# Self-Similar Outflows at the Source of the Fast Solar Wind:
A Smoking Gun of Multiscale Impulsive Reconnection?

Vadim M. Uritsky  Catholic University of America, 620 Michigan Avenue NE,
Washington, DC 20064, USA  Heliophysics Science Division, NASA Goddard Space
Flight Center, Greenbelt, MD, 20771, USA  Judith T. Karpen  Heliophysics
Science Division, NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA
Nour E. Raouafi  The John Hopkins University Applied Physics Laboratory,
Laurel, MD 20723, USA  Pankaj Kumar  Department of Physics, American
University, Washington, DC 20016, USA  Heliophysics Science Division, NASA
Goddard Space Flight Center, Greenbelt, MD, 20771, USA  C. Richard DeVore
Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt,
MD, 20771, USA  Craig E. Deforest  Southwest Research Institute, 1050 Walnut
Street, Suite 300, Boulder, CO 80302, USA

###### Abstract

We present results of a quantitative analysis of structured plasma outflows
above a polar coronal hole observed by the Atmospheric Imaging Assembly
onboard the Solar Dynamics Observatory spacecraft. In a 6-hour interval of
continuous high-cadence SDO/AIA images, we identified more than 2300 episodes
of small-scale plasma flows in the polar corona. The mean upward flow speed
measured by the surfing transform technique (Uritsky et al., 2013b) is
estimated to be 122 $\pm$ 34 km s-1, which is comparable to the local sound
speed. The typical recurrence period of the flow episodes is 10 to 30 minutes,
and the mean duration and transverse size of each episode are about 3–5 min
and 3–4 Mm, respectively. The largest identifiable episodes last for tens of
minutes and reach widths up to $40$ Mm. For the first time, we demonstrate
that the polar coronal-hole outflows obey a family of power-law probability
distributions characteristic of impulsive interchange magnetic reconnection.
Turbulent photospheric driving may play a crucial role in releasing
magnetically confined plasma onto open field. The estimated occurrence rate of
the detected self-similar coronal outflows is sufficient for them to make a
dominant contribution to the fast-wind mass and energy fluxes and to account
for the wind’s small-scale structure.

Sun: jets—Sun: corona—Sun: UV radiation—Sun: magnetic fields

##  1 Introduction

Small-scale plasma outflows in magnetically open regions of the solar corona
have been observed for a long time. In many open geometries, including mid-
latitude plumes (Raouafi & Stenborg, 2014) and plumelets (Uritsky et al.,
2021), polar plumes (Raouafi et al., 2008), erupting spicules (Sterling &
Moore, 2016), and reconnection-driven jets and jetlets (Raouafi et al., 2023a;
Kumar et al., 2022, 2023; Sterling et al., 2023), the plasma outflow tends to
organize itself into fine-scale filamentary structures constrained by the
expanding magnetic field. The characteristic transverse sizes of these
structures have been shown to lie in the range 2–10 Mm, with typical lifetimes
on the order of 5–10 minutes (Uritsky et al., 2021; Kumar et al., 2022). The
upward speed of the structured outflow is comparable to the sound speed, so
waves and bulk motions could coexist – see, e.g., Uritsky et al. (2013b) and
references therein. Numerical simulations indicate that the structured outflow
can be driven by low-lying magnetic reconnection releasing and accelerating
the frozen-in plasma (Pariat et al., 2009; Karpen et al., 2017; Wyper et al.,
2018; Drake et al., 2023). The resulting jet is composed of a relatively dense
region near the coronal base exhibiting significant pressure and density
modulations, an intermediate region at a higher altitude filled with Alfvénic
and compressional perturbations, and a purely Alfvénic front well ahead of the
slower, denser mass flow (Uritsky et al., 2017). Signatures of interchange
reconnection and the resultant compressible outflow in the lower corona have
been confirmed in coronal-hole observations (Kumar et al., 2022; Mason &
Uritsky, 2022).

A key open question is the role of the small-scale coronal outflows in the
formation of the large-scale solar wind. It has been argued (Raouafi et al.,
2023a) that the former could be building blocks of the latter by providing
enough plasma mass, momentum, and kinetic energy to fuel the transient
component of the solar wind. It has also been suggested (Wang, 2022) that
quasi-continuous interchange reconnection could drive the ambient solar wind
by a combination of nanoflare-like plasma heating (Klimchuk, 2006, 2015) and
MHD-wave deposition of momentum and kinetic energy (Cranmer & Winebarger,
2019; Zank et al., 2021). Other estimates suggest that the occurrence
frequency of the small-scale jets is at least high enough to cause significant
perturbations in the outflowing plasma (Kumar et al., 2022; Drake et al.,
2023). In either scenario, the reconnection-driven flows definitely generate
upward propagating discontinuities (Uritsky et al., 2017; Roberts et al.,
2018), which might give rise to the sudden rotations of the interplanetary
magnetic field known as switchbacks (Horbury et al., 2020; Kumar et al., 2022,
2023; Bale et al., 2021, 2023). The switchbacks have been observed in
abundance by Parker Solar Probe (Fox et al., 2016; Raouafi et al., 2023b),
revealing the highly structured nature of the solar wind.

In this paper, we report results of a quantitative statistical survey aimed at
evaluating the contribution of the structured transient plasma outflows to the
bulk solar wind. We use high-resolution extreme ultraviolet (EUV) images of
the dynamic solar atmosphere above a polar coronal hole to identify more than
2300 individual small-scale jets and measure their spatial scale, duration,
upward speed, and entrained mass. Our analysis demonstrates that the
structured plasma outflows obey a family of multiscale probability
distributions characteristic of turbulent impulsive reconnection, which may
play a crucial role in releasing the heated low-coronal plasma into the solar
wind.

##  2 Data and methods

We have analyzed a sequence of 1,801 EUV images collected by the Atmospheric
Imaging Assembly onboard the Solar Dynamics Observatory (Lemen et al., 2012;
Pesnell et al., 2012). The images covered a 6-hour interval between 2021-04-28
00:00:09 and 2021-04-28 05:59:57, with the cadence of 12 s, spatial resolution
1.5″, and plate scale 0.6″, or 435 km. We used 171 Å images dominated by Fe IX
emission at a characteristic temperature of $6.3\times 10^{5}$ K observed in
the quiet corona and the upper transition region. The selected time interval
is sufficiently long to obtain statistically representative signatures of the
studied flow structures, which are omnipresent in the open-field corona.

Our investigation was focused on the quasi-steady plasma outflow observed
above the north polar limb and launched from the polar coronal hole. The
region of interest (ROI) covered a $45^{\circ}$ limb sector centered at the
north pole (Figure 1(a)). Two ranges of altitude above the limb (4.35 Mm–174
Mm and 4.35 Mm–43.5 Mm) were used to study, respectively, the velocity and the
occurrence statistics of the outflows as explained below. The images were
transformed into plane-polar coordinates to simplify the analysis of quasi-
radial structures, with a position-angle bin size of $6.25\times 10^{-4}$
radians (0.0358∘) matching the native AIA resolution at the base of the ROI
and exceeding it by a factor of 1.26 at the upper ROI boundary, and a radial
position bin equal to the original AIA resolution. After transforming the
images into polar coordinates, we subtracted the time-averaged local radial
decay of the image intensity at each azimuthal position. The resulting
detrended polar-coordinate arrays were smoothed using a $3\times 3$-pixel
boxcar window to reduce binning errors and pixel noise (Figure 1(b)), and
transformed into a set of 1257 time-distance arrays representing plasma
propagation dynamics at each location along the limb (Figure 1(c)).

Propagating disturbances in each time-distance array were identified using the
surfing transform (ST) technique (Uritsky et al., 2013b). This technique is
designed to identify wave signals in time–distance plots with low signal-to-
noise ratios, and it has been used successfully to measure parameters of
remotely observed flows and waves (Uritsky et al., 2009; Keiling et al., 2012;
Uritsky et al., 2013b, 2021; Kumar et al., 2022; Raouafi et al., 2023a; Mondal
et al., 2023). The method is based on an analysis of a velocity-dependent
surfing signal $S(t,u)$ obtained by averaging the time-distance array along
the spatiotemporal defined $u$, starting form time $t$. The dynamic range of
the signal is maximized when the assumed surfing velocity $u$ matches the true
propagation velocity $v$, in which case the averaging is performed in the
reference frame of the propagating disturbance. The main factor limiting the
accuracy of the ST method is the the nonstationarity of the signal. For quasi-
periodic flows such those studied here, the ST uncertainty is typically in the
range 2-5 km s-1 (Uritsky et al., 2013b; Mondal et al., 2023), which is an
order of magnitude smaller than the spatial variability of the flow velocity
reported in §3. Figure 1(d) shows the maximized surfing signal meeting this
condition for the time-distance plot in Figure 1(c) at $v=128$ km s-1. The
quasi-periodic nature of the disturbance and its nonlinear waveform are
evident. Figure 1(e) shows the time-averaged root-mean-square value $S_{RMS}$
of the surfing signal measured plotted as a function of $u$, with a clear-cut
peak at $u=v$. The peak broadens in the presence of long-term trends, which
have to be removed for more accurate results.

Figure 1:  (a) An SDO/AIA 171 Å image in the middle of the studied 6-hour time
interval, with the region of interest bounded by the red lines. The solid line
shows the main ROI used for the statistical survey; the dotted line is an ROI
with an extended upper boundary used for the speed measurements based on the
ST method. A base-ten logarithmic brightness scale is used to enhance the low-
intensity structure above the limb. (b) Plane-spherical coordinate
representation of the main ROI on panel (a) after radial trend removal. The
highly filamented texture of the outflowing plasma is easily seen. (c) One of
1257 studied time-distance plots representing time evolution of the lower
corona along the radial slit marked with a red dashed line on panel (b). The
gray scales on panels (b) and (c) are calibrated in linear AIA DN units. A
spatiotemporal direction corresponding to the propagation velocity measured by
the surfing analysis is shown with a dashed line. (d) The maximized surfing
signal $S(t,u=v)$ fulfilling the velocity resonance condition, before and
after temporal detrending (gray and black lines, respectively). (e) The RMS
value $S_{RMS}$ of the surfing signal $S(t,u)$ for the time-distance plot in
panel (c), showing a well-defined peak at $u=v$.

The ST analysis illustrated by Figure 1 was applied to all positions along the
limb in the studied data set, which allowed us to estimate the spatially
dependent propagation speed characterizing the plasma outflow. Next, the
spatiotemporal coherence between the propagation dynamics at neighboring
locations was investigated to identify the individual jet events and evaluate
their amplitude and scales. Finally, we looked at the statistics of the
obtained outflow parameters in the context of the large-scale solar wind and
the reconnection-driven jet mechanics.

##  3 Results and discussion

Figure 2 shows the results of the velocity analysis of the coronal outflow
across the studied region above the limb. The top panel visualizes the
velocity dependence $S_{RMS}(u)$ of the surfing RMS for each of the 1257
studied limb positions. The blue dots mark the location of the RMS peak on
each curve used to evaluate the resonance surfing velocity $v$. The peak
values found at or near the edges of the studied $u$ interval (50 or 250 km
s-1) were excluded from further analysis as unreliable. The one-sigma range of
the flow speed variability is 88 to 156 km s-1, with a mean value of 122 km
s-1 that is roughly the local sound speed. The observed range of speeds is
somewhat lower than the characteristic speed of jetlet activity reported
previously for the same coronal region (Raouafi et al., 2023a) and is
systematically below the typical speed ($\approx$ 200 km s-1) in mid-latitude
coronal plumelets (Uritsky et al., 2021). That the detected outflows are
quasi-periodic is suggested by the power spectral analysis of the resonant
surfing signals (Figure 2(b)). The period of the propagating structures varies
across limb positions in the range of 10 to 30 min, with an average of 17 min,
which is roughly consistent with recent observations of propagating
disturbances in plumes within polar coronal holes (Cho et al., 2021). Due to
the applied temporal detrending, it is possible that the image sequence
contains time scales that are longer than those appearing on the spectrogram.
However, it is evident that no higher frequencies are present in the studied
polar coronal region. In particular, there are no signs of the ubiquitous 3–5
min $p$-mode oscillations accompanying the small-scale outflows in the
plumelets and jetlets. This is likely because the plume studied here is faint
and rooted in weak network/plage flux concentrations, whereas the plumes
studied in our previous works (Uritsky et al., 2021; Kumar et al., 2022, 2023)
are bright and rooted in strong network/plage regions. Jet activity at the
base of plumes is a function of the strength of the associated flux
concentrations, a dependence that requires further investigation.

