# CoordiQ : Coordinated Q-learning for Electric Vehicle Charging
Recommendation

Carter W. Blum1,2 Liu Hao1 Hui Xiong1,3

1 Business Intelligence Lab, Baidu Inc., Beijing, CN
2 University of Minnesota, Minneapolis, MN, US
3 Rutgers University, Newark, NJ, US Contact Author<EMAIL_ADDRESS>
(December 2019)

##  1 Abstract

Electric vehicles have been rapidly increasing in usage, but stations to
charge them have not always kept up with demand, so efficient routing of
vehicles to stations is critical to operating at maximum efficiency. Deciding
which stations to recommend drivers to is a complex problem with a multitude
of possible recommendations, volatile usage patterns and temporally extended
consequences of recommendations. Reinforcement learning offers a powerful
paradigm for solving sequential decision-making problems, but traditional
methods may struggle with sample efficiency due to the high number of possible
actions. By developing a model that allows complex representations of actions,
we improve outcomes for users of our system by over 30% when compared to
existing baselines in a simulation. If implemented widely, these better
recommendations can globally save over 4 million person-hours of waiting and
driving each year.

##  2 Introduction

As the public becomes more ecologically conscious, electric vehicles are
surging in popularity, putting up double digit growth rates year over year.
However, despite consumer enthusiasm, electric vehicles still face some major
hurdles that they must face before they can truly become mainstream. One such
hurdle is public charging stations. Because not all electric vehicle owners
have the infrastructure to charge their vehicles at home, many rely on
electric vehicle charging stations to keep their tank full. This problem is
particularly prevalent among electric car owners in urban environments, who
often do not have a space to charge their own vehicle.

Unfortunately, electric vehicle charging stations have not spread as
prolifically as the vehicles themselves, causing a potential lack of supply.
Each charging station has a number of chargers, each of which can service one
car at a time. Our data analysis shows that, in the city of Beijing, at least
20% of charging stations are at 100% occupancy - even during periods of low-
usage. Electric vehicles can take hours to charge, so stations that are full
can stay full for long periods of time, resulting in very long wait times for
any vehicles waiting for a charger.

In our increasingly digital world, consumers increasingly rely on mobile apps
to direct them to the best utilities. Perhaps the most prominent such utility
in China is Baidu Maps, which services 300 million users per month and
processes hundreds of thousands of electric vehicle charging station queries
per year. Consumers query the app, send it their location, and it sends them
back recommendations for nearby charging stations. Current applications simply
return a sorted list of the nearest highly rated stations, but this can cause
problems, as these are often full, and wait time can cause more of an
inconvenience to the user than driving time.

This project implements and evaluates a reinforcement learning agent to
intelligently respond to queries, recommending users to nearby charging
stations with minimal inconvenience. The project assumes no control over when
the app will receive queries and it assumes that there are many other electric
vehicles that it has no control over. Despite these constraints, we show that
the agent is able to learn effective recommendations that reduce the user’s
time-to-charging by over 45%.

The key contributions of this work are as follows :

  * •

We present the first machine learning based algorithm for charging station
recommendation, and implement a context-aware reinforcement algorithm that
coordinates between different recommendations. It substantially outperforms
all existing baselines.

  * •

To the best of our knowledge, there are no published works yet using Graph
Neural Networks (GNNs) to solve reinforcement learning tasks. We propose an
architecture and show it improves performance

##  3 Related Works

Reinforcement learning is a useful paradigm for solving sequential control
tasks of many times. Reinforcement learning models operate in a Markov
Decision Process, learning a probability distribution over a finite set of
actions to maximize reward Russell and Norvig (2016)Sutton and Barto (2018).
With the recent increase of processing power and the modeling power of neural
networks, Deep Q-Networks (DQNs) have shown impressive capabilities to handle
a diverse range of tasks Mnih et al. (2013). To that extent, Deep Q-Networks
have been shown to learn complex tasks, and have even been able to excede
human level control when solving many problems Mnih et al. (2015).

Reinforcement Learning has been successfully applied to improving charging of
electric vehicles. However, most of the research has focused on improving the
charging experience once a vehicle has arrived in a station, such as in
Valogianni et al. (2013), Xiong et al. (2018) and Zou et al. (2016). These
methods successfully improve load distribution equity in stations, and
schedule charging demands accordingly, as in Zhao et al. (2018). Ramachandran
et al. (2018) also analyze charging station demand to improve the layout of
stations to better handle peak demand.

