# Edge Computing Assisted Autonomous Flight for UAV: Synergies between Vision
and Communications

Quan Chen12, Hai Zhu3, Lei Yang1, Xiaoqian Chen1, Sofie Pollin2, and Evgenii
Vinogradov2
1College of Aerospace Science and Engineering, National University of Defense
Technology
2Department of Electrical Engineering, KU Leuven
3Department of Cognitive Robotics, Delft University of Technology

###### Abstract

Autonomous flight for UAVs relies on visual information for avoiding obstacles
and ensuring a safe collision-free flight. In addition to visual clues, safe
UAVs often need connectivity with the ground station. In this paper, we study
the synergies between vision and communications for edge computing-enabled UAV
flight. By proposing a framework of Edge Computing Assisted Autonomous Flight
(ECAAF), we illustrate that vision and communications can interact with and
assist each other with the aid of edge computing and offloading, and further
speed up the UAV mission completion. ECAAF consists of three functionalities
that are discussed in detail: edge computing for 3D map acquisition, radio map
construction from the 3D map, and online trajectory planning. During ECAAF,
the interactions of communication capacity, video offloading, 3D map quality,
and channel state of the trajectory form a positive feedback loop. Simulation
results verify that the proposed method can improve mission performance by
enhancing connectivity. Finally, we conclude with some future research
directions.

##  I Introduction

With the advantages of high mobility, flexible deployment, and low cost,
unmanned aerial vehicles (UAVs), or drones, have drawn great attention in
commercial, civil, and military fields. Generally, UAVs are expected to
quickly complete their mission while guaranteeing autonomy and safety.
Autonomy and safety require good environmental awareness on the one hand, but
also reliable wireless connectivity for remote control, cooperation, and
aerial deconfliction on the other hand [1]. As both visual mapping and
wireless connectivity are a given, the main question we answer is how they
could strengthen each other synergistically with the aid of edge computing.

Current UAV-ground communications mostly adopt short-range local network
techniques such as WiFi [2]. Although easy to use, they have some limitations
in operating range and co-channel interference management [3]. Recently, with
the rapid development of cellular networks (LTE and 5G), cellular-connected
UAV has become a promising solution where the UAV connects with cellular base
stations (BSs) as an aerial user along with ground users [4]. Cellular
networks allow the UAV to maintain connectivity in a large range. The 5G
Ultra-Reliable Low-Latency Communication (URLLC) service also meets the
demands of low latency, real-time communication, and robust security.

In some visual UAV applications (e.g., surveillance, monitoring), the onboard
camera is the primary payload to perform the task. It also plays a key role in
supporting subsystems and is widely adopted by UAVs for autonomous navigation
and obstacle avoidance. From the captured video, it is possible to extract
auxiliary information such as the UAV position and 3D map features of the
explored areas [5]. With this information, the UAV is able to obtain awareness
of the environment around it and autonomously adjust its trajectory. Besides,
the UAV movement is also affected by the information update rate: a higher
flying speed requires a faster update rate, which poses a challenge for the
onboard processing capability. However, in the upcoming 5G era, Mobile Edge
Computing (MEC) enables powerful computing capability at the edge of cellular
networks [4]. Similar to autonomous driving, the onboard computational tasks
can be offloaded to the ground and processed remotely, which speeds up the
information acquisition for UAV [6].

On the other hand, the explored environmental information can be exploited to
improve communication performance. Specifically, the 3D map enables the UAV to
be aware of the obstacles between the BS and UAV that influence the signal
propagation. The 3D map can be generated from the captured video by vision-
processing algorithms such as simultaneous localization and mapping (SLAM)
[5]. The key idea in this paper is to leverage the captured video to
incrementally construct a 3D map of the environment for the UAV and predict
the wireless link states around it; then the UAV adaptively designs its real-
time trajectory, trying to maintain Line-of-Sight (LoS) link with the BS. The
UAV can offload its computing tasks to the edge computing server for faster
video processing and map acquisition. Since typically the speed of a vision-
navigated UAV is limited by the map update rate, edge computing also enhances
UAV mobility and makes the trajectory more flexible. Moreover, edge computing
relies on the offloading link capacity. Thus, in turn, a good link guarantees
high-quality mapping and UAV mobility. By the positive interaction of visual
information and communications, both of them can be enhanced.

Inspired by the above idea, in this paper, we propose a framework of Edge
Computing Assisted Autonomous Flight (ECAAF) for UAV flying in urban areas.
The contributions can be summarized as follows: i) Focusing on the synergies
between vision and communications, we propose a solution to realize real-time
path planning for the UAV in urban areas without known maps. ii) By ECAAF, we
illustrate that vision and communications can interact with and assist each
other with the aid of edge computing and task offloading. The framework
architecture and trade-offs are thoroughly discussed. iii) We offer a detailed
implementation for ECAAF solution and verify it by a case study.

##  II Communications and vision for UAV flight

This section presents the communication and vision related approaches to
improve UAV flight performance. We discuss two specific techniques that are
typically considered in isolation: communication-aware, and vision-assisted
trajectory planning. For autonomous flight, communication awareness requires
the UAV to sense or predict its link states and keep the connectivity
reliable. And vision awareness obtained from SLAM enables the UAV to construct
and update the 3D map of its environment using its onboard sensors, so that it
can adaptively adjust its trajectory.

###  II-A Communication-aware flight

In order to maintain a good connection with the BS, the UAV trajectory or
placement can be deliberately designed with communication awareness. There are
several approaches to enhance communication awareness, such as LoS-based
channel modeling, channel measurement-based estimation, and map-assisted
channel estimation [4].

For the channel modeling, unexpected obstacles are the primary cause of the
degradation and uncertainty of air-to-ground link, especially in urban areas.
Due to the obstacles, the link state can be either LoS or NLoS (Non-LoS). The
latter results in much lower received signal quality. To estimate the channel
gain, a widely adopted method is to approximate link path loss (i.e., signal
attenuation in the channel) by estimating the LoS occurrence probability which
depends on the UAV-ground elevation angle [7]. Some studies adopt this model
to define the UAV altitude for maximal coverage [8], and to optimize the UAV
trajectory for maximal data collection in a UAV-enabled wireless sensor
network (WSN) application [7]. The statistical channel models are general and
provide good insights to the network designers. However, the LoS-probability-
based method does not apply to real-time flights in obstacle-dense regions.
One improved method is to combine the offline LoS modeling and online channel
measurement-based schemes [9]: firstly, an offline path is optimized using the
probabilistic LoS model; next, the UAV adaptively adjusts its speed and
communication scheduling along the offline path based on the instantaneous
channel state information (CSI). Although the leaning-based CSI approach can
infer the LoS state from the variation pattern of CSI over time, this approach
is mainly effective at the measured moments and locations, which limits its
application in large and unexplored regions. If learning approach is further
adopted to predict the CSI map of unexplored regions, it requires massive
offline training data and dedicated training model [9], which might be
unavailable in a new environment.

Another way to estimate the channel gain is by exploiting the real 3D map
instead of the above two approaches. The map-based LoS/NLoS classification
works for larger regions than the CSI approach, and it is more reliable for
locations where the UAV has never been. Once the 3D map is given, the link
state (LoS or NLoS) between any two locations can be accurately inferred [4].
This map-aided method has been implemented in several UAV applications. For
example, in [10], by combining the 3D map and a path loss model to generate
the coverage map, the shortest trajectory is designed under cellular
connectivity constraints. Note that the premise of this method is to obtain a
priori 3D map of the flight-related regions, which may be not available or
reliable in practice.

###  II-B Vision-assisted flight

Vision is another type of important supporting information for UAV flight that
can be captured by onboard vision-sensors (e.g., depth camera). To extract
valid information from the captured video, SLAM is a widely adopted technique
where the UAV is capable of online map building, while simultaneously
utilizing the generated map to estimate and correct errors in the navigation
solution obtained [11]. When video frames are captured, they are first pre-
processed, e.g., pose feature extraction, pose estimation, and keyframe
selection. Then based on the keyframes, visual tracking and dense map fusion
are executed. Finally, the updated 3D map can be obtained after a loop
correction [12]. With the map information, the UAV can not only recognize the
obstacles but also plan its trajectory to avoid them. Therefore, SLAM enables
the UAV to autonomously and safely fly in map-unavailable areas and even GPS-
denied areas [5].