The degree of modulation of the optical intensity by the propagating
disturbances is quite low, as evidenced by Figure 2(c) showing peak $S_{RMS}$
values at each position. The flow intensity varies between 0.1 and 0.5 AIA DN
units, which is a small fraction of the background intensity of the same
coronal region (see Figure 1(b)).

Figure 2:  ST analysis of the coronal outflow speed. (a) Speed dependence of
the surfing RMS for each location on the limb. Each vertical slice represents
a $S_{RMS}(u)$ dependence such as that plotted on Figure 1(e), with the blue
dot showing the position of the resonance maximum used to identify the
propagation speed. (b) Fourier power spectra of the resonance surfing signals
for different limb positions. The blue dots show the spectral peaks used to
evaluate the dominant frequency. Temporal trends longer than 20 min were
removed for a more accurate velocity estimation. (c) Resonance value
$S_{RMS}(u=v)$ representing the characteristic amplitude of the propagating
quasi-periodic disturbance at each location.

Besides being a sensitive tool for velocity analysis, the ST technique
provides a natural means to detect small-scale outflows appearing at different
times and limb locations. If a single small-scale jet spans more than one
azimuthal position, the surfing signals at all positions within the jet should
be time-correlated, reflecting coherent plasma propagation. By isolating
clusters of such correlated pixels and measuring their spatial and temporal
sizes, one can therefore evaluate the spatial and temporal scales of the
small-scale jets. Figure 3 shows a spatiotemporal map $S_{RMS}(t,x,u=\langle
v\rangle)$ of the flow activity in the studied coronal region, where $t$ is
the time, $x$ is the position along the limb, and $\langle v\rangle=122$ km
s-1 is the average outflow velocity reported in Figure 2(a). Each horizontal
row of pixes is a surfing signal at a given limb position. No temporal high-
pass filtering has been applied to this plot, to preserve all relevant time
scales. The clustered groups of red pixels are the coherent crests of the
surfing signals; likewise, the clusters of blue pixels are the coherent
troughs. Each blob can be interpreted as an individual propagating front
described by a certain transverse size and duration.

Figure 3:  Top: Spatiotemporal map of the transient outflow in the studied
region constructed using the ST technique. The compact blobs shown in red
represent starting times and locations of the quasi-periodic propagating
fronts (see text for details). Bottom: Temporal and spatial autocorrelation
functions of the surfing signals; the provided characteristic scales represent
the 50$\%$ levels and assume an exponential decay.

Based on Figure 3(a), the outflows clearly involve a wide range of spatial and
temporal scales, making it difficult to pinpoint a single combination of
scales that could be used as a hallmark of a specific morphology, such as a
jetlet or a plumelet. Some flow bursts are only 2–3 azimuthal pixels ($\sim 1$
Mm) wide, while other encompass 20–30 pixels ($\sim$ 10–15 Mm). Some flow
bursts last for several minutes only, while others are sustained for up to
30–50 minutes. While this multiscale behavior cannot be described by precisely
defined scales, it can be characterized statistically by a set of probability
distributions. To perform such a statistical analysis, we decomposed the
complex flow activity pattern shown in Figure 3(a) into a set of discrete flow
events by performing a spatiotemporal segmentation. To make this segmentation
fully objective, we based our algorithm on the intrinsic correlation scales of
the pattern and an automatically optimized detection threshold, as described
below.

Panels (b) and (c) of Figure 3 show temporal and spatial autocorrelation
functions of the $S_{RMS}(t,x)$ array. We evaluated the correlation scales
($t_{c}$ and $r_{c}$, respectively) at the half-maximum level; the values are
provided on the plots. Next, we removed the trends from the $S_{RMS}(t,x)$
array by subtracting its boxcar-averaged version, with the sliding window size
in the $t$ and $x$ directions set to $2\,t_{c}$ and $2\,r_{c}$, respectively.
The resulting detrended map contains compact flow events that can be isolated
by applying a constant detection threshold. The number $N_{s}$ of the detected
events depends upon the threshold. To find the threshold that produces the
largest population of flows, we varied the threshold in the range from $-$1 to
3 and located the position of the peak with the maximum $N_{f}=2353$ (Figure
4(a)). The detection threshold of 0.818 AIA DN units corresponding to this
peak was used for the subsequent analysis. Additional tests have confirmed
that the power-law components of the flows distributions reported on Figure 5
are insensitive to the particular choice of the threshold, although the range
of the scaling behavior gradually contracts if the threshold exceeds the
optimal value.

Figure 4(b) shows an example of a surfing signal computed at the limb position
$x=-267$ Mm before (gray line) and after (black line) the detrending
procedure. The signal values above the optimized detection threshold 0.818 are
marked with red color. Figure 4(c) presents a similar plot representing an
azimuthal profile of the flow map at $t=3.6$ min measured from the beginning
of the studied time interval.

Figure 4(d) shows a map of the detected flow bursts using the methodology
described above. The individual flow events detected automatically as
contiguous pixel clusters are marked with random colors; the black background
is below the detection threshold and represents the ambient plasma. Two
aspects of the analysis should be noted before moving forward with the
statistical analysis of the constructed flow population. First, the detected
events lack the largest and longest-duration scales exhibited by the flows in
the non-detrended, non-thresholded flow map (Figure 3(a)). This bias is an
unavoidable price paid for segmenting a continuous outflow pattern. It implies
that both the upper scaling cutoffs and the net contribution of the flow
material and energy to the solar wind could be larger than those revealed by
the flow statistics. Second, we had no unambiguous way to verify the
statistical homogeneity of the obtained set of events, although its resulting
statistics presented below speak in favor of a reasonably uniform and
representative sample.

Figure 4:  (a) The number $N_{f}$ of distinct identifiable flow fronts on the
detrended surfing map versus the detection threshold. The optimized threshold
(0.818) resulting in the largest number of identified flow events (2353) was
used for the subsequent spatiotemporal event detection. (b,c) Example of
temporal and spatial detrending of the surfing flow map $S_{RMS}(t,x)$ based
on the estimated correlation scales. (d) The processed flow map showing the
detected flow events marked with random color. Only the events involving more
than 10 spatiotemporal pixels were plotted and used for the statistical survey
presented in Figure 5, since smaller events are not informative due to the
applied image smoothing.

The probability distributions of the flows over the lifetime $T$ and the
transverse size $d$ still appear remarkably multi-scale, spanning over nearly
two orders of magnitude (Figure 5(a,b)). The mean ($\mu$) and median ($m$)
values indicated on the histogram panels are biased by the smallest events,
which have the highest occurrence rate. The reported values are approximately
consistent with earlier published estimates, but the fact that the power-law
form of the two distributions continues down to the smallest scales resolved
by our analysis could be an indication that even smaller and shorter flow
bursts are present in the corona. The standard deviation $\sigma$ of both $T$
and $d$ parameters is of the order of the mean value, reflecting the wide
statistical spread associated with heavy-tailed distributions. The log-log
slopes (dashed red lines in the insets of panels (a) and (b)) of the power-law
portions of the distributions unaffected by detrending are greater than $-2$,
which suggests that the mean flow scales could be weighted toward the largest,
rather than the smallest, outflow events. The small-scale size distribution
cutoff ($\approx 1.5$ Mm) is caused by the $3\times 3$-pixel image smoothing.
It does not reflect the smallest physical scale of the flows, which could be
lower.

Figure 5:  Probability histograms of the detected transient flow events over
the event: (a) lifetime $T$; (b) transverse flow size $L$; (c) upward flow
velocity $v$; (d) relative emission intensity $\delta I/I_{0}$; (e)
transported flow volume $V$; and (f) kinetic energy $K$ of the flow. Mean
($\mu$), median ($m$), and standard deviation ($\sigma$) values are provided
for each parameter. The dashed red lines show the log-log slopes of the $T$,
$L$, $V$, and $K$ distributions, supplemented by the evaluated power-law
exponent values.

Compared to the spatiotemporal scales, the velocity distribution of the flows
(Figure 5(c)) is relatively compact, with the mode and the median roughly
consistent with the mean as expected for Gaussian statistics. A possible
secondary peak around 70–80 km s-1 could be a signature of a separate
population of flows, pending its statistical significance being confirmed in
future studies involving larger sets of events.

Figure 5(d) presents the probability distribution of relative flow intensities
obtained by dividing the peak surfing intensity $\max(S_{RMS})$ measured at
each position on the limb by the time- and height-averaged AIA brightness
computed for the same azimuthal position. This ratio varies between about
$3\times 10^{-3}$ and $8\times 10^{-3}$, with a mean of $5\times 10^{-3}$. The
interpretation of this weak EUV emission signature in terms of plasma density
modulation $\delta n$ caused by the flow relative to the unperturbed
background density $n_{0}$ requires a line-of-sight (LOS) integration of the
luminous plasma material in the compact flow versus the ambient corona.
Assuming similar filling factors in the two plasma systems, the ratio of the
emission signals can be expressed as $\delta I/I_{0}\sim(L/D)(\delta
n^{2}/n_{0}^{2})$, where $L$ and $D$ are respectively the characteristic
transverse size of the flow and the characteristic LOS depth of the corona at
a given distance from the limb. The assumption of similar filling factors is
an important uncertainty (DeForest et al., 1991), because the observed jets
could, in principle, have a smaller filling factor than the surrounding
corona. This follows, e.g., from the comparison of plume basal electron
densities based on density sensitive spectral line ratios (Warren & Hassler,
1999) versus the EUV photometry (Walker et al., 1993). We defer verification
of this possibility to future work. Using the available empirical radial
density profile models (Cranmer et al., 1999; Guhathakurta et al., 1999), we
evaluated the e-folding $D$ value to be on the order of 0.7 solar radii
($\approx 500$ Mm) for the range of altitudes covered by the studied solar
region. Based on our flow analysis, $\langle\delta I/I_{0}\rangle\approx
5\times 10^{-3}$ and $\langle d\rangle\approx 4$ Mm, so the relative density
perturbation caused by the typical flow $\delta n/n_{0}=\sqrt{(D/L)(\delta
I/I_{0})}\approx 0.6$ and the total density of the flow $n\approx 1.6\,n_{0}$.
Therefore, despite its minuscule emission footprint, the detected flows can
cause a substantial local density enhancement. Importantly, a lower filling
factor for the observed flows would result in an even more significant $\delta
n/n_{0}$.

The volume of plasma transported by each transient flow event can be evaluated
as $V=v\,d^{\,2}\,T$. Since both $d$ and $T$ have wide probability
distributions with distinct power-law ranges, the distribution of volumes
(Figure 5(e)) exhibits an even broader interval of power-law behavior spanning
over at least four orders of magnitude. The distribution of kinetic energy in
the flows $K=(1.6\,n_{0})\,m_{p}\,v^{\,2}\,V\,/2$, in which we used a typical
background number density $n_{0}\approx 5\times 10^{8}$ cm-3 near the base of
the corona (Raouafi et al., 2008) for pure hydrogen composition, also shows a
broad interval of power-law decay (Figure 5(f)). As for the lifetime and size
distributions, the power-law exponents describing the statistics of $V$ and
$K$ are well above $-2$, implying that the characteristic values of the
transported plasma volume and energy are weighted toward large events. The
standard deviations of $V$ and $K$ parameters are several times greater than
the corresponding mean values, as expected for broad-band power-law statistics
with a shallow log-log slope.

The obtained probability distributions of multiscale polar jets provide new
evidence for the reconnection-driven mechanism of the coronal plasma outflow.
The power-law volume distribution $p(V)\propto V^{-\tau_{V}}$ can be derived
directly from the jet size distribution $p(L)\propto V^{-\tau_{L}}$ under the
assumption that the outflow speed is independent of the width $L$ of the
ejected flow observed at a distance from the reconnection site, which seems
plausible. The power-law form of $p(T)$ implies that the lifetime $T\propto
L^{\,D_{T}}$, and therefore $V(L)\propto L^{\,2+D_{T}}$, where $D_{T}$ is the
lifetime geometric exponent (Uritsky et al., 2013a). From the probability
conservation relation $p(V)dV=p(L)dL$ we obtain

$\tau_{V}=\frac{\tau_{L}+D_{T}+1}{D_{T}+2}.$ (1)

The substitution of the measured values $\tau_{L}\approx 1.33$ and
$D_{T}\approx 1$ into (1) results in $\tau_{V}\approx 1.1$, which is
consistent with our statistical analysis. Since $K\propto V$, the same
conclusion applies to the kinetic energy distribution exponent $\tau_{V}$,
also in agreement with our observations.