However, reinforcement learning has been applied to electric vehicle station
recommendation as well. Houbbadi et al. (2019) provides a thorough analysis of
bus charging recommendations. However, the methods provided focus specifically
on buses, following strict and predictable schedules, and the primary
difficulty is in choosing when to charge, not where. The question being
approached in this paper is the opposite, we assume that the user needs to
charge now and the recommender attempts to minimize the inconvenience to the
user. On the other hand, Tian et al. (2016) focuses on choosing what stations
to recommend taxis to. The authors employ a rule-based method to do
recommendation, but the method provided relies on future knowledge of the
occupancy of a charging station at the time that the user arrives.
Furthermore, it assigns each taxi in isolation, without accounting for the
impact that one taxi’s recommendation may have on the wait-time of another.

Outside of electric vehicle recommendation, there is a large body of work in
spatio-temporal reinforcement learning. In Xu et al. (2018), the authors use a
tabular method to recommend idle taxis to nearby regions and allocate
dispatches. While this approach was extremely successful for its authors, it
heavily exploits the interchangeability of taxis, a property that electric
vehicle charging stations do not have to the same degree.

A non-tabular approach has also been pursue in several works, including He and
Shin (2019), which implements a capsule network to recommend idle taxi
cruising. This approach is also modified and expanded in several similar
papers, including Jindal et al. (2018), Oda and Joe-Wong (2018), Oda and
Tachibana (2018), Shou et al. (2019), Alabbasi et al. (2019). These papers
improve taxi idle cruising to reposition them to nearby areas, either by
considering the taxis in concert or in isolation. They train reinforcement
learning models in simulations, with space discretized in to tiles, and the
use a convolutional neural network architecture to output Q-values for each
location, and then selecting the highest Q-value from the adjacent tiles.
However, most tiles are not valid actions for station recommendation, giving
this method a very high noise to signal ratio when applied to charging station
recommendation.

A couple other approaches to spatio-temporal recommendation have also been
used. One approach is to try to minimize the divergence between the
distribution of queries and taxi locations, as demonstrated in Zhou et al.
(2019). While this approach appears very promising, the number of 0 support
regions in both query and station distributions causes many distributional
distance metrics to be unstable. Another direction is explored in Verma et al.
(2017), which uses a tabular method but dynamically splits the zones whenever
it can improve Q-value accuracy. This method can be suitable for macro-
planning, but charging station recommendation requires distinguishing between
stations that are close together. Under this paradigm, those stations may be
grouped in the same zone, resulting in decreased performance.

An alternate method for handling spatially related data is to use graphical
neural networks. Graphical neural networks define the input as a graph, with
vertices edges and weights. This allows information from nodes to propagate
from one to another Scarselli et al. (2008). Initial methods put restrictions
on the weights of the graph convolution kernels to ensure that repeated
convolution would converge Pineda (1987)Defferrard et al. (2016). However,
more recently methods methods have been doing away with these restrictions
Kipf and Welling (2016). One such method that has had significant success is
GraphSage, which utilizes local graph convolution over a finite set of
iterations. This method does not guaranteed converge to a set value, but can,
in practice achieve similar, if not superior results with significantly less
computation.

##  4 Methodology

We formulate the problem as a Markov Decision Process (MDP), with a statespace
$\mathcal{S}$, action space $\mathcal{A}$, transition function $T$, and reward
function $R$. Our recommendation system is represented by a policy
$\pi:\mathcal{S}\rightarrow\mathcal{A}$ At each timestep in an MDP, an agent
must look at it’s current state and choose an action to maximize its expected
discounted rewards. The expected rewards of action $a\in\mathcal{A}$ while at
state $s\in\mathcal{S}$ is given by :

$Q(s,a)=E[R(t)]+\gamma\sum_{s^{\prime}\in\mathcal{S}}T(s^{\prime}|s,a)\max_{a^{\prime}\in\mathcal{A}}Q(s^{\prime},a^{\prime})$

here $\gamma\in[0,1)$ is the discount factor.