However, real-time SLAM requires a huge amount of computations, especially
when the vehicle is moving at high speed which requires a high SLAM update
rate. This requirement quickly exceeds the limited onboard computing power of
UAV and limits its flying speed and flexibility. A potential solution is to
offload the computational task to the ground server and send the obtained SLAM
results back to the UAV. This mode is known as cloud SLAM that has been
studied in some practical applications [12]. Robo-Earth, as a representative
project, has successfully demonstrated some cloud SLAM cases [13] in practice.
Note that here the ’cloud’ concept only refers to non-local processing, which
is not identical to that in ’cloud computing.’ In cellular networks, cloud
SLAM requires extra computing capability at BS side, which just matches the
typical scenario of the upcoming 5G with MEC [4].

Figure 1: Edge computing assisted autonomous flight (ECAAF). The UAV offloads
its captured video to the Edge Server and receives the processed 3D map. The
awareness of the local environment helps the UAV maintain a LoS link with BS
by online trajectory planning.

##  III ECAAF framework

According to the above introduction, both awareness of communication and
vision can promote UAV mission performance. In this section, by combining the
two approaches and enabling SLAM offloading, we propose a framework of edge
computing assisted autonomous flight (ECAAF) to interactively improve SLAM and
communication performances.

###  III-A Features of ECAAF: vision and communication synergies

Figure 2: Positive interaction loop of vision and communications in ECAAF.
Better communication improves environment perception, which in turn boosts
communication performance.

As illustrated in Fig. 1, a cellular-connected UAV keeps communication with a
cellular BS during the whole flight. The UAV captures real-time video and can
offload it to the BS. The BS is connected with an Edge Server (ES) where the
offloaded SLAM task is processed. Next, the obtained SLAM results are sent
back to the UAV for map construction and online trajectory planning. The goal
is to design a real-time trajectory from Starting point to Destination with i)
minimum duration; and ii) less time and distance where the UAV-BS link is NLoS
(i.e., unreliable channel).

During the ECAAF, visual perception and communications interact and promote
each other, as shown in Fig. 2. A higher-capacity link allows high-speed video
transmission between UAV and BS, which improves the 3D map quality and update
rate. Then the UAV obtains a better awareness of its wireless environment. It
can also fly faster with a higher update rate of mapping and localization.
Accordingly, the UAV can more accurately plan its real-time trajectory with
better LoS states, which in turn promotes the communication capacity and
reliability. Therefore, a positive interactive loop is formed where better
communication improves environment perception, which meanwhile boosts
communication performance.

###  III-B Architecture of ECAAF

Figure 3: ECAAF architecture. The captured video is offloaded to cellular BS
and processed at ES. Then the obtained SLAM results are sent back to the UAV
and help it perceive its wireless environment and plan its real-time
trajectory. The data flow forms a closed loop.

Fig. 3 shows the architecture of ECAAF, consisting of three parts:

i) Video transmission and processing: When the raw video is captured, it is
compressed and offloaded to the UAV’s serving BS through uplink. Then, the map
features are extracted at the ES side by the SLAM algorithm.

ii) Map construction: The obtained map features and localization results are
sent back to the UAV via downlink. Then the UAV merges new map features into
the existing local 3D map and transforms it into a radio map (a distribution
of uplink channel gain) by classifying the link state of neighboring
locations.

iii) Trajectory planning: From the map, the UAV knows which directions are
promising to achieve higher link capacity. It accordingly plans its following
trajectory as well as the offloading decisions (i.e., whether the video is
processed locally or remotely). When UAV moves to a new position, the incoming
video is processed using the same process.

From the communication aspect, the 3D map acquisition relies on both downlink
and uplink. The raw video offloading is through uplink and the generated map
features transmission is through downlink. The uplink determines the
offloading rate and thus affects the remote processing rate, while the
requirements for downlink are lower because the returned map features are much
less data. The downlink also transmits command and control signals from the
ground mission control station for mission control and necessary intervention.

The ECAAF implementation has the following requirements which have been
separately realized in practical cases [4, 11, 13, 12]: i) cellular network
connectivity; ii) an onboard visual sensor (monocular or stereo camera) to
capture video; iii) an ES and onboard computer with SLAM solving capability.

##  IV Implementation of ECAAF

This section concentrates on the specific technical details of ECAAF solution:
radio map construction, flying speed adjustment, and online trajectory
planning. The details of SLAM algorithms can be found in [11] and are not
described here.

###  IV-A Radio map construction

Upon receiving the new 3D map features from the ES, the UAV merges them into
the local map and updates its stored map, which contains the building height
data of the explored areas. Then the UAV classifies the link state of its
neighboring locations and then transforms the obstacle map into a radio map.

Figure 4: Map transformation from the 3D map to a radio map. The radio map
refers to the distribution of channel gain, which is affected by the
obstacles.

Link state classification: Knowing the BS location and UAV current location,
the UAV-BS direct line can be calculated. The UAV judges whether the line
intersects with obstacles in its explored map and classifies the link state as
LoS (no intersection) or NLoS (intersection occurs), which is marked by a
state indicator.

On the other hand, the BS can infer the LoS/NLoS states of the UAV position by
analyzing the variation pattern of the measured CSI. Then the obtained
LoS/NLoS information is sent to the UAV and helps the UAV to correct the map-
based link state classification. Because the complete map is unknown to the
UAV, there may be unexplored regions lacking map information along the UAV-BS
line. In this case, those map-missing parts are first assumed to be LoS and
then corrected by the CSI-based classification. If the link states of some
directions have been marked as NLoS earlier, they keep the NLoS state without
further estimation.

Map transformation: After obtaining the link state of current location and
surroundings, the UAV calculates the path loss and channel gain at each
estimated location and saves it to a radio map. The radio map is represented
by a 3-dimensional table which saves the channel gain corresponding to each
location. The locations are defined in a BS-centered coordinate system. When
UAV explores new space, the channel gain is calculated and added to the table.
In this way, the 3D vision map is transformed into a 3D radio map. Fig. 4
gives a diagram of the map transformation where a horizontal 2D slice of radio
map is given for simplicity.

###  IV-B Flying speed adjustment

Because of the requirements of control accuracy and flight safety, UAV speed
should be kept within a safe range, which depends on the offloading link
capacity. In a simplified model, let the maximum speed be proportional to the
mapping & localization update rate, which can be viewed as the frame rate of
SLAM processing and is limited by offloading transmission (remote processing
time is ignored). The achievable frame rate is determined by link capacity and
average data size per frame. Thus, the speed limit also depends on these two
values. For example, assume the required SLAM information for safe movement is
2 frame/m and the file size of each frame is 1 Mbit. When the offloading link
capacity is 10 Mbps; If the transmission delay is solely considered, the
achievable frame update rate is 10 frame/s, and the UAV speed can achieve a
maximum 5 m/s. Thus, the UAV should adaptively adjust its speed of the
following trajectory based on the link capacity and safe flight requirements.
Note that the remote processing mode also causes additional delay for the
results feedback loop and local action decision, which will lower the map
update rate. The additional delay includes processing delay and
uplink/downlink data transmission delay. If certain downlink message is
missing, it also lowers the map update rate as well as the speed limitation.
When the link state is bad and UAV switches to local processing mode, the
speed limitation depends on the local computing performance.

###  IV-C Online trajectory planning

At each moment, with the incrementally constructed environment map and the
corresponding transformed radio map, a local LoS-aware and collision-free
trajectory is planned online for the UAV using model predictive control (MPC)
[14]. MPC has been widely used for online trajectory generation of aerial
vehicles [15]. It formulates a constrained optimization problem with a local
short planning horizon, in which the LoS-dependent speed constraints and the
environment-dependent obstacle avoidance constraints are imposed. The optimal
local trajectory is then obtained by minimizing the flying time of the
trajectory while satisfying the imposed constraints. The above formulated
constrained optimization problem is solved at each moment in real-time, with
updated environment map and radio map, until the UAV reaches its destination.

##  V Case study

In this section, by comparison, we give a representative case to illustrate
the performance improvement of the proposed ECAAF approach, then present the
statistical results of multiple tests.