The validity of the theoretical relation (1) indirectly confirms that the
acceleration mechanism underlying the detected outflows is rooted in
reconnection physics. The observed self-similarity of the flow pattern is
likely imposed by the spatial complexity of the photospheric source regions
(Uritsky et al., 2013a; Uritsky & Davila, 2014), which contain embedded
bipoles spanning a wide range of scales (Karpen et al., 2017; Kumar et al.,
2021). In this context, it is worth noting that the slopes of the $p(V)$ and
$p(K)$ distributions are roughly consistent with the power-law slopes
describing the distribution of volumes and dissipated energies in fragmented
pieces of a full-size coronal jet in a high-resolution 3D MHD simulation (see
Figure 13 of Uritsky et al. (2017)), which leaves the possibility that at
least some of the propagating disturbances detected here are produced by
nonlinear instabilities (Yuan et al., 2019) in a large-scale turbulent jet
rather than by stand-alone small-scale jetlets.

Whether the small-scale flows constituting the majority of the studied
statistical population make a significant contribution to the solar wind
depends upon their occurrence rate and the amount of plasma mass and kinetic
energy injected into the corona. Using the mean volume and energy estimates
shown in Figure 5(e) and (f), the net volume flow rate by all the flows in the
studied polar region is $N_{f}\left\langle V\right\rangle/T_{obs}\approx
4\times 10^{26}$ cm3 s-1, where $N_{f}=2353$, $T_{obs}=6\times 3600$ seconds,
and $\left\langle\,\right\rangle$ denotes ensemble averaging. The obtained
volumetric speed translates into a particle flow rate of about $2\times
10^{35}$ protons s-1 (using the same nominal number density as in the $K$
calculation), and a mass flow rate $\approx 3\times 10^{11}$ g s-1. In turn,
the total kinetic-energy rate (the kinetic power supplied by all detected
flows) is $N_{f}\left\langle K\right\rangle/T_{obs}\approx 5\times 10^{25}$
erg s-1. The propagated standard errors of the provided estimates cause
variations of 10% or less from the mean values, although the true statistical
uncertainties accounting for data nonstationarity, event detection errors, and
observational constraints could be much larger.

The photospheric surface area of the 45∘ spherical sector defining of the
studied ROI (Figure 1(a)) is about $4\%$ of the surface of the Sun. If the
cumulative footpoint area of the open-field coronal morphologies supporting
structured plasma outflow were only 3 times larger, the particle flow rate
associated with studied transient flows would be sufficient to generate the
net solar-wind particle loss rate $6\times 10^{35}$ protons s-1 predicted for
the entire Sun (Wang, 2016, 2020). The kinetic energy rate produced by the
flows is consistent with the total kinetic energy loss rate $\approx 10^{27}$
erg s-1 estimated for the whole Sun assuming an asymptotic wind speed of 500
km s-1 (see, e.g., Raouafi et al., 2023a, and references therein), once the
difference between this nominal downstream speed and the measured flow speed
$v\approx 122$ km s-1 near the coronal base is taken into account. Together
with the scaling relation (1), these estimates strongly support a
reconnection-driven solar-wind scenario in which the nonsteady coronal plasma
outflow is composed predominantly of small-scale jets, each ejected by a
localized impulsive magnetic reconnection event in the low atmosphere, which
propagate through a much slower background coronal plasma without significant
mass and momentum exchange between the two systems. The latter condition is
crucial for the described scenario and requires definitive verification in
future studies.

##  4 Conclusions

We have conducted an in-depth quantitative investigation of the bulk plasma
outflow from a polar coronal hole as represented by a sequence of 1,801 high-
resolution, high-cadence EUV images above the north solar limb. To our
knowledge, this is the first statistical survey reporting detailed
probabilistic properties of the flows and evaluating their physical and
geometric parameters based on a large set of automatically detected flow
events. Below is a summary of our main conclusions.

  1. 1.

The upward plasma transport above a polar coronal hole is highly structured
and contains numerous embedded flows.

  2. 2.

The average flow speed estimated using the surfing transform technique is 122
$\pm$ 34 km s-1. The local speed varies significantly with position along the
limb, likely reflecting the nonuniform photospheric magnetic structure at the
base of the coronal hole.

  3. 3.

The outflows consist of transient quasi-periodic bursts, with a repetition
period of about 10 to 30 minutes and a characteristic burst duration of about
5 minutes. The typical transverse size of the individual flows is about 3–4
Mm, which is consistent with our earlier estimates (Kumar et al., 2022).

  4. 4.

The spatial and temporal scales of the collimated flows obey broadband power-
law probability distributions occupying a continuum of scales from the
smallest jetlets to full-size jets, suggesting that the physical mechanism
responsible for solar jets is inherently self-similar.

  5. 5.

The volume and kinetic-energy distributions of the flows are consistent with a
scenario in which the structured coronal outflow is formed by a large number
of impulsive reconnection events within the coronal hole, described by a wide
range of spatial and temporal scales presumably imposed by the photospheric
driver.

  6. 6.

Assuming that the multiscale jets and the inter-jet medium have comparable
filling factors, the estimated occurrence rate of the analyzed transient
events appears sufficient to produce the net mass and energy outflow for the
nonsteady fast solar wind.

Overall, our smoking-gun findings add strong circumstantial support to the
view that the nonsteady component of the fast solar wind originates from
small-scale impulsive interchange reconnection in the low corona, as has been
proposed earlier (Raouafi et al., 2023a). The quasi-steady component, on the
other hand, may be driven by quasi-continuous interchange reconnection that
adds heat and wave-associated momentum and kinetic energy to the plasma (Wang,
2022). More comprehensive high-resolution observations extending further into
the corona, supported by high-resolution MHD simulations with full coronal
thermodynamics, are needed to explore and quantify the contributions of
impulsive and continuous reconnection, the global extent and persistence of
the resulting outflows, and their ultimate connection to the heliosphere.

SDO is a mission for NASA’s Living With a Star (LWS) program. We thank J.
Drake, S. Gibson, and A. Sterling for useful discussions. VMU has been partly
supported through the Partnership for Heliophysics and Space Environment
Research (NASA grant No. 80NSSC21M0180). PK is supported by an NSF grant
(#2229336) and NASA’s Internal Scientist Funding Model (H-ISFM) program. CRD
also received H-ISFM support. The Surfing Transform code was developed by VMU
and is available upon request.

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# Irreducible Contact Curves via Graph Stratification

Giosuè Muratore  CMAFcIO, Faculdade de Ciências da ULisboa, Campo Grande
1749-016 Lisboa, Portugal<EMAIL_ADDRESS>https://sites.google.com/view/giosue-muratore

(Date: August 28, 2024)

###### Abstract.

We prove that the moduli space of contact stable maps to $\mathbb{P}^{2n+1}$
of degree $d$ admits a stratification parameterized by graphs. We use it to
determine the number of irreducible rational contact curves in
$\mathbb{P}^{2n+1}$ with any Schubert condition. We give explicitely some of
these invariants for $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$. We give another
proof of the formula for the number of plane contact curves in
$\mathbb{P}^{3}$ meeting the appropriate number of lines.

###### Key words and phrases:

Legendrian, contact, enumeration, symplectic, stable maps, rational curves

###### 2020 Mathematics Subject Classification:

Primary 14N10, 14C17, 53D10; Secondary 14H10, 14D22, 14L30

##  1\. Introduction

A fast-growing area of mathematics is Contact Geometry, that is the study of
manifold with a contact structure. First defined by [Kob59], a contact
structure on a smooth complex manifold $X$ of odd dimension is a corank $1$
non-integrable holomorphic distribution. All projective spaces of odd
dimension admit a (unique) contact structure.

The interest for these manifolds comes from real geometry. Calabi [Cal67]
proved that the study of harmonic maps of spheres $f\colon S^{2}\rightarrow
S^{4}$ is equivalent, modulo an involution, to the study of holomorphic
horizontal curves $\tilde{f}\colon\mathbb{CP}^{1}\rightarrow\mathbb{CP}^{3}$
of fixed degree. Roughly speaking, $f$ minimizes the energy functional if $f$
or $-f$ has a lift $\tilde{f}$, such that $\tilde{f}(\mathbb{P}^{1})$ is
integral with respect to the contact structure of $\mathbb{P}^{3}$.

The enumeration of contact curves (i.e., rationally connected curves integral
with respect to the contact structure) with geometric conditions started in
[LV11]. Levcovitz and Vainsencher found, among other results, the number of
contact plane curves of degree $d$ meeting $3+d$ general lines in
$\mathbb{P}^{3}$. On the other hand, the first systematic study of enumerative
invariants of those curves started in [Mur20] (strengthening results from
[Amo14]). The strategy was to consider the closed subscheme
$\mathcal{S}_{m}(\mathbb{P}^{n},d)\subset\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$
of stable maps whose image is a contact curve. Certain Gromov-Witten
invariants related to $\mathcal{S}_{m}(\mathbb{P}^{n},d)$ are enumerative.
Using localization, the author was able to give enumerative invariants of
contact curves with any Schubert condition.

The motivation to this paper is that very few enumerative invariants are known
for irreducible rational contact curves. The reason is that
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ has many irreducible components. As
already pointed out in [Mur20, Remark 3.10], the subscheme of
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ parameterizing reducible curves has
codimension $0$. That is, the “boundary” of
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ is not a divisor, but a component purely
of the same dimension. So that, the enumerative invariants get contribution
from all irreducible components.

In this paper we give a decomposition of $\mathcal{S}_{m}(\mathbb{P}^{n},d)$
in components parameterized by graphs, with a complete description of the
dimension of all components (Theorem 4.4). This stratification is induced by a
similar one of $\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$. We use this
stratification to give an explicit formula to compute the number of reducible
contact curves with any dual graph (Corollary 5.2). Moreover, we are able to
give another proof of Levcovitz and Vainsencher’s result (Proposition 5.7).

In Tables 5.1 and 5.2, we give enumerative invariants of irreducible rational
contact curves in $\mathbb{P}^{3}$ and $\mathbb{P}^{5}$ of low degree. The
strategy is to subtract, from the number of contact curves, the number of
reducibles contact curves with the same Schubert conditions. Note that the
enumerative numbers of contact curves in $\mathbb{P}^{2n+1}$ are all known
thanks to [MS22].

An alternative way to get enumerative invariants of irreducible contact curves
may be to use another moduli space, like the Hilbert scheme. We plan to persue
this research path in the future.

I would like to thank Carlos Florentino for his many suggestions. This problem
came out from one of the many conversations I had with Israel Vainsencher
during my work at UFMG. I would like to thank him for his time. I also thank
Giordano Cotti, Angelo Lopez and Filippo Viviani for useful conversations.
Finally, I thank Csaba Schneider for his computational support.

The author is supported by FCT - Fundação para a Ciência e a Tecnologia, under
the project: UIDP/04561/2020.

##  2\. Contact Structures

All varieties in this article are defined over $\mathbb{C}$. In this section
$n$ is any non-negative integer.

######  Definition 2.1.

Let $X$ be a $(2n+1)$-dimensional complex manifold. A contact structure on $X$
is an open cover $\\{U_{i}\\}_{i}$ of $X$ together with $1$-forms $\alpha_{i}$
on each $U_{i}$ such that:

  1. (1)

at every point of $U_{i}$, the form
$\alpha_{i}\wedge(\mathrm{d}\alpha_{i})^{\wedge n}$ is non-zero, and

  2. (2)

if the intersection $U_{i}\cap U_{j}$ is non empty, then there exists a non-
vanishing holomorphic function $f_{ij}$ on $U_{i}\cap U_{j}$ such that
$\alpha_{i}=f_{ij}\alpha_{j}$ on $U_{i}\cap U_{j}$.

A contact curve is a rationally connected curve $C\subset X$ such that any
local $1$-form $\alpha_{i}$ vanishes at every smooth point of $C\cap U_{i}$.