States: In this problem, the state is the information available to the station
to make recommendations with. Some information is available for each city that
may be relevant to all stations, such as estimates of traffic, the weather,
day of week, and time of day. For a given state $s$, we’ll refer to this
global information as $s_{G}$. For each of the $N$ stations, information about
the status of its chargers is available, as well as its location and id. The
information available about the $n^{th}$ station is denoted as $s_{n}$.
Finally, the agent has access to information about the queries it is
responding to, including their location and user id, as well as time of query.
For the $i^{th}$ query received at time $t$, the relevant state information is
denoted as $s_{i}$.

We discretize the each region in to a grid, with each grid cell having an area
of roughly $0.25$km2. We similarly turn the system in to a sequential decision
process by dividing time in to non-overlapping intervals of duration $\tau$.

Actions: At each timestep $t$, the agent receives a set of user queries
$Q_{t}$ for which it must provide recommendation. The station can recommend
each of the $|Q_{t}|$ user queries to any of the $N$ stations, resulting in a
possible action space of $N^{|Q_{t}|}$ combinations. This poses a problem,
because not only is the possible action space potentially huge, but the
available actions vary from timestep to timestep. To address this, the agent
makes each of its decisions for each query sequentially and executes all
actions after decisions about all queries have been made. When there is no
loss of clarity, we will instead refer to recommending a user to station $n$
as action $a_{n}$.

Rewards: After a driver is recommended to a station, they must drive there.
Subsequently, if the station is full, then they must spend time waiting for a
space to open up. To accommodate these requirements, the agent must then
satisfy the multi-modal objective of minimizing both of these. Let
$t^{i}_{drive}$ be the driving time for the user corresponding to query $i$,
as estimated by driving speed and distance. Similarly, we define
$t^{i}_{wait}$ to be the the number of timesteps that the user spends waiting
at the station before a spot opens for them. As the agent wants to find an
optimal split between users, the goal is to then minimize the following
equation

$\sum_{t}\sum_{i\in\mathcal{Q}(t)}t^{i}_{wait}+\lambda t^{i}_{drive}$

where $\lambda$ is a factor for weighting the two criteria. We define the sum
of these two items (with $\lambda=1$) to be the inconvenience time for a user.
To achieve this, the reward function is defined as

$R(t)=KN_{arrive}-N_{wait}-\lambda N_{drive}$

Where $N_{arrive},N_{wait}$ and $N_{drive}$ are defined as the number of users
who arrived at a station, waited there or were driving at the current
timestep, respectively. $K$ is the reward that the agent receives for
successfully completing a dispatch. Trivially, as $\gamma\rightarrow 1$,
maximizing this reward corresponds to minimizing the previous equation.

Transition Function:  In this study, the transition function is modeled by a
simulation, which is further described in the ‘Analysis & Results’ section. In
practice, the transition function will be the real-world movement of cars to
and from stations, as well as the updated information that the agent receives
from these actions. Note that this simulation assumes that there are many
vehicles that leave and enter the station outside of our control, and these
vehicles must be considered part of the transition function. Once a car
recommended by our station begins charging, we have no further influence over
it and it becomes part of the environment.

However, he above formulation does not quite fulfill the requirements for a
Markov decision process, which requires that the transition function is solely
a function of the current state, because of several reasons. First, how many
cars enter and leave a given charging station may be influenced by how long
those cars have already been charging. In the analysis section, we show that,
counter-intuitively, this is not the case.

Second, how many cars enter a station is partially influenced by our previous
recommendations. To account for this, we add an additional engineered feature
included in the station information ($s_{n}$). For each $\delta t\in[1,k]$, we
include information about the number of cars that are estimated to arrive at
timestep $t+\delta t$ as a result of our recommendations. As this information
can be updated by the agent every timestep and whenever it makes a decision,
it allows the agent to remember its queued decisions that it hasn’t executed
yet. As a result of this, it is able to coordinate between the many decisions
it can be forced to make at each timestep. Koenig et Al. show that an optimal
policy learned this way will obtain no less than half of the rewards of an
optimal policy that considers all $\mathcal{Q}(t)$ at once, while avoiding the
exponential complexity of such an algorithm.