###  V-A Scenario settings

Figure 5: Trajectories of ECAAF (left solid line) and baseline without LoS
awareness (right dashed line). Blue and red color indicate LoS and NLoS parts,
respectively. The BS with blue star represents the associated BS. Gray boxes
are buildings.

We consider a 400$\times$400$\text{m}^{2}$ city map with dense Manhattan-like
building distribution. The building height follows the Rayleigh distribution
[8] with a scale parameter of 35 m. The distance between the start and end
points is 320m. The maximum flying speed and altitude of UAV are 15 m/s and
50m, respectively. Three BSs are deployed in the scenario and the UAV
maintains connection with one BS as shown in Fig. 5.

The BS height is 25m. The UAV antenna adopts a directional pattern as in [4]
and the transmit power is 30 dBm. Other detailed settings and parameters can
be found in the simulation setup in [2]. The required SLAM update rate for
safe movement is 2 frame/m. The offloading model and SLAM processing settings
refer to [6], and the camera field of view is set as 120 deg. In the online
trajectory planning phase, the planning horizon is set as 10 seconds, and the
algorithm parameters refer to [15]. The objective is to minimize the flight
duration as well as NLoS duration with the constraints stated in Section III.

For comparison, a baseline trajectory without LoS awareness and an optimal
trajectory (denoted as GlobeMap) with global environmental awareness are
solved. In the baseline, the link capacity is estimated using the
probabilistic LoS model [8], and an extra obstacle avoidance mechanism [15] is
added to the trajectory. The baseline only uses the local map to avoid
obstacles but not estimates LoS states. In the GlobeMap case, the trajectory
is obtained under a known global map for both LoS awareness and obstacle
avoidance. The baseline, ECAAF, and GlobeMap correspond to the without map,
with self-explored local map, and with global map cases, respectively.

TABLE I: Comparison results of ECAAF, baseline and GlobeMap in 20 tests. | Total flight distance | Total flight duration | Average uplink capacity | NLoS distance ratio
---|---|---|---|---
Baseline | 6.1 km | 971 s | 14.8 Mbps | 49.2 %
ECAAF | 6.5 km | 755 s | 16.3 Mbps | 27.3 %
GlobeMap | 6.4 km | 742 s | 16.4 Mbps | 26.8 %

###  V-B Results

Fig. 5 illustrates an ECAAF trajectory compared with the baseline where the
LoS and NLoS parts are marked on the trajectories. The trajectory of GlobeMap
is much close to ECAAF trajectory and thus not drawn. Compared to the
baseline, with the environmental awareness, the ECAAF trajectory avoids many
obstacles between the UAV and BS. Thus a higher link capacity and faster speed
are achieved. The flight distance of the baseline, ECAAF, and GlobeMap is
373m, 402m, and 397m, while the flight duration is respectively 74.5s, 48.3s,
and 46.8s. Besides, the average uplink capacity of the three planned
trajectories is respectively 13.6 Mbps, 16.7 Mbps, and 16.8 Mbps. In this
example, the performance of the proposed ECAAF is much promoted compared to
the conventional non-map-aware method, and close to the global-map-aware
method. With the local map and LoS awareness, the UAV manages to fly in good-
channel regions and achieves higher link capacity. Although the detour causes
some extra journeys, the UAV can fly faster with the improved link capacity,
thus the total flight duration is significantly reduced. Because the locations
of non-associated BSs are not known to the UAV, the UAV does not proactively
predict the interference. When it flies towards the non-associated BS, the
degraded downlink leads to extra delays and a lower flying speed.

We also further repeat the simulation for 20 times with different start and
end points and with the straight-line distance ranging from 200 to 400 m.
Before each test, a new map is initialized. Table I lists the comparison
results of the average performance metrics of the 20 runs between the three
types of trajectory. In the table, the NLoS distance ratio is the percentage
of NLoS path length in the total flight path. When the UAV falls into NLoS
link, the interference shows more significant impacts on the transmission
delay and the UAV tends to switch to local processing mode. It can be seen
from the table that our proposed ECAAF can achieve faster flying and higher
overall uplink capacity while keeping a lower NLoS distance radio compared
with the baseline where LoS information is not used in trajectory planning.
Besides, although the environmental awareness is local, the performance of
ECAAF is close to the GlobeMap approach. In some cases, when both ECAAF and
GlobeMap maintain LoS with the associated BS, the trajectory in GlobeMap
deflects away from the non-associated BSs to mitigate the downlink
interference. Note that in the randomly generated scenarios, sometimes the
three approaches can obtain all-LoS trajectories, which causes little
difference between their performances and lowers the averaged promotion.

##  VI Potential trade-offs and Future Work

The performance of ECAAF framework will be affected by many factors in
practice. In this section, we list some potential trade-offs that can be
explored in order to achieve even better performance than the current version
of the ECAAF framework. We also present future research directions.

###  VI-A Potential trade-offs

Local processing vs. cloud processing: As an autonomous system, the UAV also
has local computation capability [11]. Since a higher map update rate relaxes
the UAV velocity limitation, in the proposed framework, the UAV always selects
the processing mode with the higher achievable map update rate. If UAV falls
into a bad communication state, the offloading-constrained cloud processing
may be even worse than local processing. Then the UAV stops offloading and
switches to local processing mode. Compared with cloud SLAM, the local SLAM
algorithm is simpler and has inferior map quality, but it avoids the delay
overhead of uplink/downlink transmission.

UAV altitude and map accuracy: UAVs may tend to fly higher for better LoS
probability, but their altitude is constrained by path loss, regulations, as
well as the map accuracy. A higher-altitude means a lower resolution of the
explored map, and thus the radio map estimation accuracy and trajectory
planning are also affected.

Multi-BS interference mitigation: In the multi-BS scenario, interference from
non-associated BSs also affects the UAV downlink channel quality. The UAV can
change its trajectory to avoid the non-associated BSs. But in some cases, the
UAV needs to make a trade-off between NLoS link and link with strong
interference. On the other hand, in the uplink communication the UAV could
also cause interference to those non-associated BSs. If the uplink capacity is
sufficient, the UAV can flexibly adjust transmit power to mitigate the
interference. Moreover, UAVs can also exploit the obstacles as barriers to
mitigate interference by dedicated trajectory design. When interference
exists, the environment and ECAAF bring new challenges and opportunities for
UAV communications.

Energy consumption: When LoS links are achieved, a trajectory with detour may
consume extra flying energy. But the good channel also improves the UAV motion
flexibility, so that the UAV can fly at an optimal energy-efficient speed for
energy saving [4]. In an energy-constrained scenario, the path and velocity
should be jointly designed.

###  VI-B Future research directions

Multi-BS handover and cooperation: In a multi-BS scenario, the UAV can switch
its associated BS for optimal communication during its flight. Thus when
designing the trajectory, the UAV should also schedule the handover between
different BSs to achieve better LoS channel and avoid interference. On the
other hand, the UAV can also benefit from the multi-BS cooperation. The
communication resources are jointly optimized across different BSs to avoid
co-channel interference and improve link reliability. The multi-BS handover
and cooperation lead to new issues such as BS selection, handover scheme, map
sharing and migration, and trade-offs between performance and backhaul
overhead.

Map reuse: In this work, we consider the UAV always flying over an unknown
environment and thereby always relying on video processing. In some complex
missions, the UAV is likely to fly by some explored areas. Then the obtained
map can be reused so that the video processing is avoided and the UAV mobility
is not limited by the uplink. In these cases, the trajectory and offloading
decision can be jointly planned.

##  VII Conclusion

In this article, a framework of edge computing assisted autonomous flight is
proposed to achieve high-capacity and reliable UAV-ground communications. With
an onboard camera and cloud SLAM, the UAV is able to explore and exploit the
environment and adaptively plan its real-time trajectory for better channel
states, which improves the link reliability. The interaction of vision and
communications boosts both of them. Finally, the case study verifies the
proposed approach. ECAAF is a complex mission in practice. A number of
practical factors and trade-offs are worthy of further investigation.

##  VIII Acknowledgments

This research is supported by the Research Foundation Flanders (FWO), project
no. S003817N (OmniDrone) and the National Natural Science Foundation of China
no. 11725211. Q. Chen appreciates the financial aid from China Scholarship
Council. Q. Chen and H. Zhu contributed equally to this work.