######  Remark 2.2.

An equivalent definition is the following: A contact structure on $X$ is a
pair $(X,L)$ where $L$ is a line subbundle of $\Omega_{X}^{1}$ such that if
$s$ is a non trivial local section of $L$, then $s\wedge(\mathrm{d}s)^{\wedge
n}$ is everywhere non-zero.

Any symplectic $1$-form on $\mathbb{C}^{2n+2}$ defines a contact structure on
$\mathbb{P}^{2n+1}$ by the projection
$\mathbb{C}^{2n+2}\backslash\\{0\\}\rightarrow\mathbb{P}^{2n+1}$. Viceversa
any contact structure on $\mathbb{P}^{2n+1}$ is induced by a symplectic form
on $\mathbb{C}^{2n+2}$. Let $\\{x_{0},\ldots,x_{n},y_{0},\ldots,y_{n}\\}$ be
local coordinates of $\mathbb{C}^{2n+2}$. All symplectic $1$-forms on
$\mathbb{C}^{2n+2}$ are conjugated to

$\alpha:=\sum_{i=0}^{n}x_{i}\mathrm{d}y_{i}-y_{i}\mathrm{d}x_{i}.$

In particular, there exists a unique contact structure on $\mathbb{P}^{2n+1}$
modulo conjugation. See [OSS11, 1.4.2].

######  Example 2.3.

On $\mathbb{P}^{3}$, the distribution induced by $\alpha$ assigns to
$p=[a:b:c:d]$ the plane with equation

$H_{p}:=\\{[X:Y:Z:W]\in\mathbb{P}^{3}/bX-aY-dZ+cW=0\\}.$

A curve is said to be a contact curve if the tangent line at each smooth point
$p$ is contained in the plane $H_{p}$. The contact curves of degree $1$ in
$\mathbb{P}^{2n+1}$ are parameterized by the symplectic Grassmannian of
isotropic $2$-spaces of $\mathbb{C}^{2n+2}$. See [LV11] for other examples.

######  Remark 2.4.

We do not require that a contact curve $C$ is irreducible. Since the property
of being contact is local, every component of $C$ is contact in its own.

###  2.1. Symplectic group

Let us denote by $\mathrm{Sp}(2n+2,\mathbb{C})$ the set of square matrices $M$
of order $2n+2$ with complex coefficients such that $M^{T}\Omega M=\Omega$,
where $\Omega$ is the block matrix

$\Omega=\left(\begin{array}[]{cc}0&I\\\ -I&0\end{array}\right),$

and $I$ is the unit matrix of order $n+1$. The following facts are well-known:

  * •

The set $\mathrm{Sp}(2n+2,\mathbb{C})$ is a subgroup of the group of matrices
with determinant $1$. The two groups coincide if and only if $n=0$.

  * •

The standard action of $\mathrm{Sp}(2n+2,\mathbb{C})$ on $\mathbb{C}^{2n+2}$
is transitive, and preserves the $1$-form $\alpha$.

There exists an explicit description of $\mathrm{Sp}(2n+2,\mathbb{C})$. Let
$M$ be a block matrix

$M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix},$

where $A,B,C,D$ are matrices of order $n+1$. Then $M$ is in
$\mathrm{Sp}(2n+2,\mathbb{C})$ if and only if

(2.1) $\displaystyle C^{T}A$ $\displaystyle=$ $\displaystyle A^{T}C$ (2.2)
$\displaystyle A^{T}D-C^{T}B$ $\displaystyle=$ $\displaystyle I$ (2.3)
$\displaystyle D^{T}B$ $\displaystyle=$ $\displaystyle B^{T}D.$

Main references are [FH91, Lecture 16] and [O’M78]. It follows that the
standard action of $\mathrm{Sp}(2n+2,\mathbb{C})$ on $\mathbb{P}^{2n+1}$
preserves the contact structure given by $\alpha$. More generally, for any
contact structure on $\mathbb{P}^{2n+1}$ there is an action of
$\mathrm{Sp}(2n+2,\mathbb{C})$ preserving it. This follows from the fact that
all contact structures are equivalent by a change of coordinates.

Let $\\{U_{\imath}\\}_{\imath=0}^{n}$ be the standard open cover of
$\mathbb{P}^{n}$. It is well known that there exist subgroups
$\\{H_{\imath}\\}_{\imath=0}^{n}$ of $\mathrm{GL}(n+1,\mathbb{C})$ isomorphic
to $\mathbb{C}^{n}$, such that the action $H_{\imath}\curvearrowright
U_{\imath}$ is the translation (see [GH78, 1.5]). Unfortunely, such subgroups
do not preserve the contact form. The following proposition may be known to
experts, but I could not find any reference.

######  Proposition 2.5.

Consider the standard action of $\mathrm{Sp}(2n+2,\mathbb{C})$ on
$\mathbb{P}^{2n+1}$. For every $\imath=0,\ldots,2n+1$ there exist

  * •

a subgroup $H_{\imath}<\mathrm{Sp}(2n+2,\mathbb{C})$,

  * •

an open subset $U_{\imath}\subset\mathbb{P}^{2n+1}$, and

  * •

two isomorphisms $\varphi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow
H_{\imath}$ and $\phi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow U_{\imath}$,

such that

  1. (1)

$U_{\imath}$ is $H_{\imath}$-invariant,

  2. (2)

the action of $H_{\imath}$ on $U_{\imath}$ is free and transitive, and

  3. (3)

the open subsets $\\{U_{\imath}\\}_{\imath=0}^{2n+1}$ form an open cover of
$\mathbb{P}^{2n+1}$.

###### Proof.

Let $\\{x_{0},\ldots,x_{n},y_{0},\ldots,y_{n}\\}$ be a basis of
$\mathbb{C}^{2n+2}$, and $\imath\in\\{0,\ldots,n\\}$. Let $U_{\imath}$ be the
open subset of $\mathbb{P}^{2n+1}$ of vectors with coefficient of $x_{\imath}$
non-zero, and let $\phi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow U_{\imath}$
be the standard isomorphism.

Let us define the map
$\varphi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow\mathrm{Sp}(2n+2,\mathbb{C})$
in the following way. For every
$(\vec{a},\vec{c})=(a_{0},a_{1},\ldots,\widehat{a_{\imath}},\ldots,a_{n},c_{0},c_{1},\ldots,c_{n})\in\mathbb{C}^{2n+1}$,
$\varphi_{\imath}(\vec{a},\vec{c})$ is the linear map sending

$\begin{array}[]{ccl}x_{j}&\mapsto&x_{j}+c_{j}y_{\imath}\\\
x_{\imath}&\mapsto&x_{\imath}+c_{\imath}y_{\imath}+\sum_{k\neq\imath}a_{k}x_{k}+c_{k}y_{k}\\\
y_{j}&\mapsto&y_{j}-a_{j}y_{\imath}\\\
y_{\imath}&\mapsto&y_{\imath}\end{array}$

where $j=0,\ldots,\hat{\imath},\ldots,n$. It is an easy exercise to prove that
$\varphi_{\imath}(\vec{a},\vec{c})$ defines a linear map, whose inverse is
$\varphi_{\imath}(-\vec{a},-\vec{c})$ and
$\varphi_{\imath}(\vec{a},\vec{c})\varphi_{\imath}(\vec{b},\vec{d})=\varphi_{\imath}(\vec{q},\vec{w})$
where

(2.4) $\begin{array}[]{ccl}q_{j}&=&a_{j}+b_{j}\\\ w_{j}&=&c_{j}+d_{j}\\\
w_{\imath}&=&c_{\imath}+d_{\imath}+\sum_{k\neq\imath}(c_{k}b_{k}-a_{k}d_{k}).\end{array}$

By direct computation, $\varphi_{\imath}(\vec{a},\vec{c})$ fixes the $1$-form
$\sum_{k=0}^{n}x_{k}\mathrm{d}y_{k}-y_{k}\mathrm{d}x_{k}$ (see also Remark
2.6). Finally the coefficient of $x_{\imath}$ is preserved, hence $U_{\imath}$
is $\varphi_{\imath}(\vec{a},\vec{c})$-invariant. We proved that the map
$\varphi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow\mathrm{Sp}(2n+2,\mathbb{C})$
is an injective morphism and $H_{\imath}=\varphi_{\imath}(\mathbb{C}^{2n+1})$
is a subgroup acting on $U_{\imath}$.

Note that
$\varphi_{\imath}(\vec{a},\vec{c})\cdot\phi_{\imath}(\vec{b},\vec{d})=\phi_{\imath}(\vec{q},\vec{w})$
where $\vec{q}$ and $\vec{w}$ are the vectors whose coordinates are in Eq.
(2.4). It follows that the isomorphism
$\phi_{\imath}\circ\varphi_{\imath}^{-1}\colon H_{\imath}\rightarrow
U_{\imath}$ is $H_{\imath}$-linear. Hence the action of $H_{\imath}$ on
$U_{\imath}$ is free and transitive.

Now, we prove that such a construction works for every $\imath$. Suppose
$n<\imath\leq 2n+1$ and let $f$ be the involutive linear automorphism of
$\mathbb{C}^{2n+2}$ such that $f(x_{k})=y_{k}$ for $k=0,\ldots,n$. Let
$U_{\imath}$ be the open subset of $\mathbb{P}^{2n+1}$ of vectors with
coefficient of $y_{\imath-n-1}$ non-zero, and let
$\phi_{\imath}\colon\mathbb{C}^{2n+1}\rightarrow U_{\imath}$ be the standard
isomorphism. Let us define $\varphi_{\imath}:=f\circ\varphi_{\imath-n-1}\circ
f$. It is clear that such a morphism satisfies the conditions, and
$\\{U_{\imath}\\}_{\imath=0}^{2n+1}$ is the standard open cover. ∎

######  Remark 2.6.

We can see that $\varphi_{\imath}(\vec{a},\vec{c})$ is a symplectic linear
isomorphism by looking at its matrix description. With respect to the basis
$\\{x_{0},\ldots,x_{n},y_{0},\ldots,y_{n}\\}$, let us consider the
representation

$\varphi_{\imath}(\vec{a},\vec{c})=\begin{pmatrix}A&B\\\ C&D\end{pmatrix},$

where $A,B,C,D$ are square matrices of order $(n+1)$. For example, when
$\imath=0$,

$A=\begin{pmatrix}1\\\ a_{1}&1\\\ \vdots&&\ddots\\\
a_{n}&&&1\end{pmatrix}C=\begin{pmatrix}c_{0}&c_{1}&\cdots&c_{n}\\\ c_{1}\\\
\vdots\\\ c_{n}\end{pmatrix}D=\begin{pmatrix}1&-a_{1}&\cdots&-a_{n}\\\ &1\\\
&&\ddots\\\ &&&1\end{pmatrix}$

and $B$ is the zero matrix. It is easy to see that they satisfy Eqs (2.1),
(2.2), and (2.3).

##  3\. Contact Stable Maps

###  3.1. Stable trees

We give an introduction to stable trees as in [BM96, Part I]. A graph $\tau$
is a quadruple $(F_{\tau},V_{\tau},\partial_{\tau},j_{\tau})$, where
$F_{\tau}$ is a finite set of flags, $V_{\tau}$ is a finite set of vertices,
$\partial_{\tau}\colon F_{\tau}\rightarrow V_{\tau}$ is a map, and
$j_{\tau}\colon F_{\tau}\rightarrow F_{\tau}$ is an involution. The elements
of $E_{\tau}:=\\{\\{f,f^{\prime}\\}\subseteq
F_{\tau}/f^{\prime}=j_{\tau}(f)\neq f\\}$ are called the edges of $\tau$, and
the elements of $L_{\tau}:=\\{f\in F_{\tau}/j_{\tau}(f)=f\\}$ are called the
leaves (or tails) of $\tau$. For each vertex $v\in V_{\tau}$, its valence
$n(v)$ is the cardinality of $F_{\tau}(v):=\partial_{\tau}^{-1}(\\{v\\})$.

