3)$ | $\frac{1}{16}q(q-2)^{2}$
$[4^{1}]$ | $[1^{3},4^{1}]$ | $-1$ | $2$ | $\frac{1}{8}(q+1)q(q-1)$ | $\frac{1}{8}q^{3}$

####  6.4.4 Two $k$-conics intersect in three points.

con pts |  $[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---|---
$[1^{2}]$ | $[1^{7}]$ | $1$ | $6$ | $\frac{1}{16}(q-3)^{2}(q-5)$ | $\frac{1}{16}(q-2)(q-4)(q-5)$
| $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{8}(q-1)^{2}(q-3)$ | $\frac{1}{8}q(q-2)(q-3)$
| $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{16}(q-1)^{3}$ | $\frac{1}{16}q(q-1)(q-2)$
$[2^{1}]$ | $[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{16}(q-1)^{2}(q-3)$ | $\frac{1}{16}q(q-2)(q-3)$
| $[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{8}(q-1)^{3}$ | $\frac{1}{8}q(q-1)(q-2)$

####  6.4.5 A conjugate pair of conics intersects in three point.

$[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---
$[1^{3},2^{2}]$ | $1$ | $2$ | $\frac{1}{8}(q^{2}-2q-1)(q-1)$ | $\frac{1}{8}q(q-1)(q-2)$
$[1^{3},4^{1}]$ | $-1$ | $2$ | $\frac{1}{8}(q+1)(q-1)^{2}$ | $\frac{1}{8}q^{2}(q-1)$

###  6.5 A cubic and two lines.

The cubic can be smooth or singular. If the cubic is singular then one of the
lines can be tangent to the cubic. Any other type will have fever than seven
singularities.

####  6.5.1 A singular cubic and no lines are tangent to it.

If the lines are defined over $k_{2}$ then $\sigma_{5}(\lambda)$ will always
be zero so we need only consider the types where the lines are defined over
$k$.

######  Example 6.10.

We count the number of curves of the type where all singularities are defined
over $k$. Take two $k$-lines and a $k$-point $P$ outside the lines. Then
choose three $k$-points on each line, such that no two of these six points are
on a line through $P$ and none of the points is the intersection of the two
lines. There is precisely one irreducible cubic through the six points that
has a singularity at $P$ of delta-invariant one. We get

$\frac{1}{|\text{PGL}_{3}(k)|}\binom{q^{2}+q+1}{2}(q^{2}-q)\binom{q}{3}\binom{q-3}{3}=\frac{1}{72}(q-2)(q-3)(q-4)(q-5).$

$[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---
$[1^{8}]$ | $6$ | $8$ | $\frac{1}{72}(q-2)(q-3)(q-4)(q-5)$
$[1^{6},2^{1}]$ | $-4$ | $6$ | $\frac{1}{12}q(q-1)(q-2)(q-3)$
$[1^{4},2^{2}]$ | $2$ | $4$ | $\frac{1}{8}(q+1)q(q-1)(q-2)$
$[1^{5},3^{1}]$ | $3$ | $5$ | $\frac{1}{18}(q+1)q(q-1)(q-2)$
$[1^{3},2^{1},3^{1}]$ | $-1$ | $3$ | $\frac{1}{6}(q+1)q^{2}(q-1)$

####  6.5.2 A smooth cubic.

######  Example 6.11.

We count the number of curves of the type where the components and
singularities are all defined over $k$. Take two $k$-lines $L,L^{\prime}$ and
three $k$-points on each line that are not the intersection point. There is a
$\mathbb{P}^{3}$ of cubics through the six points. From this we subtract the
reducible cubics through the six points. We get a reducible cubic by taking
any $k$-line together with the lines $L,L^{\prime}$. Or we can take a line
though two of the points and an irreducible conic through the other four
points. Finally we can take three lines connecting the six points such that
none of the lines is $L$ or $L^{\prime}$.