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## Biography

Quan Chen received the B.E. (2015) and M.E. (2017) degrees from National
University of Defense Technology (NUDT), China, where he is currently pursuing
the PhD degree. He is also a visiting scholar with the Department of
Electrical Engineering, KU Leuven, Belgium. His research interests include UAV
networks, mega-constellation networks, and integrated space-terrestrial
networks.

Hai Zhu is a PhD student at the Department of Cognitive Robotics at the Delft
University of Technology. He received the B.E. (2015) and M.Sc. (2017) degrees
from NUDT. His current research focuses on motion planning for multi-robot
systems with an emphasis on real-time algorithms for aerial vehicles and
planning under uncertainty. He received the IEEE ICRA best paper award on
multi-robot systems in 2019.

Lei Yang received his Ph.D. degree from NUDT, China, in 2008. He is currently
a professor at NUDT. His current research interests are focused on space
communication networks, onboard computer, spacecraft system modeling and
simulation.

Xiaoqian Chen received his Ph.D. degree from NUDT, China, in 2001. He is a
professor at NUDT and National Innovation Institute of Defense Technology. His
current research interests include spacecraft systems engineering, advanced
digital design methods of space systems, and multidisciplinary design
optimization.

Sofie Pollin is an associate professor with the Electrical Engineering
Department, KU Leuven, Belgium. Her research centers around networked systems,
which require networks that are ever more dense, heterogeneous, battery
powered, and spectrum constrained.

Evgenii Vinogradov is with the Department of Electrical Engineering, KU
Leuven, Belgium, where he works on wireless communications with unmanned
aerial vehicles (UAVs) and UAV detection. His research interests include UAV
communications and multidimensional radio propagation channel modeling.


# The Coble-Mukai lattice from ${\mathbb{Q}}$-Gorenstein deformations

Giancarlo Urzúa<EMAIL_ADDRESS>Facultad de Matemáticas, Pontificia
Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna
4860, Santiago, Chile.

###### Abstract.

We show some geometric properties of Enriques surfaces via
${\mathbb{Q}}$-Gorenstein smoothings of Coble surfaces. In particular, we
explicitly identify the Enriques lattice of the general fiber with the Coble-
Mukai lattice. At the end, we discuss applications to Gorenstein
${\mathbb{Q}}$-Homology projective planes with trivial canonical class.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Class and Picard groups
  4. 4 Coble-Mukai lattice and Enriques lattice
  5. 5 Explicit computations for the first degeneration
  6. 6 Further degenerations and ${\mathbb{Q}}$-Homology projective planes

##  1\. Introduction

Our ground field is ${\mathbb{C}}$. An Enriques surface is a nonsingular
projective surface $Z$ with $q(Z)=h^{1}({\mathcal{O}}_{Z})=0$,
$p_{g}(Z)=h^{2}({\mathcal{O}}_{Z})=0$, and $2K_{Z}\sim 0$. (An excellent
survey on Enriques surfaces is [Do16], our main reference will be the two
volumes book [CDL23, DK23].) Degenerations of Enriques surfaces have been
studied in e.g. [Ku77], [M81], [P77], [K92], [CM94]. We are interested in
${\mathbb{Q}}$-Gorenstein degenerations (called moderate degenerations in
[K92]).

An Enriques W-surface is a normal projective surface $X$ together with a
proper deformation $(X\subset{\mathcal{X}})\to(0\in{\mathbb{D}})$, where
${\mathbb{D}}$ is a smooth curve germ, such that

  1. (1)

$X$ has at most Wahl singularities, this is, cyclic quotient singularities of
type $\frac{1}{n^{2}}(1,na-1)$ with gcd$(a,n)=1$,

  2. (2)

${\mathcal{X}}$ is a normal $3$-fold with $K_{{\mathcal{X}}}$
${\mathbb{Q}}$-Cartier,

  3. (3)

the fiber $X_{0}$ is reduced and isomorphic to $X$,

  4. (4)

the fiber $X_{t}$ is an Enrique surface for $t\neq 0$.

(For more general W-surfaces and their birational geometry see [U16a].)
Kawamata proves that the monodromy for this type of degenerations is trivial
[K92, Section 2], and that an Enriques W-surface $X$ with $K_{X}$ nef can have
only singularities of type $\frac{1}{4}(1,1)$ [K92, Theorem 4.1] (i.e. these
are flower pot degenerations [P77]). In addition, if $X$ is indeed singular,
then it must be rational. Let $\phi\colon V\to X$ be the minimal resolution,
and let $\\{C_{1},\ldots,C_{s}\\}$ be the disjoint exceptional $(-4)$-curves.
Then

$-2K_{V}\sim C_{1}+\ldots+C_{s},$

and so $V$ is a Coble surface of K3 type [DK23, Section 9.1]. By [DK23,
Corollary 9.1.5], we have that $s\leq 10$.

For any given Enriques W-surface $X$, the purpose of this note is to directly
compute some geometric properties of the Enriques surfaces $X_{t}$. For
example, this point of view gives a direct way to identify the Coble-Mukai
lattice of $V$ with the Picard group of $X_{t}$ modulo canonical class (see
Theorem 4.2). The Coble-Mukai lattice was introduced by Mukai with no
reference to a degeneration of Enriques surfaces. In [DK23, Theorem 9.2.15] it
is proved that the Coble-Mukai lattice is isomorphic to the Enriques lattice,
but there are no degenerations of Enriques surfaces involved either. We also
explicitly describe $\operatorname{Pic}(X_{t})$ from classes of curves in the
singular surface $X$. As an application, we discuss nodal Enriques surfaces
from Coble surfaces, and in particular some open questions on
${\mathbb{Q}}$-homology projective planes via Enriques W-surfaces.

### Acknowledgments

The author would like to thank Igor Dolgachev for the motivation to write this
note, and for various useful comments and references. The author also thanks
Matthias Schütt for his interest and useful comments. The author is grateful
to Phillip Griffiths and Carlos Simpson for the invitation to talk at the
“Moduli and Hodge theory” online conference at the Institute of the
Mathematical Sciences of the Americas (IMSA) during February 1-5 2021, and for
the invitation to write a paper for the corresponding proceedings. The author
would like to thank the anonymous referee for interesting and useful
observations. The author was supported by the FONDECYT regular grant 1230065.

##  2\. Preliminaries

Let $X$ be an Enriques W-surface with $K_{X}$ nef. Let

$\phi\colon V\to X$

be the minimal resolution of $X$. As it was said in the introduction, the
surface $V$ must be a Coble surface of K3 type. In this way, we have two
options for $V$ (see [DK23, Prop. 9.1.3]): it is the blow-up of either:

  * 1.

an Halphen surface of index two 111An Halphen surface of index two is a
rational elliptic surface with a multiplicity two fiber, and no $(-1)$-curves
in the fibers. over the singularities of one reduced fiber, which is of type
$II$, $III$, $IV$, or $I_{n}$, of , or

  * 2.

a Jacobian rational minimal elliptic fibration over the singularities of two
reduced fibers of type $II$, $III$, $IV$, or $I_{n}$.

The blow-ups are as follows. Over a type $II$ fiber we only blow-up the cusp
to obtain one $(-4)$-curve. For type $III$ we blow-up twice over the tangent
point, and so we obtain two $(-4)$-curves over the fiber. In the case $IV$ we
blow-up $4$ times over the triple point, so that we obtain four $(-4)$-curves.
Finally, for any $I_{n}$ fiber we blow-up the $n$ nodes, so that we obtain $n$
$(-4)$-curves.

Let $\pi\colon V\to S$ be the composition of the blow-ups. We obtain the
boundary curves $\\{C_{1},\ldots,C_{s}\\}$ in $V$ from the distinguished
singular fiber(s). We recall that $C_{i}\simeq{\mathbb{P}}^{1}$ and
$C_{i}^{2}=-4$, i.e. they are the $(-4)$-curves over the fibers. The
contraction of the boundary curves gives the surface $X$.

We will only consider the case when the singular fibers are of type $I_{n}$,
the other cases are degenerations. Hence for one singular fiber we have
$I_{s}$, and for two we have $I_{s_{1}}$ and $I_{s_{2}}$ with $s_{1}+s_{2}=s$.
We have that $1\leq s\leq 10$ by [DK23, Corollary 9.1.4].