An isomorphism of graphs $\varphi\colon\tau\rightarrow\sigma$ is a pair
$(\varphi_{F},\varphi_{V})$, where $\varphi_{F}\colon F_{\tau}\rightarrow
F_{\sigma}$ and $\varphi_{V}\colon V_{\tau}\rightarrow V_{\sigma}$ are
bijective maps such that
$\varphi_{V}\circ\partial_{\tau}=\partial_{\sigma}\circ\varphi_{F}$ and
$\varphi_{F}\circ j_{\tau}=j_{\sigma}\circ\varphi_{F}$. A graph is connected
if for every two vertices $v,v^{\prime}\in V_{\tau}$, there exists a sequence
of vertices $\\{v_{i}\\}_{i=0}^{r}$ such that

  * •

for each $i=1,\ldots r$, there exists an edge $e=\\{f,f^{\prime}\\}$ such that
$\partial_{\tau}(f)=v_{i-1}$, $\partial_{\tau}(f^{\prime})=v_{i}$, and

  * •

$v_{0}=v$, $v_{r}=v^{\prime}$.

######  Definition 3.1.

A tree is a connected graph such that
$|E_{\tau}|+|L_{\tau}|+|V_{\tau}|=1+|F_{\tau}|$.

For any pair $(m,d)$ of non negative integers, a $(m,d)$-tree is the datum
$(\tau,l_{\tau},d_{\tau})$ where $\tau$ is a tree, $A_{\tau}$ is a set,
$|A_{\tau}|=m$, $l_{\tau}\colon L_{\tau}\rightarrow A_{\tau}$ is a bijective
map, and $d_{\tau}\colon V_{\tau}\rightarrow\mathbb{Z}$ is a map such that
$\sum_{v\in V_{\tau}}d_{\tau}(v)=d$ and $d_{\tau}(v)\geq 0$ for every $v\in
V_{\tau}$.

######  Definition 3.2.

A $(m,d)$-tree $(\tau,l_{\tau},d_{\tau})$ is stable if for each $v\in
V_{\tau}$, either $d_{\tau}(v)\neq 0$ or $n(v)\geq 3$.

Two stable $(m,d)$-trees $(\tau,l_{\tau},d_{\tau})$ and
$(\sigma,l_{\sigma},d_{\sigma})$ are isomorphic if there exist isomorphisms
$\varphi\colon\tau\rightarrow\sigma$, $\theta\colon A_{\tau}\rightarrow
A_{\sigma}$ such that $\theta\circ l_{\tau}=l_{\sigma}\circ\varphi_{F}$ and
$d_{\tau}=d_{\sigma}\circ\varphi_{V}$. By abuse of notation, we may denote a
stable $(m,d)$-tree $(\tau,l_{\tau},d_{\tau})$ simply by $\tau$.

######  Definition 3.3.

The group of automorphisms of the stable $(m,d)$-tree $\tau$ is denoted by
$\mathrm{Aut}(\tau)$.

The set of all stable $(m,d)$-tree, modulo isomorphism, is denoted by
$\Gamma(m,d)$. It is finite by [GP06, Proposition 4.3]. The subset of those
trees such that $d_{\tau}(v)>0$ for every $v\in V_{\tau}$ is denoted by
$\Gamma(m,d)^{+}$.

######  Definition 3.4.

The unique stable $(m,d)$-tree with only one vertex $v$ is denoted by
$\tau_{v}$.

Since each edge joints two vertices, in a stable $(m,d)$-tree $\tau$ we have

(3.1) $|F_{\tau}|-2|V_{\tau}|=m-2.$

###  3.2. Stable maps

In this section by $n$ we will denote any odd number.

A stable map is the datum $(C,f,p_{1},\ldots,p_{m})$ where $C$ is a
projective, connected, nodal curve of arithmetic genus $0$, the markings
$p_{1},\ldots,p_{m}$ are distinct nonsingular points of $C$, and $f\colon
C\rightarrow\mathbb{P}^{n}$ is a morphism such that $f_{*}([C])$ is a
$1$-cycle of degree $d$. Moreover, for every rational component $E\subseteq C$
mapped to a point, $E$ contains at least three points among marked points and
nodal points.

The dual graph $\tau$ of the stable map $(C,f,p_{1},\ldots,p_{m})$ is defined
as follows.

  * •

For each vertex $v\in V_{\tau}$, there is an irreducible component $C_{v}$ of
$C$.

  * •

For each nodal point of $C$, there is an edge in $\tau$ connecting the
vertices corresponding to those components.

  * •

For each $p_{i}$, there is a leaf of $\tau$ attached to the vertex
corresponding to the irreducible component containing $p_{i}$.

  * •

There is a map $d_{\tau}\colon V_{\tau}\rightarrow\mathbb{Z}$ such that
$d_{\tau}(v)$ is the degree of the restriction of $f$ to $C_{v}$.

A map $f\colon C\rightarrow\mathbb{P}^{n}$ of genus $0$ with $m$ marked points
and degree $d$ is stable if and only if its dual graph is a stable
$(m,d)$-tree [GP06, Proposition 4.4].

Let $\mathcal{M}_{0,m}(\mathbb{P}^{n},d)$ be the coarse moduli space of
irreducible stable maps of genus $0$ to $\mathbb{P}^{n}$, and let
$\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$ be its compactification. Let
us denote by $\mathrm{ev}_{i}$ the evaluation map
$\mathrm{ev}_{i}\colon\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)\rightarrow\mathbb{P}^{n}$
at the marked point $p_{i}$. Given an isomorphism class $\tau$ of stable
$(m,d)$-trees, the locus
$\mathcal{M}(\tau)\subset\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$
parameterizes maps whose dual graph is isomorphic to $\tau$. It is locally
closed and of codimension $|E_{\tau}|$. For example, the unique stable
$(m,d)$-tree with only one vertex corresponds to
$\mathcal{M}_{0,m}(\mathbb{P}^{n},d)$. There is a stratification of
$\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$ given by $\mathcal{M}(\tau)$
for all $\tau\in\Gamma(m,d)$. Finally, there is a canonical isomorphism
$\mathcal{M}(\tau)\cong\mathcal{M}_{\square}(\tau)/\mathrm{Aut}(\tau)$ where
$\mathcal{M}_{\square}(\tau)$ is the fibred product

(3.2)
$\textstyle{\mathcal{M}_{\square}(\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V_{\tau}}\mathcal{M}_{0,F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{E_{\tau}\sqcup
L_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{F_{\tau}}.}$

We denote by $\overline{\mathcal{M}(\tau)}$ the topological closure of
$\mathcal{M}(\tau)$ in $\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$.

######  Remark 3.5.

We may define a scheme $\overline{\mathcal{M}_{\square}(\tau)}$ using the same
square of Eq. (3.2), but using
$\overline{\mathcal{M}}_{0,F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))$ instead
of $\mathcal{M}_{0,F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))$. Such a scheme
has a natural ramified map to $\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$
of degree $\mathrm{Aut}(\tau)$. The image is the closure of
$\mathcal{M}(\tau)$. See [BM96] or [MM08].

Let us now focus on contact curves.

######  Definition 3.6.

Let $(X,L)$ be a contact structure. A contact stable map is a stable map
$(C,f,p_{1},\ldots,p_{m})$ such that for any local section $s$ of $L$,
$f^{*}s=0$.

In [Mur20] we proved the following properties:

  * •

The moduli space of contact stable maps
$\mathcal{S}_{m}(\mathbb{P}^{n},d)\subset\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$
is the zero locus of a vector bundle $\mathcal{E}$ of rank $2d-1$.

  * •

If $d>0$, the irreducible components of $\mathcal{S}_{m}(\mathbb{P}^{n},d)$
are purely of dimension $d(n-1)+n+m-2$.

  * •

The Gromov-Witten invariant

(3.3)
$\int_{\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})\cdot
c_{2d-1}(\mathcal{E})=\int_{\mathcal{S}_{m}(\mathbb{P}^{n},d)}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})$

is enumerative.

Enumerative means that if $\\{\Gamma_{i}\\}_{i=1}^{m}$ are subvarieties of
$\mathbb{P}^{n}$ in general position such that
$\sum_{i=1}^{m}\mathrm{codim}(\Gamma_{i})=\dim\mathcal{S}_{m}(\mathbb{P}^{n},d)$,
then (3.3) equals the number of contact curves passing through all
$\Gamma_{i}$.

If the stable map $(C,f,p_{1},\ldots,p_{m})$ contracts $C$, then it is
contact. It follows that
$\mathcal{S}_{m}(\mathbb{P}^{n},0)=\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},0)\cong\overline{\mathcal{M}}_{0,m}\times\mathbb{P}^{n}$.
Hence

(3.4)
$\int_{\mathcal{S}_{m}(\mathbb{P}^{n},0)}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})=\begin{cases}\Gamma_{1}\cdots\Gamma_{m}&m=3\\\
0&m\neq 3.\end{cases}$

######  Remark 3.7.

Let $H$ be the generator of $H^{2}(\mathbb{P}^{n},\mathbb{Z})$. It can be
defined a “contact potential” $\Phi\in\mathbb{Q}\left\llbracket
h_{0},\ldots,h_{n}\right\rrbracket$, like in the usual GW-potential:

$\displaystyle I_{d}(m_{0},\cdots,m_{n})$ $\displaystyle=$
$\displaystyle\int_{\mathcal{S}_{m_{0}+\ldots+m_{n}}(\mathbb{P}^{n},d)}\prod_{i=0}^{n}\mathrm{ev}^{*}(H^{i})^{\cdot
m_{i}}$ $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sum_{m_{0}+\ldots+m_{n}\geq
3}\sum_{d=0}^{\infty}I_{d}(m_{0},\cdots,m_{n})\frac{h_{0}^{m_{0}}}{m_{0}!}\cdots\frac{h_{n}^{m_{n}}}{m_{n}!}.$

For example, in the case $n=3$,

$\Phi=h_{0}h_{1}h_{2}+\frac{h_{1}^{3}}{3!}+\frac{h_{0}^{2}h_{3}}{2!}+2\frac{h_{1}^{3}}{3!}+h_{1}h_{2}h_{3}+2\frac{h_{2}h_{3}^{2}}{2}+\cdots.$

The WDVV differential equations is the system of PDE:

$\sum_{e=0}^{n}\Phi_{ije}\Phi_{kl(n-e)}=\sum_{e=0}^{n}\Phi_{jke}\Phi_{il(n-e)},$

where $\Phi_{ijk}:=\frac{\partial^{3}}{\partial h_{i}\partial h_{j}\partial
h_{k}}\Phi$. The usual GW-potential satisfies the WDVV equations [KM94]. The
contact potential does not. In order to see this, it is enough to compute all
derivatives $\Phi_{ijk}$, and evaluate them at $0$. There are not known
differential equations satisfied by $\Phi$.

##  4\. Stratification of the Moduli of Contact Stable Maps

######  Definition 4.1.

For every $\tau\in\Gamma(m,d)$, we denote by $\mathcal{S}(\tau)$ the space
given by the intersection
$\mathcal{S}_{m}(\mathbb{P}^{n},d)\cap\mathcal{M}(\tau)$.

We will denote by $\overline{\mathcal{S}(\tau)}$ the topological closure of
$\mathcal{S}(\tau)$ in $\mathcal{S}_{m}(\mathbb{P}^{n},d)$. The restriction of
the evaluation maps
$\mathrm{ev}_{i}\colon\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)\rightarrow\mathbb{P}^{n}$
to $\mathcal{S}(\tau)$ or $\overline{\mathcal{S}(\tau)}$ will still be denoted
by $\mathrm{ev}_{i}$. We denote by $\mathcal{S}_{m}(\mathbb{P}^{n},d)^{\circ}$
the open subset
$\mathcal{S}_{m}(\mathbb{P}^{n},d)\cap\mathcal{M}_{m}(\mathbb{P}^{n},d)$. The
following result follows from Proposition 2.5. See also [Bag19, Proposition
3.8].

######  Proposition 4.2.

Let $\tau\in\Gamma(m,d)$. The map
$\mathrm{ev}_{i}\colon\mathcal{S}(\tau)\rightarrow\mathbb{P}^{n}$ is Zariski
locally trivial.