Classical Model: A classical method of solving such problems is to use a feed
forward Deep Q-Network (DQN). In this case, the state is input as a
concatenation of all of the parts of the station $s=\\{s_{n}\forall
n,s_{i}\forall i\in\mathcal{Q}(t),s_{G}\\}$. The DQN, parameterized by
$\theta$, outputs a vector $\vec{q}$, such that $\vec{q}_{n}$ denotes the
Q-value for action $a_{n}$. The policy $\pi$, is to then usually select the
station with the highest Q-value. However, to incentivize exploration, the
agent sometimes chooses a random agent with probability $\epsilon$. The
network is then trained via temporal differencing, such that the update is
calculated as follows:

$\Delta\theta=\eta\nabla_{\theta}\Big{(}R(t)+\max_{a^{\prime}\in
A}Q(s^{\prime},a^{\prime}|\Theta^{\prime})-Q(s,a|\Theta)\Big{)}$

To improve convergence, a replay memory buffer is used to sample old
experiences to perform the TD-update on.

Convolutional Model: There can be hundreds of stations, meaning that this
single network must take in all of the hundreds on inputs at once and map this
to the hundreds of appropriate Q-values. However, he occupancy has of a
station on the southern side of the city likely should have little influence
on the agent’s evaluation of an arbitrary station on the other side of the
city. Furthermore, the evaluation process should be roughly the same for each
station - look at how far it is, look at whether or not we anticipate a demand
in the area, etcetera. As a result, the network can be improved by evaluating
each station individually. In this case, rather than concatenating all of the
states, when calculating $Q(s,a_{n})$ for the $i^{th}$ query, the agent is
only input with the values of $\\{s_{n},s_{i},s_{G}\\}$. In this formulation,
the evaluation of all of the stations shares the same parameters.This also
enables the agent to operate with far fewer parameters, speeding up
computation. Mathematically, this operation is equivalent to inputting the
state as an $Nx1$ image, with depth equal to the number of features for each
station plus the number of features for the global state. Processing each
station one at a time then corresponds to doing 1x1 convolution over the
stations, hence the name ‘convolutional model’.

Graphical Model: Despite these advantages, this analysis can be slightly
reductive. Evaluating each station in isolation could potentially miss out on
important information - if all other nearby stations are full, the remaining
station with vacancies can expect a lot of their demand to be redirected to
it. It is then useful for the station to have a method to view information
about nearby states.

Given that the spread of stations is non-uniform, traditional convolutional
methods do not work. As a result, this paper employs graph convolution to add
context Kipf and Welling (2016). Specifically, this paper employs a form of
graph convolutional networks similar to GraphSage. Specifically, we define a
graph $G=\\{V,E,W\\}$ over the state space. Specifically, we let the set of
vertices $V$ to be the set of stations and add edges between two nodes if the
corresponding stations are within distance $D$ from each other. Letting
$d(n,m)$ be the distance between stations $n$ and $m$, the corresponding
weight is defined as $\alpha_{n}me^{-\beta d(n,m)}$, where $\alpha$ is a
hyperparameter.

Denoting $E(n)$ as the stations that share an edge with $n$, the Q-values of
action $n$ are computed as follows.

$h_{n}=f(s,a_{n}|\theta_{1})$ $x_{n}=\\{h_{n},\sum_{m\in
E(n)}\alpha_{m}h_{m}\\}$ $q_{n}=g(x_{n}|\theta_{2})$

where $f,g$ are feed-forward networks. A diagram explaining each of the three
model types is provided in figure (1) and the pseudocode provides an overview
of the sequential decision process.

Algorithm 1 Code for agent actions

ACT($\pi$, $S$, $\mathcal{Q}(t)$, $E$)

1: SHUFFLE($\mathcal{Q}(t)$)

2: for $q^{(j)}\in\mathcal{Q}(t)$ do

3: for $s^{(i)}\in\text{NEARBY}(q^{(j)},S)$ do

4: bid${}^{(i)}=Q(s^{(i)},q^{(j)},E)$

5: end for

6: to = $\text{argmax}_{j}\Big{[}\text{bid}^{(i)}\Big{]}$

7: dist = DISTANCE($s^{(to)}$, $q^{(j)}$)

8: $s^{(i)}_{d_{dist}}+=1$

9: end for

Figure 1: Depictions of easy model type. Top-Left: Classical FFDQN network
Bottom-Left: Convolutional Model Bottom-Right: GraphSage Model

An alternate way to formulate the problem is as a multi-agent problem. In this
formulation, each station could be formulated as its own agent, and for each
query, it has a decision to accept or not accepted the user to be recommended
to it. In this way, each station receives reward for the queries that are
dispatched to it. However, our MDP requires that the agent can only recommend
one station at a time, so there needs to be consensus between stations, such
that exactly one station ‘accepts’ each query.