One can actually prove that such an $X$ has no local-to-global obstructions to
deform, and that any ${\mathbb{Q}}$-Gorenstein smoothing of it is an Enriques
surface (see [U16b, Theorem 4.2(0)], where there is a result for more general
elliptic fibrations). The ${\mathbb{Q}}$-Gorenstein deformation space of $X$
has dimension $10$ (see [H16, Section 3]).

######  Example 2.1 (Blow-ups of Enriques W-surfaces).

As above, consider an elliptic fibration $S\to{\mathbb{P}}^{1}$ with sections
and at least two $I_{1}$ fibers $F_{1}$ and $F_{2}$. Let $V\to S$ be the blow-
up of both nodes. We now blow-up over a general point in, say, $F_{2}$ to
obtain the chain $[6,2,2]$ with a $(-1)$-curve intersecting the middle
$(-2)$-curve, where the $(-6)$-curve is the proper transform of $F_{2}$ (see
Figure 1). We contract $[4]$ and $[6,2,2]$ to obtain a surface $Z$. One can
prove that $Z$ has no local-to-global obstructions to deform. We consider a
${\mathbb{Q}}$-Gorenstein smoothing of $Z$. One can show that the general
fiber is the blow-up of an Enriques surface at one point. We note that $K_{Z}$
is not nef because the image $A$ of the $(-1)$-curve intersecting the middle
$(-2)$-curve is a negative curve for $K_{Z}$. There is a divisorial
contraction of the family which contracts the $(-1)$-curves in the general
fibers and $A$ in $Z$, and the resulting surface is an Enriques W-surface.

Figure 1. The Wahl chains $[4]$ and $[6,2,2]$

These blow-ups of Enriques W-surfaces come from the divisorial contraction
universal family [HTU13] defined by $\frac{1}{4}(1,1)$. This is explained in
[U16b, Section 2], precisely this particular divisorial contraction family is
[U16b, Example 2.13], where the combinatorial data with all the infinitely
many possible Wahl singularities is given by

$[4]-[2,\bar{2},6]-[2,2,2,\bar{2},8]-[2,2,2,2,2,\bar{2},10]-\cdots.$

For example, in Figure 2 we have a way to produce a similar situation but with
three Wahl singularities, where the $(-1)$-curve in the general fiber (blow-up
of an Enriques surface at one point) degenerates into the image of the
$(-1)$-curve between $[8,2,2,2,2]$ and $[6,2,2]$.

Figure 2. The Wahl chains $[4]$, $[6,2,2]$, and $[8,2,2,2,2]$

######  Example 2.2.

In [K92, Example 4.5], Kawamata shows the local picture of a
${\mathbb{Q}}$-Gorenstein smoothing of an elliptic fibration with a singular
$I_{d}$ fiber with Wahl singularities $\frac{1}{n^{2}}(1,na-1)$ at the nodes.
Kawamata explains how these local elliptic surface ${\mathbb{Q}}$-Gorenstein
smoothens into other elliptic surfaces, and the degeneration is explained as
the concurrence of a smooth fibre with multiplicity $n$ with $d$ $I_{1}$
fibers.

Let $\beta_{i}$ be the class of $C_{i}$ in $\operatorname{Pic}(V)$. As in
[DK23, Chapter 9], we define the Coble-Mukai lattice of $V$ as

$\operatorname{CM}(V):=\\{x\in\widetilde{\operatorname{Pic}}(V)\colon
x\cdot\beta_{i}=0\ \text{for all}\ i\\}$

where $\widetilde{\operatorname{Pic}}(V)$ is the lattice in
$\operatorname{Pic}(V)_{{\mathbb{Q}}}$ generated by $\operatorname{Pic}(V)$
and the rational classes $\frac{1}{2}\beta_{i}$.

##  3\. Class and Picard groups

Let $X$ be an Enriques W-surface with $K_{X}$ nef. We only consider Coble
surfaces $V$ of K3 type from a rational minimal elliptic fibration
$S\to{\mathbb{P}}^{1}$ with sections. Hence we have blow-ups at the nodes of
two singular fibers $I_{s_{1}}$, $I_{s_{2}}$ with $s_{1}+s_{2}=s$. We have the
diagram

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\textstyle{S}$$\textstyle{X}$

where the morphism $\phi$ is the minimal resolution of $X$, and $\pi$ is a
composition of blow ups from $S$. The $(-4)$-curves $\\{C_{1},\ldots,C_{s}\\}$
are the exceptional curves of $\phi$. We choose in $V$ a chain of smooth
rational curves

$C_{1}-(-1)-C_{2}-(-1)-\ldots-(-1)-C_{s}$

using one section from $S\to{\mathbb{P}}^{1}$, and $(-1)$-curves from the
blow-up $\pi\colon V\to S$. The contraction of that chain defines a surface
$W$ with one singularity of type $\frac{1}{4s}(1,2s-1)$, whose Hirzebruch-Jung
continued fraction is $[3,2,\ldots,2,3]$ (with $(s-2)$ $2$’s). Hence we have a
contraction $\sigma\colon X\to W$, and $W$ has no local-to-global obstructions
to deform (same proof as for $X$).

We now look at ${\mathbb{Q}}$-Gorenstein deformations of $W$, and then we will
do it for $X$. All deformations below happen over ${\mathbb{D}}$, and
${\mathbb{Q}}$-Gorenstein deformation means that the canonical class for the
corresponding $3$-fold is ${\mathbb{Q}}$-Cartier.

######  Proposition 3.1.

The surface $W$ can be ${\mathbb{Q}}$-Gorenstein deformed into a surface
$W_{t}$ with singularities of types either $A_{e_{1}-1}$, $\ldots$,
$A_{e_{r}-1}$ or $\frac{1}{4e_{1}}(1,2e_{1}-1),A_{e_{2}-1}$, $\ldots$,
$A_{e_{r}-1}$, where $e_{1}$, $\ldots$, $e_{r}$ is a partition of $s$. In the
case when we have only singularities of type $A_{e_{i}-1}$, the minimal
resolution of $W_{t}$ is an Enriques surface.

###### Proof.

We know that there are no local-to-global obstructions to deform $W$. The rest
is [HP10, Proposition 2.3]. ∎

The partial resolution $\sigma\colon X\to W$ is locally an M-resolution of the
singularity in $W$ (cf. [BC94]). Any ${\mathbb{Q}}$-Gorenstein deformation of
the singularity in $W$ is the blowing-down deformation (see [KM92, (11.4)]) of
a ${\mathbb{Q}}$-Gorenstein deformation of the exceptional divisor of
$\sigma$. The deformations in Proposition 3.1 into singular surfaces happen
over a proper analytic closed set of the deformation space of the singularity
[HP10], and so for almost all ${\mathbb{Q}}$-Gorenstein smoothings of $X$, the
blowing-down deformation gives an isomorphism between general fibers.

######  Lemma 3.2.

Let $(W\subset\mathcal{W})\to(0\in{\mathbb{D}})$ be a
${\mathbb{Q}}$-Gorenstein smoothing. Let $P$ be the singularity
$\frac{1}{4s}(1,2s-1)$ in $W$. Let $M_{P}$ be the Milnor fiber corresponding
to the induced local smoothing of $P$. Then, there is an exact sequence

$0\to H_{2}(M_{P})\simeq{\mathbb{Z}}^{s-1}\to H_{2}(W_{t})\to H_{2}(W)\to 0,$

where $W_{t}$ is a smooth fiber (and so an Enriques surface).