###### Proof.

We start by proving that
$\mathrm{ev}_{i}\colon\mathcal{S}_{m}(\mathbb{P}^{n},d)\rightarrow\mathbb{P}^{n}$
is Zariski locally trivial. As we saw in Proposition 2.5, there exists an open
cover of $\mathbb{P}^{n}$ such that each open set $U$ is isomorphic to a
subgroup $H$ of $\mathrm{Sp}(n+1,\mathbb{C})$, in such a way that $U$ is
$H$-invariant and the action of $H$ on $U$ is free and transitive. This
induces an action $\beta\colon
H\times\mathrm{ev}_{i}^{-1}(U)\rightarrow\mathrm{ev}_{i}^{-1}(U)$ sending
$(h,(C,f,p_{1},\ldots,p_{m}))$ to $(C,h\cdot f,p_{1},\ldots,p_{m})$. Let $o\in
U$ be a point. There exists a map $\gamma\colon
H\times\mathrm{ev}_{i}^{-1}(\\{o\\})\rightarrow\mathrm{ev}_{i}^{-1}(U)$ given
by

$H\times\mathrm{ev}_{i}^{-1}(\\{o\\})\rightarrow
H\times\mathrm{ev}_{i}^{-1}(U)\rightarrow\mathrm{ev}_{i}^{-1}(U),$

where the first map is the embedding and the second is $\beta$. Let us prove
that $\gamma$ is invertible. Since the action of $H$ on $U$ is free and
transitive, there exists a unique isomorphism of varieties $g\colon
U\rightarrow H$ such that $g(x)\cdot x=o$ for all $x\in U$. Let us consider
the map
$((g\circ\mathrm{ev}_{i}),\mathrm{id})\colon\mathrm{ev}_{i}^{-1}(U)\rightarrow
H\times\mathrm{ev}_{i}^{-1}(U)$. The composition of that map with $\beta$ has
image contained in $H\times\mathrm{ev}_{i}^{-1}(\\{o\\})$ and it is the
inverse of $f$.

Since $H$ preserves the dual graph of any contact stable map, it induces an
action on any $\mathcal{S}(\tau)$, in such a way that
$\mathrm{ev}_{i}\colon\mathcal{S}(\tau)\rightarrow\mathbb{P}^{n}$ is Zariski
locally trivial by restriction. ∎

######  Proposition 4.3.

Let $\tau\in\Gamma(m,d)$. Then
$\mathcal{S}(\tau)\cong\mathcal{S}_{\square}(\tau)/\mathrm{Aut}(\tau)$ where
$\mathcal{S}_{\square}(\tau)$ is the fibred product

(4.1)
$\textstyle{\mathcal{S}_{\square}(\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V_{\tau}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{E_{\tau}\sqcup
L_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{F_{\tau}}.}$

Moreover, $\mathcal{S}(\tau)$ and $\mathcal{S}_{\square}(\tau)$ are purely of
the same dimension.

###### Proof.

By standard properties of fibred product (e.g., [GW20, Proposition 4.16]),
there is a double fibred product diagram involving
$\mathcal{S}_{\square}(\tau)$

$\textstyle{\mathcal{S}_{\square}(\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}_{\square}(\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{E_{\tau}\sqcup
L_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V_{\tau}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V_{\tau}}\mathcal{M}_{0,F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{F_{\tau}}.}$

That is, $\mathcal{S}_{\square}(\tau)$ is embedded in
$\mathcal{M}_{\square}(\tau)$ and it is $\mathrm{Aut}(\tau)$-invariant. This
induces an action of $\mathrm{Aut}(\tau)$ on $\mathcal{S}_{\square}(\tau)$, in
particular
$\mathcal{S}_{\square}(\tau)/\mathrm{Aut}(\tau)\subseteq\mathcal{M}(\tau)$. As
we said in Remark 2.4, a reducible curve is contact if and only if each one of
its components is contact in its own. That is, if
$(C,f,p_{1},\ldots,p_{m})\in\mathcal{M}(\tau)$ is contact then for every
irreducible component $C_{i}\subseteq C$, the restriction $f_{|C_{i}}\colon
C_{i}\rightarrow\mathbb{P}^{n}$ is contact too. Thus both
$\mathcal{S}_{\square}(\tau)/\mathrm{Aut}(\tau)\subseteq\mathcal{S}(\tau)$ and
$\mathcal{S}(\tau)\subseteq\mathcal{S}_{\square}(\tau)/\mathrm{Aut}(\tau)$ are
true.

The fact that $\mathcal{S}_{\square}(\tau)$ is purely dimensional follows from
Proposition 4.2. Indeed, each
$\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}$ is purely
dimensional, so each fiber $F$ of the locally trivial map
$\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}\rightarrow\mathbb{P}^{n}$
is also. It follows that $\mathcal{S}_{\square}$ is locally isomorphic to
$F\times U$ where $U$ is an open subset of $(\mathbb{P}^{n})^{E_{\tau}\sqcup
L_{\tau}}$. Since $\mathrm{Aut}(\tau)$ is finite, $\mathcal{S}(\tau)$ is also
purely dimensional and
$\dim\mathcal{S}(\tau)=\dim\mathcal{S}_{\square}(\tau)$. ∎

Even if we use the fact that $\mathcal{S}_{m}(\mathbb{P}^{n},d)^{\circ}$ is
purely dimensional, we do not say that
$\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}$ is
irreducible. As already noted in the proof of [Mur20, Lemma 3.7],
$\mathcal{S}_{m}(\mathbb{P}^{n},d)^{\circ}$ is a quotient
$\mathcal{M}_{d}/\mathrm{Aut}(\mathbb{P}^{1})$ where $\mathcal{M}_{d}$ is the
moduli space of harmonic maps
$\mathbb{C}\mathbb{P}^{1}\rightarrow\mathbb{C}\mathbb{P}^{3}$ we cited in
Introduction. The manifold $\mathcal{M}_{d}$ need to be irreducible [Loo99,
KL98]. In the same Lemma, it is claimed that

$\dim(\mathcal{S}_{1}(\mathbb{P}^{n},d_{1})\times_{\mathbb{P}^{n}}\mathcal{S}_{1}(\mathbb{P}^{n},d_{2}))=\dim\mathcal{S}_{1}(\mathbb{P}^{n},d_{1})+\dim\mathcal{S}_{1}(\mathbb{P}^{n},d_{2})-n.$

This follows from the fact the action of $\mathrm{Sp}(n+1,\mathbb{C})$ on
$\mathbb{P}^{n}$ is transitive and preserve the fibers of
$\mathrm{ev}_{1}\colon\mathcal{S}_{1}(\mathbb{P}^{n},d)\rightarrow\mathbb{P}^{n}$.
So $\mathrm{ev}_{1}$ is flat.

We are now ready to state the first important result of the paper.

######  Theorem 4.4.

Let $n$ be an odd number and $\mathcal{S}_{m}(\mathbb{P}^{n},d)$ be the moduli
space of contact stable maps.

  1. (1)

Every subspace $\mathcal{S}(\tau)$ is locally closed in
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$.

  2. (2)

There exists a stratification of $\mathcal{S}_{m}(\mathbb{P}^{n},d)$ given by
the union of all $\mathcal{S}(\tau)$.

  3. (3)

For every $\tau\in\Gamma(m,d)$, the spaces $\mathcal{S}(\tau)$ are of pure
codimension equal to $|V_{\tau}^{0}|$, where $V_{\tau}^{0}=\\{v\in
V_{\tau}/d_{\tau}(v)=0\\}$.

In particular there exists a decomposition of
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ as union of components
$\overline{\mathcal{S}(\tau)}$ for $\tau\in\Gamma(m,d)^{+}$, so that

(4.2)
$\int_{\mathcal{S}_{m}(\mathbb{P}^{n},d)}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})=\sum_{\tau\in\Gamma(m,d)^{+}}\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m}).$

###### Proof.

The first two points follow from analogous properties of
$\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$, so let us prove Point $(3)$.
By Proposition 4.3 we have

$\dim\mathcal{S}(\tau)=n(|E_{\tau}|+|L_{\tau}|-|F_{\tau}|)+\sum_{v\in
V_{\tau}}\dim\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}.$

Recall that if $d>0$ (resp., $d=0$) then
$\dim\mathcal{S}_{m}(\mathbb{P}^{n},d)^{\circ}=d(n-1)+n+m-2$ (resp., $n+m-3$).
Thus the dimension of $\mathcal{S}(\tau)$ is, using Eq. (3.1),

$\displaystyle n(1-|V_{\tau}|)+\sum_{v\in
V_{\tau}^{0}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}+\sum_{v\in
V_{\tau}\setminus
V_{\tau}^{0}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}$
$\displaystyle=$ $\displaystyle
n(1-|V_{\tau}|)+n|V_{\tau}|+|F_{\tau}|-3|V_{\tau}^{0}|+d(n-1)-2(|V_{\tau}|-|V_{\tau}^{0}|)$
$\displaystyle=$ $\displaystyle
n+|F_{\tau}|-2|V_{\tau}|-|V_{\tau}^{0}|+d(n-1)$ $\displaystyle=$
$\displaystyle d(n-1)+n+m-2-|V_{\tau}^{0}|$ $\displaystyle=$
$\displaystyle\dim\mathcal{S}_{m}(\mathbb{P}^{n},d)-|V_{\tau}^{0}|.$

In particular for those $\tau\in\Gamma(m,d)^{+}$ we have
$\dim\mathcal{S}(\tau)=\dim\mathcal{S}_{m}(\mathbb{P}^{n},d)$. Since
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ is purely of the expected dimension, it
can be decomposed as union of $\overline{\mathcal{S}(\tau)}$ for
$\tau\in\Gamma(m,d)^{+}$. Equation (4.2) follows from linearity of the
intersection product [Ful98, Example 1.8.1]. ∎

In the next section, we will see how to use the theorem to compute all
enumerative invariants of contact curves.

##  5\. Applications

Let us briefly introduce the decomposition of stable trees. Given a stable
$(m,d)$-tree $(\tau,l_{\tau},d_{\tau})$, for every edge
$\\{f,f^{\prime}\\}=e\in E_{\tau}$ there exist two unique stable trees
$(\sigma,l_{\sigma},d_{\sigma})$ and
$(\sigma^{\prime},l_{\sigma^{\prime}},d_{\sigma^{\prime}})$ with the following
properties:

  * •

there is a partition $F_{\tau}=F_{\sigma}\sqcup F_{\sigma^{\prime}}$ such that
$f\in F_{\sigma}$ and $f^{\prime}\in F_{\sigma^{\prime}}$,

  * •

for every $x\in F_{\tau}\backslash\\{f,f^{\prime}\\}$ such that $x\in
F_{\sigma}$ (resp., $F_{\sigma^{\prime}}$), $j_{\sigma}(x)$ (resp.,
$j_{\sigma^{\prime}}(x)$) is equal to $j_{\tau}(x)$,

  * •

$f$ and $f^{\prime}$ are fixed points of, respectively, $j_{\sigma}$ and
$j_{\sigma^{\prime}}$,

  * •

$V_{\sigma}=\partial_{\tau}(F_{\sigma})$ and
$V_{\sigma^{\prime}}=\partial_{\tau}(F_{\sigma^{\prime}})$, and,

  * •

$d_{\sigma}$ (resp., $d_{\sigma^{\prime}}$) is the restriction of $d_{\tau}$
to $V_{\sigma}$ (resp., $V_{\sigma^{\prime}}$).

It is immediate to deduce that $E_{\tau}\backslash\\{e\\}=E_{\sigma}\sqcup
E_{\sigma^{\prime}}$ and $L_{\tau}\sqcup\\{f,f^{\prime}\\}=L_{\sigma}\sqcup
L_{\sigma^{\prime}}$. We will use the following notation:
$E_{\tau}^{*}=E_{\tau}\backslash\\{e\\}$,
$L_{\sigma}^{*}=L_{\sigma}\backslash\\{f\\}$ and
$L_{\sigma^{\prime}}^{*}=L_{\sigma^{\prime}}\backslash\\{f^{\prime}\\}$. The
maps $l_{\sigma}$ and $l_{\sigma^{\prime}}$ are defined in the obvious way.

We will denote by $\tau=\sigma\times_{e}\sigma^{\prime}$ the decomposition of
$\tau$ with respect to the edge $e$. Note that every vertex $v$ of $\sigma$ or
$\sigma^{\prime}$ has the same valence as a vertex of $\tau$, so $\sigma$ and
$\sigma^{\prime}$ are always stable.