We also subtract the singular irreducible cubics we counted in Example 6.10.
This gives us

$\frac{1}{|\text{PGL}_{3}(k)|}\binom{q^{2}+q+1}{2}\binom{q}{3}^{2}\left(|\mathbb{P}^{3}|-|\mathbb{P}^{2}|-9(q-2)-6\right)\\\
-\frac{1}{72}(q-2)(q-3)(q-4)(q-5)=\frac{1}{72}(q^{4}-3q^{3}+3q^{2}-17q+36)(q-2).$

lines |  $[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---|---
$[1^{2}]$ | $[1^{7}]$ | $1$ | $7$ | $\frac{1}{72}(q^{4}-3q^{3}+3q^{2}-17q+36)(q-2)$
| $[1^{5},2^{1}]$ | $-1$ | $5$ | $\frac{1}{12}(q^{3}-q^{2}+q-3)q(q-2)$
| $[1^{3},2^{2}]$ | $1$ | $3$ | $\frac{1}{8}(q^{2}-q+2)(q+1)q(q-1)$
| $[1^{4},3^{1}]$ | $1$ | $4$ | $\frac{1}{18}(q^{2}-q+1)(q+1)q(q-2)$
| $[1^{1},3^{2}]$ | $1$ | $1$ | $\frac{1}{18}(q^{3}-2)(q+1)q$
| $[1^{2},2^{1},3^{1}]$ | $-1$ | $2$ | $\frac{1}{6}(q^{2}-q+1)(q+1)q^{2}$
$[2^{1}]$ | $[1^{1},2^{3}]$ | $-1$ | $1$ | $\frac{1}{12}(q^{3}-q^{2}+4)(q^{2}-3)$
| $[1^{1},2^{1},4^{1}]$ | $1$ | $1$ | $\frac{1}{4}(q+1)q^{2}(q-1)^{2}$
| $[1^{1},6^{1}]$ | $-1$ | $1$ | $\frac{1}{6}(q^{3}-q^{2}-2)q^{2}$

####  6.5.3 A singular cubic and one line is tangent to it.

$[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---
$[1^{7}]$ | $1$ | $6$ | $\frac{1}{6}(q-2)(q-3)(q-4)$
$[1^{5},2^{1}]$ | $-1$ | $4$ | $\frac{1}{2}q(q-1)(q-2)$
$[1^{4},3^{1}]$ | $1$ | $3$ | $\frac{1}{3}(q+1)q(q-1)$

###  6.6 A cubic and a conic.

To get seven singularities the cubic must be singular and it has to intersect
the conic in six points.

######  Example 6.12.

We count the number of curves of the type where the singularities are all
defined over $k$ for $\text{char}(k)\neq 2$. Choose an irreducible conic and
two $k$-points $A,B$ on the conic. Take the tangents to the conic at $A$ and
$B$ and denote their intersection by $P$. We want to pick six points on the
conic such that there are no two on a line through $P$. There are three ways
we can do this: We can take $A,B$ and four of the other points, either $A$ or
$B$ and five of the other points, or six of the other points. There is exactly
one irreducible cubic through the six points with a singularity at $P$. We can
also choose an irreducible conic and a conjugate pair of points on the conic.
We then again take $P$ to be the intersection point of the corresponding
tangent lines and choose six $k$-points on the conic such that there are no
two on a line through $P$. Combining these cases gives us

$\frac{1}{|\text{PGL}_{3}(k)|}(q^{2}+q+1)q^{2}(q-1)\\\
\Bigg{(}\binom{q+1}{2}(q-1)(q-3)(q-5)(q-7)\left(\frac{1}{4!}+2\frac{1}{5!}(q-9)+\frac{1}{6!}(q-9)(q-11)\right)\\\
+\frac{q^{2}-q}{2}\frac{1}{6!}(q+1)(q-1)(q-3)(q-5)(q-7)(q-9)\Bigg{)}=\frac{1}{720}(q^{2}-9q+15)(q-3)(q-5)(q-7).$