###### Proof.

Let $W^{o}=W\setminus\\{P\\}$. By [Ko05, Prop.4.2(3)], we have $H_{1}(W^{o})$
is torsion, so $H^{1}(W^{o})=0$. By [Ko05, Prop.4.2(1)], we have
$H^{1}(W^{o})\simeq H_{3}(W)$ and so $H_{3}(W)=0$. By [H16, Lemma 2.2.3], we
have then the exact sequence

$0\to H_{2}(M_{P})\to H_{2}(W_{t})\to H_{2}(W)\to H_{1}(M_{P})\to
H_{1}(W_{t})\to 0,$

since $H_{1}(W)=0$. On the other hand, we know that
$H_{2}(M_{P})={\mathbb{Z}}^{s-1}$, $H_{1}(M_{P})={\mathbb{Z}}/2$, and
$H_{1}(W_{t})={\mathbb{Z}}/2$, and so we have the wanted exact sequence. We
recall that locally at $P$, a ${\mathbb{Q}}$-Gorenstein smoothing of
$\frac{1}{4s}(1,2s-1)$ is a ${\mathbb{Z}}/2$-quotient of a smoothing of the
canonical cover $A_{2s-1}\to\frac{1}{4s}(1,2s-1)$, and on the smooth fibers is
a topological covering (see e.g. [HP10, Section 2.2]). ∎

######  Theorem 3.3.

Consider the partial resolution $\sigma\colon X\to W$ of $P$ (hence $X$ has
$s$ singularities of type $\frac{1}{4}(1,1)$ in a chain of
${\mathbb{P}}^{1}$’s), and let us consider an Enriques W-surface $X$. Let
$(W\subset\mathcal{W})\to(0\in{\mathbb{D}})$ be the induced blowing-down
${\mathbb{Q}}$-Gorenstein deformation, and assume that it is isomorphism on
general fibers (i.e. $X_{t}\simeq W_{t}$).

Then, there is an exact sequence

$0\to H_{2}(X_{t})\to H_{2}(X)\to({\mathbb{Z}}/2)^{s-1}\to 0.$

The group $({\mathbb{Z}}/2)^{s-1}$ has as generators the images of the
$(-1)$-curves between the $\frac{1}{4}(1,1)$ singularities. We have
$H_{2}(X_{t})=\operatorname{Pic}(X_{t})$, and $H_{2}(W)=\operatorname{Cl}(W)$
(the class group of $W$).

###### Proof.

Let $\Gamma_{i}$ be the images in $X$ of the $(-1)$-curves between the
$C_{i}$. This short exact sequence is achieved through Lemma 3.2, starting
with the restriction morphism

$H_{2}(X_{t})\to H_{2}(X)\simeq
H_{2}(W)\oplus_{i=1}^{s-1}{\mathbb{Z}}\Gamma_{i}.$

For more details see [TU22, Section 8], where we work out a more general
picture for ${\mathbb{Q}}$-Gorenstein smoothings. For the last part see [H13,
p.1196]. ∎

######  Remark 3.4.

Since $q(X)=p_{g}(X)=0$, we have identifications
$\operatorname{Pic}(X_{t})=H^{2}(X_{t})=H_{2}(X_{t})$,
$\operatorname{Pic}(X)=H^{2}(X)\subset H_{2}(X)=\operatorname{Cl}(X)$ (see
[Ko05]), and Mumford’s intersection theory on $X$ which respects the
topological intersection theory when restricted to $H^{2}(X)$. Hence the
restriction map $H_{2}(X_{t})\to H_{2}(X)$ of the sequence in Theorem 3.3
respects intersections.

######  Remark 3.5.

Let $\Gamma$ be the image in $X$ of a $(-1)$-curve from the chain

$C_{1}-(-1)-C_{2}-(-1)-\ldots-(-1)-C_{s}.$

Any divisor $D$ in $\operatorname{Cl}(X)$ corresponds to
$D=\sum_{i=1}^{s-1}d_{i}\Gamma_{i}$ in $({\mathbb{Z}}/2)^{s-1}$, where $d_{i}$
is the local multiplicity of $D$ at any of the singularities in $\Gamma_{i}$.
Note that numerically

$\phi^{*}(D)\equiv D^{\prime}+\sum_{i=1}^{s}\mu_{i}C_{i},$

where $D^{\prime}$ is the strict transform, and so $D^{\prime}\cdot
C_{i}=4\mu_{i}=d_{i}$ for $i=1,\ldots,s-1$ as $H_{1}={\mathbb{Z}}/4$ for the
link at each singularity.

##  4\. Coble-Mukai lattice and Enriques lattice

Let $X$ be an Enriques W-surface as in Theorem 3.3, and let $V$ be the
associated Coble surface of K3 type. Let $\sigma\colon V\to X$ be the
contraction of the boundary $\\{C_{1},\ldots,C_{s}\\}$. We have the short
exact sequence

$0\to\operatorname{Pic}(X_{t})\to\operatorname{Cl}(X)\to({\mathbb{Z}}/2)^{s-1}\to
0.$

(Note that for $s=1$ it also works, and so
$\operatorname{Pic}(X_{t})\simeq\operatorname{Cl}(X)$ in that case.)

######  Lemma 4.1.

The image of $\operatorname{Pic}(X_{t})$ in $\operatorname{Cl}(X)$ is the set
of classes whose proper transform have even intersection number with each
$C_{i}$.

###### Proof.

In Remark 3.5 we have

$\phi^{*}(D)\equiv D^{\prime}+\sum_{i=1}^{s}\mu_{i}C_{i},$

where $D^{\prime}\cdot C_{i}=4\mu_{i}=d_{i}$ and $(d_{1},\ldots d_{s-1})$ is
the image of $D$ in $({\mathbb{Z}}/2)^{s-1}$. But $\operatorname{Pic}(X_{t})$
is exactly the kernel. ∎

######  Theorem 4.2.

The image of $\operatorname{Pic}(X_{t})$ in $\operatorname{Cl}(X)$ quotient by
$\langle{\mathcal{O}}_{X_{t}}(K_{X_{t}})\rangle$ is isomorphic to
$\operatorname{CM}(V)$.

###### Proof.

We prove this through the pull-back morphism

$\phi^{*}\colon\operatorname{Cl}(X)\to\widetilde{\operatorname{Pic}}(V).$

First, the pull-back on any class in $\operatorname{Cl}(X)$ is orthogonal to
all $\beta_{i}$. So we now restrict to
$\phi^{*}\colon\operatorname{Pic}(X_{t})\to\operatorname{CM}(V)$.

Let $D=D^{\prime}+\sum_{i=1}^{s}\frac{a_{i}}{2}\beta_{i}$ in
$\operatorname{CM}(V)$ with $D^{\prime}$ not supported at the $\beta_{i}$, and
$a_{i}\in{\mathbb{Z}}$. Then by definition $D\cdot\beta_{i}=0$ and so
$D^{\prime}\cdot C_{i}$ is even for all $i$, and so $\phi^{*}$ is onto, by
Lemma 4.1. If
$\phi^{*}(D)=D^{\prime}+\sum_{i=1}^{s}\frac{a_{i}}{2}\beta_{i}=0$, then
$\phi^{*}(2D)=0$ in $\operatorname{Pic}(V)$. Say that $D\neq 0$ in
$\operatorname{Pic}(X_{t})$, and so we have a numerical 2-torsion class, and
this implies $D\sim K_{X_{t}}$ by Riemann-Roch on $D$. ∎

In [DK23, Theorem 9.2.15], it is proved that the Coble-Mukai lattice of a
Coble surface is isomorphic to the Enriques lattice over ${\mathbb{C}}$ by a
different method.

##  5\. Explicit computations for the first degeneration

In this section, we explicitly compute the Picard group of an Enriques surface
from the class group of $X$, which has two $\frac{1}{4}(1,1)$ singularities
and it is constructed as follows. Consider a pencil of cubic curves generated
by two nodal cubics $C_{1},C_{2}$ which intersect at nine distinct points. Let
$S\to{\mathbb{P}}^{2}$ be the blow-up at those points, and so $S$ has an
elliptic fibration with two $I_{1}$ fibers $C_{1}$, $C_{2}$ (proper transforms
of $C_{1},C_{2}$), and at least nine sections $R_{1},\ldots,R_{9}$ from the
nine points blown-up. Let $\pi\colon V\to S$ be the blow-up at the nodes of
$C_{1}$ and $C_{2}$, and let $E_{1},E_{2}$ be the exceptional curves. Let $H$
be the pull-back in $V$ of a general line in ${\mathbb{P}}^{2}$.