######  Corollary 5.1.

Let $\tau\in\Gamma(m,d)$ such that $\tau=\sigma\times_{e}\sigma^{\prime}$ for
an edge $e=\\{f,f^{\prime}\\}$. Then

  1. (1)

There exists a projective variety $\overline{\mathcal{S}_{\square}(\tau)}$ and
a map
$\overline{\mathcal{S}_{\square}(\tau)}\rightarrow\overline{\mathcal{S}(\tau)}$
extending $\mathcal{S}_{\square}(\tau)\rightarrow\mathcal{S}(\tau)$.

  2. (2)

There exist isomorphisms

$\begin{array}[]{cc}\mathcal{S}_{\square}(\tau)\cong\mathcal{S}_{\square}(\sigma)\times_{\mathbb{P}^{n}}\mathcal{S}_{\square}(\sigma^{\prime}),&\overline{\mathcal{S}_{\square}(\tau)}\cong\overline{\mathcal{S}_{\square}(\sigma)}\times_{\mathbb{P}^{n}}\overline{\mathcal{S}_{\square}(\sigma^{\prime})}.\end{array}$

###### Proof.

Let us define $\overline{\mathcal{S}_{\square}(\tau)}$ as the fiber product

$\textstyle{\overline{\mathcal{S}_{\square}(\tau)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod_{v\in
V_{\tau}}\overline{\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{E_{\tau}\sqcup
L_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{F_{\tau}}.}$

Point $(1)$ follows from Remark 3.5: The image of the natural map
$\mathcal{S}_{\square}(\tau)\hookrightarrow\overline{\mathcal{S}_{\square}(\tau)}$
is dense in every irreducible component of
$\overline{\mathcal{S}_{\square}(\tau)}$, and it factorizes the inclusion
$\mathcal{S}_{\square}(\tau)\hookrightarrow\overline{\mathcal{M}_{\square}(\tau)}$.
It follows that the restriction of
$\overline{\mathcal{M}_{\square}(\tau)}\rightarrow\overline{\mathcal{M}}_{0,m}(\mathbb{P}^{n},d)$
to $\overline{\mathcal{S}_{\square}(\tau)}$ maps to
$\overline{\mathcal{S}(\tau)}$.

Let us prove Point $(2)$. The isomorphism $V_{\tau}\cong V_{\sigma}\sqcup
V_{\sigma^{\prime}}$ preserves the valence of each vertex $v\in V_{\tau}$. So
the exists an isomorphism

$X:=\prod_{v\in
V_{\tau}}\mathcal{S}_{F_{\tau}(v)}(\mathbb{P}^{n},d_{\tau}(v))^{\circ}\cong\prod_{v\in
V_{\sigma}}\mathcal{S}_{F_{\sigma}(v)}(\mathbb{P}^{n},d_{\sigma}(v))^{\circ}\times\prod_{v\in
V_{\sigma^{\prime}}}\mathcal{S}_{F_{\sigma^{\prime}}(v)}(\mathbb{P}^{n},d_{\sigma^{\prime}}(v))^{\circ}.$

Using the equality $F_{\tau}=F_{\sigma}\sqcup F_{\sigma^{\prime}}$, it follows
that we have a diagram

(5.1)
$\textstyle{\mathcal{S}_{\square}(\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{\square}(\sigma)\times\mathcal{S}_{\square}(\sigma^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{e}\times(\mathbb{P}^{n})^{E_{\tau}^{*}\sqcup
L_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{f}\times(\mathbb{P}^{n})^{f^{\prime}}\times(\mathbb{P}^{n})^{E_{\sigma}\sqcup
L_{\sigma}^{*}\sqcup E_{\sigma^{\prime}}\sqcup
L_{\sigma^{\prime}}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{P}^{n})^{F_{\tau}}.}$

The right-hand square is a fibred product diagram, it is obtained by (4.1)
applied to $\sigma$ and $\sigma^{\prime}$. By applying again [GW20,
Proposition 4.16], we deduce that also the left-hand square is a fibred
product diagram. The immersion
$(\mathbb{P}^{n})^{e}\rightarrow(\mathbb{P}^{n})^{f}\times(\mathbb{P}^{n})^{f^{\prime}}$
is the diagonal, hence
$\mathcal{S}_{\square}(\tau)\cong\mathcal{S}_{\square}(\sigma)\times_{\mathbb{P}^{n}}\mathcal{S}_{\square}(\sigma^{\prime})$.
The same proof applies equally well in
$\overline{\mathcal{S}_{\square}(\tau)}\cong\overline{\mathcal{S}_{\square}(\sigma)}\times_{\mathbb{P}^{n}}\overline{\mathcal{S}_{\square}(\sigma^{\prime})}$.
∎

The next corollary allow us to compute intersections in
$\overline{\mathcal{S}(\tau)}$ recursively.

######  Corollary 5.2.

Let $\tau\in\Gamma(m,d)$ such that $\tau=\sigma\times_{e}\sigma^{\prime}$ for
an edge $e=\\{f,f^{\prime}\\}$. Then

$\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})$

is equal to

(5.2)
$\frac{|\mathrm{Aut}(\sigma)||\mathrm{Aut}(\sigma^{\prime})|}{|\mathrm{Aut}(\tau)|}\sum_{j=0}^{n}\int_{\overline{\mathcal{S}(\sigma)}}\prod_{i\in
L_{\sigma}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f}^{*}(H^{n-j})\,\int_{\overline{\mathcal{S}(\sigma^{\prime})}}\prod_{i\in
L_{\sigma^{\prime}}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{j}).$

###### Proof.

In Corollary 5.1 we proved that the map
$\overline{\mathcal{S}_{\square}(\tau)}\rightarrow\overline{\mathcal{S}(\tau)}$
is generically of degree $|\mathrm{Aut}(\tau)|$, so by projection formula (see
[Ful98, 1.4])

$\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})=\frac{1}{|\mathrm{Aut}(\tau)|}\int_{\overline{\mathcal{S}_{\square}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m}).$

Let us denote again by $H$ the generator of
$H^{2}(\mathbb{P}^{n},\mathbb{Z})$. The class of the diagonal of
$\mathbb{P}^{n}\times\mathbb{P}^{n}$ is $\sum_{j=0}^{n}H^{n-j}\boxtimes
H^{j}$, so using again the corollary:

$\displaystyle\int_{\overline{\mathcal{S}_{\square}(\sigma)}\times_{\mathbb{P}^{n}}\overline{\mathcal{S}_{\square}(\sigma^{\prime})}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})$
$\displaystyle=$
$\displaystyle\int_{\overline{\mathcal{S}_{\square}(\sigma)}\times\overline{\mathcal{S}_{\square}(\sigma^{\prime})}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})\cdot\sum_{j=0}^{n}\mathrm{ev}_{f}^{*}(H^{n-j})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{j})$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\int_{\overline{\mathcal{S}_{\square}(\sigma)}}\prod_{i\in
L_{\sigma}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f}^{*}(H^{n-j})\int_{\overline{\mathcal{S}_{\square}(\sigma^{\prime})}}\prod_{i\in
L_{\sigma^{\prime}}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{j})$
$\displaystyle=$
$\displaystyle\frac{|\mathrm{Aut}(\sigma)|}{|\mathrm{Aut}(\sigma^{\prime})|^{-1}}\sum_{j=0}^{n}\int_{\overline{\mathcal{S}(\sigma)}}\prod_{i\in
L_{\sigma}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f}^{*}(H^{n-j})\int_{\overline{\mathcal{S}(\sigma^{\prime})}}\prod_{i\in
L_{\sigma^{\prime}}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{j}).$

The last equation implies (5.2). ∎

######  Corollary 5.3.

Let $\tau\in\Gamma(m,d)$ and let $\Gamma_{1},...,\Gamma_{m}$ be subvarieties
of $\mathbb{P}^{n}$ in general position such that

(5.3) $\sum_{i=1}^{m}\mathrm{codim}(\Gamma_{i})=\dim\mathcal{S}(\tau).$

Then the number of contact stable maps in $\mathbb{P}^{n}$ of degree $d$ whose
images meet all $\Gamma_{1},...,\Gamma_{m}$, and with dual graph $\tau$, is

(5.4)
$\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m}).$

In particular, if $\sum_{i=1}^{m}\mathrm{codim}(\Gamma_{i})=d(n-1)+n+m-2$, the
number of irreducible contact curves in $\mathbb{P}^{n}$ of degree $d$ meeting
all $\Gamma_{1},...,\Gamma_{m}$ is

(5.5)
$\int_{\mathcal{S}_{m}(\mathbb{P}^{n},d)}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m})-\sum_{\tau\in\Gamma(m,d)^{+}\backslash\\{\tau_{v}\\}}\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(\Gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\Gamma_{m}).$

###### Proof.

This proof follows the same line of [Mur20, Theorem 3.8], let us give a
sketch. Let $G$ be the group of invertible complex matrices of order $n+1$,
and let $X=\mathbb{P}^{n}$. Consider the diagram

$\textstyle{\overline{\mathcal{S}(\tau)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{ev}_{1}\times\cdots\times\mathrm{ev}_{m})}$$\textstyle{\Gamma_{1}\times\cdots\times\Gamma_{m}=:\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\times
m}.}$

By applying the Kleiman-Bertini Theorem [Kle74], we deduce that for a general
$\sigma\in G^{\times m}$, $\Gamma^{\sigma}\times_{X^{\times
m}}\overline{\mathcal{S}(\tau)}$ is either empty, or of dimension $0$. We
assume, without loss of generality, that $\Gamma^{\sigma}$ is smooth by
applying again Kleiman-Bertini. In the same way we may prove111Details can be
found in [Alu97, Lemma p. 5] or [FP97, Lemma 14]. that the intersection

$\overline{\mathcal{S}(\tau)}\cap\mathrm{ev}_{1}^{-1}(g_{1}\Gamma_{1})\cap\cdots\cap\mathrm{ev}_{m}^{-1}(g_{m}\Gamma_{m})$

is supported in the intersection between the automorphisms-free part of
$\overline{\mathcal{M}}_{0,m}(X,d)$ and the smooth part of
$\overline{\mathcal{S}(\tau)}$. So, when $\Gamma^{\sigma}\times_{X^{\times
m}}\overline{\mathcal{S}(\tau)}$ is not empty, it is reduced of dimension $0$.
Points of that intersection represent maps whose images meet all
$\Gamma_{1},...,\Gamma_{m}$, and it is equal to (5.4).

Finally, $\dim\mathcal{S}(\tau)=d(n-1)+n+m-2$ implies that $\tau$ has no
contracted components by Theorem 4.4. So if we are interested in a unique non-
contracted component, it must be $\tau=\tau_{v}$. Equation (5.5) follows from
(4.2). ∎

######  Remark 5.4.

The last results imply that for every $\tau\in\Gamma(m,d)$, we can compute
every integral like in (5.4). Indeed, if $d=1$ then
$\overline{\mathcal{S}(\tau)}=\mathcal{S}_{m}(\mathbb{P}^{n},1)$ and we can
use the package `AtiyahBott` of [MS22]. If $d>1$, then Eqs. (5.2) and (5.5)
allow us to reduce the computation to intersections in
$\mathcal{S}_{m}(\mathbb{P}^{n},d)$ and $\mathcal{S}(\tau^{\prime})$, for some
$\tau^{\prime}\in\Gamma(m^{\prime},d^{\prime})$ where
$m^{\prime}\in\mathbb{Z}$ and $d^{\prime}<d$. The result follows by using
again the package and recursion on $d$.

Let us see an example of these applications.