$[\lambda,1^{1}]$ pts | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ |  $\\#\\{C\\}$ for $\text{char}(k)\neq 2$ |  $\\#\\{C\\}$ for $\text{char}(k)=2$
---|---|---|---|---
$[1^{7}]$ | $1$ | $7$ | $\frac{1}{720}(q^{2}-9q+15)(q-3)(q-5)(q-7)$ | $\frac{1}{720}(q-2)(q-4)^{2}(q-6)(q-8)$
$[1^{5},2^{1}]$ | $-1$ | $5$ | $\frac{1}{48}(q^{3}-6q^{2}+10q-1)(q-1)(q-3)$ | $\frac{1}{48}(q-2)^{3}q(q-4)$
$[1^{3},2^{2}]$ | $1$ | $3$ | $\frac{1}{16}(q^{2}-q-1)(q+1)(q-1)(q-3)$ | $\frac{1}{16}(q^{2}-2q-4)q^{2}(q-2)$
$[1^{1},2^{3}]$ | $-1$ | $1$ | $\frac{1}{48}(q^{2}-q-3)(q^{2}-2q-7)(q-3)$ | $\frac{1}{48}(q^{2}-2q-4)(q+2)(q-2)(q-4)$
$[1^{4},3^{1}]$ | $1$ | $4$ | $\frac{1}{18}(q^{2}-3q+3)(q+1)q(q-1)$ | $\frac{1}{18}(q+1)q(q-1)^{2}(q-2)$
$[1^{1},3^{2}]$ | $1$ | $1$ | $\frac{1}{18}(q^{2}+2q+3)q^{2}(q-2)$ | $\frac{1}{18}(q^{2}+2q+3)(q+1)(q-1)(q-2)$
$[1^{2},2^{1},3^{1}]$ | $-1$ | $2$ | $\frac{1}{6}(q^{2}-q-1)(q+1)q(q-1)$ | $\frac{1}{6}(q+1)^{2}q(q-1)(q-2)$
$[1^{3},4^{1}]$ | $-1$ | $3$ | $\frac{1}{8}(q^{2}-q+1)(q+1)^{2}(q-1)$ | $\frac{1}{8}(q^{2}-2)q^{3}$
$[1^{1},2^{1},4^{1}]$ | $1$ | $1$ | $\frac{1}{8}(q^{2}-q-1)(q+1)(q-1)^{2}$ | $\frac{1}{8}(q^{2}-2)q^{2}(q-2)$
$[1^{2},5^{1}]$ | $1$ | $2$ | $\frac{1}{5}(q^{2}+1)q^{2}(q+1)$ | $\frac{1}{5}(q^{2}+1)(q+1)^{2}(q-1)$
$[1^{1},6^{1}]$ | $-1$ | $1$ | $\frac{1}{6}(q^{2}+q+2)q^{2}(q-1)$ | $\frac{1}{6}(q^{2}+q+2)(q+1)(q-1)^{2}$

###  6.7 A quartic and a line.

To get seven singularities the quartic must have three singularities and it
has to intersect the line in four points.

quart. sing. |  $[\lambda,1^{1}]$ points | $\sigma_{5}(\lambda)$ | $\\#\delta_{k}^{1}$ | $\\#\\{C\\}$
---|---|---|---|---
$[1^{3}]$ | $[1^{7}]$ | $1$ | $7$ | $\frac{1}{144}(q-2)(q-3)^{2}(q-4)(q-5)$
| $[1^{5},2^{1}]$ | $-1$ | $5$ | $\frac{1}{24}q(q-1)^{2}(q-2)(q-3)$
| $[1^{3},2^{2}]$ | $1$ | $3$ | $\frac{1}{48}(q+1)q(q-1)(q-2)(q-3)$
| $[1^{4},3^{1}]$ | $1$ | $4$ | $\frac{1}{18}(q+1)q^{2}(q-1)(q-2)$
| $[1^{3},4^{1}]$ | $-1$ | $3$ | $\frac{1}{24}(q+1)q^{2}(q-1)^{2}$
$[1^{1},2^{1}]$ | $[1^{5},2^{1}]$ | $-1$ | $5$ | $\frac{1}{48}q(q-1)(q-2)(q-3)^{2}$
| $[1^{3},2^{2}]$ | $1$ | $3$ | $\frac{1}{8}(q+1)q(q-1)^{2}(q-2)$
| $[1^{1},2^{3}]$ | $-1$ | $1$ | $\frac{1}{16}(q^{2}-q-4)(q+1)(q-2)(q-3)$
| $[1^{2},2^{1},3^{1}]$ | $-1$ | $2$ | $\frac{1}{6}(q+1)q^{3}(q-1)$
| $[1^{1},2^{1},4^{1}]$ | $1$ | $1$ | $\frac{1}{8}(q+1)q^{2}(q-1)^{2}$
$[3^{1}]$ | $[1^{4},3^{1}]$ | $1$ | $4$ | $\frac{1}{72}(q+1)q(q-1)(q-2)(q-3)$
| $[1^{2},2^{1},3^{1}]$ | $-1$ | $2$ | $\frac{1}{12}(q+1)q^{2}(q-1)^{2}$
| $[1^{1},3^{2}]$ | $1$ | $1$ | $\frac{1}{9}(q^{3}-q-3)(q+1)q$