Let $Q_{n}(s,1)$ be the discounted expected reward of station $n$ if it
accepts, and $Q_{n}(s,0)$ if it rejects. Then maximizing the total reward over
all stations is maximized by recommending the user to station $n$ such that

$n=\text{argmax}_{n\in N}Q_{n}(s,1)-Q_{n}(s,0)$

We explore this model in analysis and results below.

To make any of these models learn anything at all, several small improvements
are also necessary. First, the agent can only recommend agents to stations
within distance $D$. If there are less than $5$ stations within distance $D$,
the agent can recommend to the 5 nearest stations instead. Additionally, to
stabilize training, this paper uses the Double DQN architecture, as pioneered
in DDQN . Finally, a dueling DQN architecture was implemented, using a
convolutional neural network to learn the value function.

##  5 Experiment Design

Data Analysis: The algorithm is trained and tested in a simulation based on
data from over 6 months of data. As noted above, the region is divided in to a
rectangular grid. In the case of Beijing, the city is divided in to a grid of
size $144\times 126$. Time is discretized in to non-overlapping periods, each
with duration $\tau=15$ minutes. Charger data is collected 514 chargers across
73 stations and 96,000 timesteps. This is used to measure how long cars stay
at a station by measuring the time between when when a charger becomes
occupied and when it first becomes free again. From this, a probability
distribution of charging durations is derived, as shown in figure (2).

Figure 2: Counts of charging durations by number of observations

We note that the distribution appears to roughly follow a geometric
distribution (p ¡ 5e-4), which has the convenient property of being
memoryless. This means that knowing how long a vehicle has been charging for
does not help predict how much longer it will continue charging, so the model
does not need to keep track of this information.

As a result, at each timestep, we can model the number of remaining vehicles
as a geometric distribution. The next question is the number of incoming cars.
Naturally, one would expect that the number of cars coming in to the station
and the number of cars querying would be proportional to the traffic in the
surrounding area. However, this assumption does not match the data, as shown
in figure (3). As nearby traffic doesn’t seem to correlate strongly with any
core pieces of the environment, it is also not included in the state.

Figure 3: The number of queries received in an area doesn’t correlate strongly
with the number of cars in the area. Note that there is higher variance for
higher-traffic data values, as there are fewer datapoints.

Simulation: With these considerations, the simulation is defined as follows.
The agent is trained in an episodic manner, with each simulation starting and
ending at midnight. The number of cars that remain at the station at each
timestep is sampled from a geometric distribution. From there, a number of
cars incoming from outside for the system is sampled from a distribution
modeled off of historical data. Finally, the dispatches made by our system are
handled. If possible, all of the vehicles that are within 1 timestep of their
target station move to that station and begin charging. If there are not
enough chargers, then those vehicles that couldn’t be accommodated wait until
the next timestep (at which point they will get priority over arriving
vehicles). Finally, the position of all other vehicles is updated. A summary
of this algorithm is provided in pseudo code.

Algorithm 2 Code for training simulation

SIMULATE($f$, $g$, $\pi$, $S$)

1: Let $t=0$.

2: while $t<T_{END}$ do

3: for $s^{(i)}\in S$ do

4: $N_{cars}^{(i)}(t+1)=N^{(i)}_{rem}+N^{(i)}_{inc}+N^{(i)}_{rec}$

5: if $s^{(i)}_{T}<N_{cars}^{(i)}(t+1)$ then

6: $s^{(i)}_{d_{0}}+=N_{cars}^{(i)}(t+1)-s^{(i)}_{T}$

7: $N_{cars}^{(i)}(t+1)=s^{(i)}_{T}$

8: end if

9: $s^{(i)}_{O}=s^{(i)}_{T}-N_{cars}^{(i)}(t+1)$

10: end for

11: $\mathcal{Q}(t)\sim g(t,dow)$

12: ACT($\pi$, $S$, $\mathcal{Q}(t)$)

13: end while

##  6 Results & Analysis

We evaluate the proposed models in the simulation defined above. The
convolutional and graphical models are build with a small network size of only
100 nodes per hidden layer and 3 hidden layers. A variety of network sizes
were tried for the classical DQN, but the best performing network was
substantially larger, with 250 nodes per hidden layer. The exploration
probability, $\epsilon$, is initially set at 0.9 and decreases exponentially
to 0.1 throughout training.