As before, we contract the $(-4)$-curves $C_{1}$ and $C_{2}$ (proper
transforms of $C_{1}$ and $C_{2}$) to obtain $X$ which has no local-to-global
obstructions to deform. Let $H$, $E_{1}$, $E_{2}$, $R_{1}$, $\ldots$, $R_{9}$
be the images of the corresponding curves in $X$. We see them as classes in
$\operatorname{Cl}(X)$. In this way, we can easily compute

$\operatorname{Cl}(X)=\frac{\langle
H,E_{1},E_{2},R_{1},\ldots,R_{9}\rangle}{\langle
2E_{1}-2E_{2},2E_{1}-3H+R_{1}+R_{2}+\ldots+R_{9}\rangle}.$

Indeed, in $\operatorname{Cl}(V)$ we have that
$C_{i}=3H-2E_{i}-R_{1}-R_{2}-\ldots-R_{9}$, $C_{1}-C_{2}=-2E_{1}+2E_{2}$, and
$C_{1},C_{2}$ are contracted by $V\to X$.

We have $H^{2}=11/2$, $H\cdot R_{i}=3/2$, $R_{i}^{2}=-1/2$, $E_{i}^{2}=0$, and
$R_{i}\cdot R_{j}=1/2$ for $i\neq j$. Consider an Enriques W-surface $X$ as in
Theorem 3.3, so that we have the short exact sequence

$0\to\operatorname{Pic}(X_{t})\to\operatorname{Cl}(X)\to{\mathbb{Z}}/2\to 0.$
(5.1)

We can think of ${\mathbb{Z}}/2$ as generated by the image (say) of $R_{9}$
(any $R_{i}$ works in this case, since they are locally toric boundaries for
both singularities). Note that

$K_{X}=E_{1}-E_{2},$

and it represents the canonical class of $X_{t}$.

Using that $\operatorname{Pic}(W_{t})$ is the kernel in the short exact
sequence (5.1) and numerical independence of classes, we can compute

$\operatorname{Pic}(X_{t})=\langle
E_{1},E_{1}+2R_{9},R_{1}-R_{2},R_{5}-R_{6},R_{6}-R_{7},\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
R_{2}-R_{3},R_{3}-R_{4},R_{4}-R_{5},R_{7}-R_{8},H-R_{1}-R_{2}-R_{3}\rangle+K_{W},$

and these generators give a basis for the Enriques lattice ${\mathbb{H}}\oplus
E_{8}(-1)$, where

${\mathbb{H}}=\langle E_{1},E_{1}+2R_{9}\rangle$

and

$E_{8}(-1)=\langle R_{1}-R_{2},R_{2}-R_{3},R_{3}-R_{4},\ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
R_{4}-R_{5},R_{5}-R_{6},R_{6}-R_{7},R_{7}-R_{8},H-R_{1}-R_{2}-R_{3}\rangle.$

Following the recipe and notation in [CDL23, Section 1.5], we have the root
basis for the Enriques lattice

$\alpha_{0}:=H-R_{1}-R_{2}-R_{3},\ \ \alpha_{1}:=R_{1}-R_{2},$ $\
\alpha_{2}:=R_{2}-R_{3},\ \ \alpha_{3}:=R_{3}-R_{4},$
$\alpha_{4}:=R_{4}-R_{5},\ \ \alpha_{5}:=R_{5}-R_{6},$ $\ \
\alpha_{6}:=R_{6}-R_{7},\ \ \alpha_{7}:=R_{7}-R_{8},$
$\alpha_{8}:=-3H+E_{1}+\sum_{j=1}^{7}R_{j}+2R_{8},\ \ \ \alpha_{9}:=2R_{9}.$

We also write down the following isotropic sequence ([CDL23, Section 1.5]):

$f_{10}=E_{1},\ \ \ f_{9}=E_{1}+2R_{9},\ \ \
f_{i}=-3H+2E_{1}+\sum_{j=1}^{8}E_{j}+R_{i}+2R_{9}$

for $i=1,\ldots,7$, and so
$\Delta=\frac{1}{3}(f_{1}+\ldots+f_{10})=-8H+6E_{1}+3\sum_{j=1}^{9}R_{j}$.

##  6\. Further degenerations and ${\mathbb{Q}}$-Homology projective planes

As explained in [CDL23, 5.9], the Baily-Borel compactification of the moduli
space of Enriques surfaces is formed by a Coble divisor, which is the moduli
space of Coble surfaces, and two smooth rational curves [CDL23, Theorem
5.9.8]. Hence the Baily-Borel compactification suggests that we should see a
big part “closer to the boundary” of the moduli space of Enriques surfaces via
Enriques W-surfaces (see [CDL23, 5.10]). In this section we show what sort of
things one can actually see.

We start with the most degenerate Coble surface (see [DK23, Example 9.2.7]).
It has $10$ $\frac{1}{4}(1,1)$ singularities, and it is unique up to
isomorphism (see [DK23, Chapter 9]). This surface corresponds to one of the
two most algebraic K3 surfaces [V83] under the canonical double cover. (In
fact, the K3 surface is $X_{4}$ in Vinberg’s notation, which is birational to
the $4$-th cyclic cover of ${\mathbb{P}}^{2}$ branched at $xyz(x+y+z)=0$.) We
take as a model the following construction. We start with the rational
elliptic fibration $S^{\prime}\to{\mathbb{P}}^{1}$ which has exactly $4$
singular fibers: one $I_{9}$ and three $I_{1}$. This surface has Mordell-Weil
group isomorphic to ${\mathbb{Z}}/3$ (see e.g. [P90]). Figure 3 gives the
notation for all relevant curves.

Figure 3. Relevant curves in $S^{\prime}$

Let $\pi^{\prime}\colon V^{\prime}\to S^{\prime}$ be the blow-up at all $9$
nodes in $I_{9}$, and at the node in $I_{1}=C^{\prime}_{10}$. As we have done
it before, we keep the notation for proper transforms of curves. Let
$E^{\prime}_{i}$ be the $(-1)$-curves from the blow-up. Hence we consider in
$V^{\prime}$ the chain

$C^{\prime}_{10}-A-C^{\prime}_{1}-E^{\prime}_{1}-C^{\prime}_{2}-E^{\prime}_{2}-\ldots-E^{\prime}_{8}-C^{\prime}_{9}$

where ${C^{\prime}_{i}}^{2}=-4$ and $A^{2}=-1$. Hense $E^{\prime}_{9}$ is
between $C^{\prime}_{1}$ and $C^{\prime}_{9}$, and $E^{\prime}_{10}$ is
intersecting $C^{\prime}_{10}$ at two distinct points. We now contract all the
$C^{\prime}_{i}$ via $\phi^{\prime}\colon V^{\prime}\to X^{\prime}$, so
$X^{\prime}$ has $10$ $\frac{1}{4}(1,1)$ singularities, and
$K_{X^{\prime}}\equiv 0$. Indeed, we have that

$C^{\prime}_{10}+2E^{\prime}_{10}\sim
F\sim\sum_{i=1}^{9}C^{\prime}_{i}+2\sum_{i=1}^{9}E^{\prime}_{i},$

where $F$ is a general fiber of the elliptic fibration
$V^{\prime}\to{\mathbb{P}}^{1}$, and

$\phi^{\prime*}(K_{X^{\prime}})-\frac{1}{2}\sum_{i=1}^{10}C^{\prime}_{i}\equiv
K_{V^{\prime}}\sim-F+\sum_{i=1}^{10}E^{\prime}_{i}.$

The surface $X^{\prime}$ has no local-to-global obstructions to deform,
because we are contracting curves over two fibers of
$S^{\prime}\to{\mathbb{P}}^{1}$ (see e.g. [PSU13, Section 4]). Our notation
for relevant curves in $X^{\prime}$ is $\Gamma_{0}:=E^{\prime}_{1}$,
$\Gamma_{1}:=E^{\prime}_{2}$, $\Gamma_{2}:=E^{\prime}_{3}$,
$\Gamma_{3}:=E^{\prime}_{4}$, $\Gamma_{4}:=E^{\prime}_{5}$,
$\Gamma_{5}:=E^{\prime}_{6}$, $\Gamma_{6}:=E^{\prime}_{7}$,
$\Gamma_{7}:=E^{\prime}_{8}$, $\Gamma_{8}:=E^{\prime}_{9}$, $\Gamma_{9}:=A$,
$\Gamma_{10}:=B$, $\Gamma_{11}:=C$, and $\Gamma_{12}:=E^{\prime}_{10}$, which
is represented in Figure 4.