##  7 Results.

When we add the results together then for all characteristics of the field $k$
we get

$\sum\sigma(\lambda)\cdot\\#\delta_{k}^{1}\cdot\\#\\{C\\}=1,$

where the sum is over the rows of all the tables in Section 6.

######  Remark 7.1.

If we compute $\sum\sigma(\lambda)\cdot\\#\delta_{k}^{1}\cdot\\#\\{C\\}$ where
the sum is only over the rows of one the tables then for most tables we get
the same result for all characteristics of $k$. The only exceptions are the
tables in Sections 6.3.1 and 6.3.2.

Combining our results from the sieving part and the explicit part we get

$\frac{|T(k)|}{|\text{PGL}_{3}(k)|}=\frac{1}{2}(q^{11}+q^{10})+\frac{1}{2}(q^{11}-q^{10})+q^{10}-q^{8}+1,$

which proves our theorem.

######  Theorem 7.2.

The number of smooth trigonal curves of genus five over a finite field
$\mathbb{F}_{q}$ is given by

$\\#\mathcal{T}_{5}(\mathbb{F}_{q})=q^{11}+q^{10}-q^{8}+1.$

We have written computer programs that loop over all plane quintics over
$\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ that have an ordinary split node/non-
split node/cusp with fixed tangents at $\mathbf{P}$. For each curve we test
for all points besides $\mathbf{P}$ whether they are singular or not. This way
we count

$|\mathbf{C}_{\text{split}}(\mathbf{P},\lambda)|,\quad|\mathbf{C}_{\text{non-
split}}(\mathbf{P},\lambda)|\quad\text{and}\quad|\mathbf{C}_{\text{cusp}}(\mathbf{P},\lambda)|$
(2)

for all partitions $\lambda$. From the results for $\lambda=[]$ we can now
easily compute $\\#\mathcal{T}_{5}(\mathbb{F}_{2})$ and
$\\#\mathcal{T}_{5}(\mathbb{F}_{3})$. As an extra check the programs use the
information from (2) to count the subresults

$\sum_{|\lambda|=w}\left((-1)^{\sum_{i}\lambda_{i}}\cdot\sum_{S\in(\mathbb{P}^{2}-\\{\mathbf{P}\\})(\lambda)}|\mathbf{C}_{\text{split}}(S)|\right)$

for $0\leq w\leq 5$ and

$\sum_{|\lambda|=w}|\mathbf{C}_{\text{split}}(\mathbf{P},\lambda)|\cdot\sigma_{N}(\lambda)$

for $6\leq w\leq 9$.

The results of these computer counts agree with the results in this article.
The programs are written in the C programming language and the source code is
available online together with lists of the results for (2) and the above
subresults.
The address is: https://github.com/Wennink/countingtrigonalcurves

## References

  * [1] T. van den Bogaart and B. Edixhoven, _Algebraic stacks whose number of points over finite fields is a polynomial_ , Number Fields and Function Fields — Two Parallel Worlds, Progress in Mathematics, Vol. 239, Birkhäuser Boston, Boston, 2005\.
  * [2] Gaëtan Chenevier and Jean Lannes, _Formes automorphes et voisins de Kneser des réseaux de Niemeier_ , http://arxiv.org/abs/1409.7616, 2014\.
  * [3] J. Bergström, _Cohomology of moduli spaces of curves of genus three via point counts_ , J. Reine Angew. Math., 622:155–187, 2008\.