######  Example 5.5.

Suppose we want to compute the number of irreducible contact curves of degree
$2$ in $\mathbb{P}^{3}$ passing through two points and a line in general
position. Let $H$ be the class of a hyperplane in $\mathbb{P}^{3}$. The number
of contact curves is given by

$\int_{\mathcal{S}_{3}(\mathbb{P}^{3},2)}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{3})=2.$

The set $\Gamma(3,2)^{+}$ consists of the following stable trees:

$2$$\tau_{1}$$\\{1,2,3\\}$$1$$\\{1,2\\}$$\\{3\\}$$\tau_{2}$$1$$1$$\\{1,3\\}$$\\{2\\}$$\tau_{3}$$1$$1$$\\{2,3\\}$$\\{1\\}$$\tau_{4}$$1$$1$$\\{1,2,3\\}$$\tau_{5}$$1$

The vertices are labelled by their degree, under any vertex $v$ we put its
leaves. We want to compute
$\int_{\overline{\mathcal{S}(\tau_{1})}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{3})$.
Note that $\tau_{2}=\sigma\times_{e}\sigma^{\prime}$ where

$1$$\sigma$$\\{1,2,f\\}$$1$$\\{3,f^{\prime}\\}$$\sigma^{\prime}$

Using Corollary 5.2 and Remark 5.4, we get

$\displaystyle\int_{\overline{\mathcal{S}(\tau_{2})}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{3})$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{3}\int_{\overline{\mathcal{S}(\sigma)}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{f}^{*}(H^{3-j})\int_{\overline{\mathcal{S}(\sigma^{\prime})}}\mathrm{ev}_{3}^{*}(H^{3})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{j})$
$\displaystyle=$
$\displaystyle\int_{\overline{\mathcal{S}(\sigma)}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{f}^{*}(H^{1})\int_{\overline{\mathcal{S}(\sigma^{\prime})}}\mathrm{ev}_{3}^{*}(H^{3})\cdot\mathrm{ev}_{f^{\prime}}^{*}(H^{2})$
$\displaystyle=$ $\displaystyle 1\cdot 1=1.$

The computation of
$\int_{\overline{\mathcal{S}(\tau_{3})}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{3})=1$
is similar. Finally, one can easily figure out that
$\int_{\overline{\mathcal{S}(\tau)}}\mathrm{ev}_{1}^{*}(H^{2})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{3})=0$
for $\tau\in\\{\tau_{4},\tau_{5}\\}$. It follows that

$\int_{\overline{\mathcal{S}(\tau_{1})}}\mathrm{ev}_{1}^{*}(H^{3})\cdot\mathrm{ev}_{2}^{*}(H^{3})\cdot\mathrm{ev}_{3}^{*}(H^{2})=2-1-1-0-0=0.$

So, there are no irreducible contact conics through two points and a line in
general position. This result is completely expected since there are no
irreducible contact conics in $\mathbb{P}^{3}$ ([LM07, Proposition 17.1]).
Anyway, this shows that our method works.

######  Remark 5.6.

If the value of (5.2) is non-zero, then $|\mathrm{Aut}(\tau)|=1$. Indeed, if
$\sigma^{\prime}=\tau_{v}$ where $v$ is an extremal vertex with no leaves,
then Eq. (5.2) becomes

$\frac{|\mathrm{Aut}(\sigma)|}{|\mathrm{Aut}(\tau)|}\sum_{j=0}^{n}\int_{\overline{\mathcal{S}(\sigma)}}\prod_{i\in
L_{\sigma}}\mathrm{ev}_{i}^{*}(\Gamma_{i})\cdot\mathrm{ev}_{f}^{*}(H^{n-j})\,\int_{\overline{\mathcal{S}(\tau_{v})}}\mathrm{ev}_{f^{\prime}}^{*}(H^{j}).$

The value of
$\int_{\overline{\mathcal{S}(\tau_{v})}}\mathrm{ev}_{f^{\prime}}^{*}(H^{j})$
is zero for dimensional reasons, meaning that also (5.2) is zero. This implies
that every vertex of $\tau$ with only one edge must have leaves. Since any
automorphism preserves the leaves, all vertices with only one edge must be
fixed by any automorphism, hence the identity is the only possible
automorphism. Moreover, the sum (5.2) is non zero for at most one value of
$j$, which is $j_{0}=\dim\mathcal{S}(\tau_{v})-\sum_{i\in
L_{\tau_{v}}}\mathrm{codim}(\Gamma_{i})$, if $0\leq j_{0}\leq n$.

The following result is [LV11, Proposition 2.3(i)]. We provide another proof.

######  Proposition 5.7.

The number of contact plane curves of degree $d$ meeting $d+3$ general lines
in $\mathbb{P}^{3}$ is

$\frac{d^{2}}{6}(d+3)(d+2)(d+1)(d-1)=20d\binom{d+3}{5}.$

###### Proof.

We give a sketch of the proof, showing that it can be proved using graphs. A
contact plane curve is linearly degenerate in $\mathbb{P}^{3}$, then it is
cone [LM07, Proposition 17.1]. On the other hand, cones in $\mathbb{P}^{3}$
are contact if they are linearly degenerate222Otherwise, the symplectic form
defining the contact structure in $\mathbb{C}^{4}$ would be degenerate.. The
cases $d=1,2$ are trivial, so let us focus on the case $d\geq 3$. We are
looking for curves given by the union of $d$ contact lines, all of them
passing through a fixed point. Let $X$ be one of those contact curve. Any
stable map $(C,f,p_{1},\ldots,p_{m})$ with dual graph $\tau$ whose image is
$X$ must have $m=d+3$ marked points (since we are imposing $d+3$ Schubert
conditions). By condition (5.3), the number of contracted vertices of degree
$0$ of $\tau$ is $d-2$. The inverse image under $f$ of the singular point of
$X$ must be connected. Hence, the degree $0$ vertices form a connected
subtree. Since we are not imposing any condition on the singular point, no
leaf is attached to a degree 0 vertex. So all leaves lie on $d$ non-contracted
components of degree $1$.

The dual graph $\tau$ with these properties is not univoquely determinated. An
example is $\tau_{1}$ in Figure 5.1 (we use the same notation of Example 5.5).

$1$$\\{1,2,3\\}$$0$$\\{4,5\\}$$1$$0$$1$$\\{6\\}$$0$$1$$\\{7\\}$$0$$1$$\\{d+2\\}$$1$$\\{d+3\\}$
Figure 5.1. Stable tree $\tau_{1}$.

Equation (5.2) applied to any of these stable trees is equal to $4$ (e.g., by
induction on $d$), so there are exactly $4$ maps having $\tau_{1}$ as dual
graph (Corollary 5.3). The image of $\tau_{1}$ only depends on the vertices
with $2$ and $3$ leaves. There are exactly $\binom{d+3}{3}\binom{d}{2}$
different dual graphs with the same support as $\tau_{1}$, but with different
leaves in those vertices.

If we consider another stable tree $\tau_{1}^{\prime}$ in Figure 5.2, this
gives different stable maps but with the same images as $\tau_{1}$.

$1$$\\{1,2,3\\}$$0$$\\{6\\}$$1$$0$$1$$\\{4,5\\}$$0$$1$$\\{7\\}$$0$$1$$\\{d+2\\}$$1$$\\{d+3\\}$
Figure 5.2. Stable tree $\tau_{1}^{\prime}$.

Since we are interested in the contact stable curves rather than in the stable
maps, we can ignore all other stable trees if they give the same image. There
is another family of graphs interesting to us, it is $\tau_{2}$ in Figure 5.3.

$1$$\\{1,2\\}$$0$$\\{3,4\\}$$1$$0$$1$$\\{5,6\\}$$0$$1$$\\{7\\}$$0$$1$$\\{d+2\\}$$1$$\\{d+3\\}$
Figure 5.3. Stable tree $\tau_{2}$.

Arguing like before, we get
$\frac{1}{3!}\binom{d+3}{2}\binom{d+1}{2}\binom{d-1}{2}$ different dual graphs
with different images, each one with a contribution equal to $8$. It is easy
to see that there are no more interesting stable trees, for several reasons.
Each non-contracted component must have at least one leaf by Remark 5.6, and
no more than three leaves (there are no contact lines through $4$ general
lines). Each degree zero vertex $v$ must have valence $n(v)=3$ by Eq. (3.4).
Moreover, it is not important the stable tree itself but how the $d+3$ leaves
are distribuited among the $d$ vertices of degree $1$. Hence $\tau_{1}$ and
$\tau_{2}$ are the only interesting configurations. It follows that the number
$N$ of curves we are looking for is

$\binom{d+3}{3}\binom{d}{2}\int_{\overline{\mathcal{S}(\tau_{1})}}\prod_{i=1}^{d+3}\mathrm{ev}_{i}^{*}(H^{2})+\frac{1}{3!}\binom{d+3}{2}\binom{d+1}{2}\binom{d-1}{2}\int_{\overline{\mathcal{S}(\tau_{2})}}\prod_{i=1}^{d+3}\mathrm{ev}_{i}^{*}(H^{2}).$

Using

$\int_{\overline{\mathcal{S}(\tau_{1})}}\prod_{i=1}^{d+3}\mathrm{ev}_{i}^{*}(H^{2})=4,\,\,\,\,\,\,\,\int_{\overline{\mathcal{S}(\tau_{2})}}\prod_{i=1}^{d+3}\mathrm{ev}_{i}^{*}(H^{2})=8,$

we get

$\displaystyle N$ $\displaystyle=$
$\displaystyle\binom{d+3}{3}\binom{d}{2}4+\frac{1}{3!}\binom{d+3}{2}\binom{d+1}{2}\binom{d-1}{2}8$
$\displaystyle=$ $\displaystyle 20d\binom{d+3}{5},$

as expected. ∎

###  5.1. Number of irreducible contact curves

We denote by $N_{d}^{irr}(a_{2},\ldots,a_{n})$ the number of irreducible
contact curves of degree $d$ meeting $a_{i}$ general linear subspaces of
codimension $i$ in $\mathbb{P}^{n}$. As application of Corollary 5.3, we give
tables containing such numbers for $n=3,5$ and low values of $d$. We subtract
from the number of contact curves given in [Mur20, MS22], the number of
reducible ones. Using Julia ([BEKS17]), we implemented Eq. (5.2) in [Mur22].
Csaba Schneider wrote the part based on the algorithm for generation of trees
of [WROM86].

######  Remark 5.8.

The number $N_{3}^{irr}(1,3)=3$ was first computed by [Kal18]. The number
$N_{3}^{irr}(7,0)=1080$ was first computed by [Amo14] using hand-done
computations of reducibles contact curves of degree $3$. He also claimed,
using the same strategy, that $N_{4}^{irr}(9,0)=378944$. Unfortunely, his
computation is not complete.

$d$ | $a=(a_{2},a_{3})$ | $N_{3}^{irr}(a)$ | $d$ | $a$ | $N_{4}^{irr}(a)$ | $d$ | $a$ | $N_{5}^{irr}(a)$
---|---|---|---|---|---|---|---|---
$3$ | $(7,0)$ | $1080$ | $4$ | $(9,0)$ | $145664$ | $5$ | $(11,0)$ | $65619360$
$(5,1)$ | $132$ | $(7,1)$ | $12800$ | $(9,1)$ | $4501008$
$(3,2)$ | $18$ | $(5,2)$ | $1216$ | $(7,2)$ | $328824$
$(1,3)$ | $3$ | $(3,3)$ | $128$ | $(5,3)$ | $25884$
|  |  | $(1,4)$ | $16$ | $(3,4)$ | $2250$
|  |  |  |  |  | $(1,5)$ | $225$
Table 5.1. Irreducible contact curves in $\mathbb{P}^{3}$ of degree $d\leq 5$. $a=(a_{2},a_{3},a_{4},a_{5})$ | $N_{2}^{irr}(a)$ | $a$ | $N_{2}^{irr}(a)$ | $a$ | $N_{2}^{irr}(a)$
---|---|---|---|---|---
$(11,0,0,0)$ | $27184$ | $(4,2,1,0)$ | $100$ | $(1,5,0,0)$ | $48$
$(9,1,0,0)$ | $7554$ | $(4,0,1,1)$ | $30$ | $(1,3,0,1)$ | $2$
$(8,0,1,0)$ | $1262$ | $(3,4,0,0)$ | $168$ | $(1,2,2,0)$ | $4$
$(7,2,0,0)$ | $2112$ | $(3,2,0,1)$ | $22$ | $(1,1,0,2)$ | $0$
$(7,0,0,1)$ | $432$ | $(3,1,2,0)$ | $16$ | $(1,0,2,1)$ | $0$
$(6,1,1,0)$ | $355$ | $(3,0,0,2)$ | $8$ | $(0,4,1,0)$ | $8$
$(5,3,0,0)$ | $594$ | $(2,3,1,0)$ | $28$ | $(0,2,1,1)$ | $0$
$(5,1,0,1)$ | $119$ | $(2,1,1,1)$ | $3$ | $(0,1,3,0)$ | $0$
$(5,0,2,0)$ | $58$ | $(2,0,3,0)$ | $2$ | $(0,0,1,2)$ | $0$
Table 5.2. Irreducible contact conics in $\mathbb{P}^{5}$.

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