Aside from the classic DQN model, we evaluate our model against several rule-
based metrics, defined as follows:

  * •

Nearby: Always recommends the user to the nearest station in a greedy manner.
This should minimize $t_{drive}$ for all users.

  * •

Open: Always recommends the user to the station that has the highest number of
open chargers, breaking ties by distance. This would ideally minimize
$t_{wait}$ for all users.

  * •

NearestOpen: An approximation of the method described in Tian et al. (2016),
but avoids using future information. Will usually choose nearest station, but
will choose other stations if the desired station is full.

We test 4 different deep-learning based methods: Graphical, Convolutional and
Classical DQN are all as described above. ‘Grouping’ denotes a convolutional
model that first recommends the user to a group of stations and then uses
greedy nearby recommendation once the user arrives. Figure (4) shows a plot of
the average rewards received on the test seed by epoch. Note that ’Open’ is
not shown on graphs because it performs so poorly that it makes the other
models hard to compare.

Figure 4: Convolutional and Graphical Models Substantially Outperform Other
Strategies

The results shown strongly suggest that classical DQN methods are not able to
solve this problem to any satisfying degree, as it performs significantly
worse than both rule-based baselines. However, both the convolutional and
graphical models substantially outperform the baselines. However, there are a
couple of interesting things to note about their performance.

Firstly, there is an interesting phenomenon where the average reward nearly
monotonically increases each epoch, but then there is a slight downward dip
after roughly epoch 200, before the rewards start climbing upwards again.
There are two possible reasons for this. The more interesting reason, is that
it may be an instance of a broader phenomenon, called ‘Deep Double Descent’,
where neural networks often undergo a period where their performance decreases
after an initial rapid decrease Mei and Montanari (2019).

The second conceivable reason is a bit more banal - it has to do with random
seed. The data is tested on random seeds zero through ten. The agent initially
plays the simulation with random seed zero, and each time it plays, it
increases the random seed by one. Because of this, it does initially see the
random seeds that it is eventually tested on, once for each. However, this
experiment uses a experience replay buffer to stabilize convergence, and epoch
200 is roughly where those ‘memories’ would leave the buffer. It is fairly
dubious that this effect, if present, is very strong, because the policy that
the agent followed on its first attempt is likely very different from its
final policy, but it is something to be aware of for future experiments.

The other thing to note is that the graphical model does indeed outperform the
baseline convolutional model, but it’s not the only one to do so. Adding
grouping to the convolutional model appears to achieve roughly the same final
results, although it doesn’t converge as quickly. Future work will involve
testing whether or not combining the graphical model and the grouping
mechanism results in an even better model. The natural initial response to
this question might be a hesitant yes, but, when looking at the wait times,
it’s not totally clear that it is possible to improve.

Figure 5: Graphical Model Surprisingly Performs Near-Optimally

For this, we reference figure (5). We again note that the theoretical minimum
drive distance is equal to the drive distance given by the ‘nearest’
recommender. Looking at the graph, we see that the graphical and grouping
models are actually very close in this regard - the average drive time is only
0.1 and 0.3 timesteps more than that of the ‘nearest’ recommender. Despite
this, the graphical model achieves near-zero wait time. Impressively, the
graphing model achieves a result that is nearly as good as a fantasy world
where every user drives to the nearest station and there’s always a spot for
them - it’s average wait time is only 0.2 more than that!

| Graphical | Convolutional | FFDQN | Nearest | Open | Nearest Open
---|---|---|---|---|---|---
Reward | 291 | 242 | 97 | 148 | -910 | 173
Inconvenience Time (m) | 8.80 | 11.55 | 19.46 | 16.51 | 80.15 | 15.62
Wait Time (m) | 0.4 | 2.15 | 7.30 | 8.50 | 16.90 | 4.81
Drive Time(m) | 8.40 | 9.41 | 12.15 | 8.00 | 63.25 | 10.80
Table 1: Averages of useful metrics in simulation organized by algorithm used
for routing. Inconvenience time represents the sum of time spent waiting and
driving.