Figure 4. Relevant curves in $X^{\prime}$

In this case we have that

$\operatorname{Cl}(X^{\prime})=H_{2}(X^{\prime})=\langle\Gamma_{0},\Gamma_{1},\ldots,\Gamma_{12}\rangle/\langle{\mathcal{R}}_{1},{\mathcal{R}}_{2},{\mathcal{R}}_{3}\rangle$

where the relations are:

${\mathcal{R}}_{1}:=-2\Gamma_{0}-2\Gamma_{1}-2\Gamma_{2}-\Gamma_{3}+\Gamma_{4}+3\Gamma_{5}+3\Gamma_{6}+\Gamma_{7}-\Gamma_{8}-\Gamma_{9}-\Gamma_{10}+2\Gamma_{11}$

${\mathcal{R}}_{2}:=-\Gamma_{0}-3\Gamma_{1}-5\Gamma_{2}-4\Gamma_{3}+4\Gamma_{5}+5\Gamma_{6}+3\Gamma_{7}+\Gamma_{8}-3\Gamma_{10}+3\Gamma_{11}$

${\mathcal{R}}_{3}:=-\Gamma_{0}-3\Gamma_{1}-5\Gamma_{2}-3\Gamma_{3}+3\Gamma_{4}+9\Gamma_{5}+10\Gamma_{6}+6\Gamma_{7}+2\Gamma_{8}-\Gamma_{9}-4\Gamma_{10}+5\Gamma_{11}-2\Gamma_{12}$

Let $X$ be the surface constructed in Section 5. We have the curves
$H,E_{1},E_{2},R_{1},\dots,R_{9}$. One can see $X$ as a
${\mathbb{Q}}$-Gorenstein smoothing of $X^{\prime}$. Indeed we could choose a
general ${\mathbb{Q}}$-Gorenstein smoothing of $X^{\prime}$ which smooths all
singularities except the singularities at $\Gamma_{0}\cap\Gamma_{8}$ and at
the node of $\Gamma_{12}$. We can recover all curves
$H,E_{1},E_{2},R_{1},\dots,$ $R_{9}$ from linear combinations of the curves
$\Gamma_{0}$, $\ldots$, $\Gamma_{12}$, and in particular we can read
$\operatorname{Pic}(X_{t})$ from $\operatorname{Cl}(X^{\prime})$. Details in a
more general setup will be given in [TU22]. We note that in this case

$0\to\operatorname{Pic}(X_{t})\to\operatorname{Cl}(X^{\prime})\to({\mathbb{Z}}/2)^{9}\to
0.$

On the other hand, we note that we have many subchains in $\Gamma_{0},$
$\dots,$ $\Gamma_{11}$ which can be contracted to singularities of type
$[3,2,\dots,2,3]$. The relevance of that is the application of Proposition 3.1
to particular deformations of $W^{\prime}$, which is the contraction of some
disjoint chains of $\Gamma_{i}$’s. Let us take the maximal chain
$\Gamma_{1}+\dots+\Gamma_{9}$. Let $X^{\prime}\to W^{\prime}$ be the
contraction of that chain. In this way, the surface $W^{\prime}$ has one
T-singularity, whose minimal resolution corresponds to the Hirzebruch-Jung
continued fraction

$\frac{40}{19}=[3,2,2,2,2,2,2,2,2,3].$

We recall that $W^{\prime}$ has no local-to-global obstructions to deform
(same reason as for $X^{\prime}$), and so we can choose from Proposition 3.1 a
global ${\mathbb{Q}}$-Gorenstein deformation which deforms
$\frac{1}{40}(1,19)$ into an $A_{9}$ rational double point. This is a one
dimensional family. In this way, the general fiber $W^{\prime}_{t}$ is a
Gorenstein ${\mathbb{Q}}$-Homology projective plane with $K\equiv 0$. We
recall that a normal projective surface with only quotient singularities whose
second Betti number is $1$ is called a ${\mathbb{Q}}$-Homology projective
plane, and it is Gorenstein if all singularities are rational double points.
In [HKO15], Hwang, Keum and Ohashi classified all possible configurations of
singularities for Gorenstein ${\mathbb{Q}}$-Homology projective planes. In the
case of $K\equiv 0$ there are $31$ possible configurations, and they were able
to produce examples for $29$ of them. We can have at most $5$ singularities in
these configurations. Of course, the minimal resolution of all of them are
Enriques surfaces with particular ADE configurations of nine $(-2)$-curves.

In [S21], Schütt classifies all Gorenstein ${\mathbb{Q}}$-Homology projective
planes. It turns out that for each of the $31$ types we have a one dimensional
moduli space, and the number of components for each type varies from $1$ to
$3$:

  * •

$3$ components for $A_{7}+2A_{1}$, and $3A_{3}$;

  * •

$2$ components for $A_{8}+A_{1}$, $A_{7}+A_{2}$, $A_{5}+A_{3}+A_{1}$,
$A_{5}+2A_{2}$, $D_{8}+A_{1}$, $D_{6}+A_{3}$, $D_{4}+A_{3}+2A_{1}$,
$E_{8}+A_{1}$, $E_{7}+A_{2}$, $E_{6}+A_{3}$, $E_{7}+A_{2}$, $E_{8}+A_{1}$,
$D_{9}$, and $D_{7}+2A_{1}$;

  * •

all the other $15$ root types.

######  Theorem 6.1.

The one dimensional family from $W^{\prime}$ whose general fiber has an
$A_{9}$ singularity is in closure of the Schütt irreducible moduli
corresponding to $A_{9}$.

We can actually obtain almost all of the $31$ types from Enriques W-surfaces
just as explained for $A_{9}$, and a small variant when we meet the type $D$
or $E$ singularities. We only give two examples below, but it is possible to
realize at least $27$ types.

######  Example 6.2.

Consider the rational elliptic fibration with sections $S\to{\mathbb{P}}^{1}$
whose singular fibers are $I_{4}^{*}+2I_{1}$ (see e.g. [P90, p.7]). It has
exactly two sections. Let us fix one of these sections, we denote it by $A$.
Let $B$ and $C$ be the $I_{1}$ fibers, and let $D$ be $I_{4}^{*}$ minus the
component intersecting $A$. We blow-up the node at $B$ and the node at $C$,
and then we contract the section $A$. We obtain a surface $V$ with
$B^{2}=C^{2}=-3$, and $D$ is a $D_{8}$ configuration, i.e., the minimal
resolution of a $D_{8}$ singularity. The contraction of $B+C$ produces a
surface $W$ with a singularity $[3,3]$, and it has no-global-to-local
obstructions. Moreover, since $D$ is disjoint to $B+C$, the contraction
$W^{\prime}$ of $D$ in $W$ has also no local-to-global obstructions [PSU13,
Section 4]. Therefore, by Proposition 3.1, we can choose to
${\mathbb{Q}}$-Gorenstein deform $W^{\prime}$ so that we keep the $D_{8}$
singularity and we deform $[3,3]$ into $A_{1}$. In this way, we obtain a
${\mathbb{Q}}$-Homology projective plane with $K\equiv 0$ and root type
$D_{8}+A_{1}$. According to Schütt’s result, this root type corresponds to a
moduli curve with two components.

Similarly, we can construct a surface for the root type $D_{5}+A_{4}$ from the
rational elliptic surface with sections whose singular fibers are
$I_{1}^{*}+I_{4}+I_{1}$. Here the Mordell-Weil group is ${\mathbb{Z}}/4$
[P90]. Just choose one of the four sections, and blow-up a suitable node in
$I_{4}$ and the node in $I_{1}$, and then consider $D_{5}$ and $[3,2,2,2,3]$
to construct what we want through Proposition 3.1. This root type has an
irreducible moduli. To get other root types, we need to consider more involved
situations. For example, we may start with a special Halphen surface of index
$2$ and some suitable double sections.

Compactifications of Schütt moduli curves should correspond to degenerations
of Enriques surfaces, since there are no non-isotrivial smooth families of
Enriques surfaces over proper bases by [CDL23, Corollary 5.9.13]. Hence, a
natural question is: Find compactifications of Schütt moduli curves and the
corresponding limit singular surfaces. Which root types hit Enriques
W-surfaces? What else?

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