This indicates that, while the graphical model might not be perfect, there
likely isn’t much to gain from improving it further, as you could only reduce
wait times by another 10% at most. To contrast this, the graphical model
reduces wait times by about 45% over all of the baselines, which is much more
substantial. These results are condensed in table (1).

However, there are a number of caveats to these statements. The first, and
biggest one, is that these are only single runs for each model. Reinforcement
learning models are notorious for being unstable, and that has been my
experience with this project as well. Its very possible that the random seeds
gave good initial weights that helped things converge smoothly. With more
resources and time, further runs would prove helpful to demonstrate the
effectiveness of this method.

The second caveat is that these results are within a simulation, and it may be
significantly easier to optimize the simulation than the real world. This
simulation-gap is a common difficulty that many reinforcement learning
projects face. One potentially impactful gap is that the nature of the data
led the simulation to use a larger timestep than it would use after
deployment. Additionally, the simulation did not involve other factors such as
inclement weather or unexpected traffic issues.

The last set of experiments to touch upon are the multi-agent experiments.
Unlike the graphical model, these heavily underperformed. While it did
outperform the baseline, it still wasn’t anywhere close to the convolutional
model and it trained extremely slowly. It is difficult to solid justification
for this, and is worth investigating in the future. A plot of the reward
functions is provide in figure (6).

Figure 6: Multi-Agent Model Mysteriously Underperforms

One possible explanation for this behaviour is that the multi-agent model
doesn’t quite fully satisfy the MDP property. Whenever it does an auction, the
model doesn’t ‘know’ if it will have another query this timestep or not, even
though that is pre-determined, therefore making it suboptimal. That being
said, this same effect should apply to all of the other models as well, but
those performed fine. Another possibility is that it simply takes longer to
converge, or that the non-stationarity of the other agents makes the
environment itself non-stationarity. According to Foerster et al. (2017), the
fact that all agents share parameters should prevent that from happening, but
it’s possible that something still went wrong.

##  7 Conclusion and Future Work

In total, this report presents a framework for significantly improving
charging station recommendations in response to dynamic, user-generated
queries. The graphical model proposed is the first of its kind, and it
balances between providing the canvas for complex interactions to be
discovered and limiting the search space to allow it to tackle more difficult
problems.

There are two primary directions for future work that immediately stand out.
The first is strictly application focused. This model needs to be stress-
tested on multiple cities over multiple runs, and could be implemented in a
real-world system. To do so, we would likely want to collect more statistics,
such as what percentage of users have to wait, and how long the longest wait
times are. There are always more experiments to be run.

The other direction is the most theoretical, but has a lot of potential. In
theory, everything done by this network could be down by a fully-connected
network, but no network ever came close to achieving these results. As I see
it, feature-engineering and simulation aside, all that this paper did is tie
weights together and prune weights in the fully-connected network. It would be
wonderful if there were a system that could detect the pattern in the inputs
(that every $k$ dimensions represented a station) and automatically do this
pruning and weight-tying. This direction still involves a large amount of work
before coming to fruition, but it has many possible applications.

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##  8 Appendix

Calculations on time savings done using the assumption that the average
individual drives 13,000 miles per year and that the average electric vehicle
range is 110 miles (comparable to the Nissan Leaf or BAIC EC180). Of 560,000
electric vehicles on the road worldwide, if half of them see use, that results
in 33 million fill-ups per year. Our simulation estimates a 7.7 minute
reduction in average wait time per fill-up.


# The stable and augmented base locus under finite morphisms

Tanuj Gomez  Tanuj Gomez, Mathematisches Institut, Albert-Ludwigs-Universität
Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany
<EMAIL_ADDRESS>
###### Abstract.

We study the pullback of the stable and augmented base locus under a finite
surjective morphism between normal varieties over a perfect field.

###### Key words and phrases:

augmented base locus, stable base locus, finite morphisms, foliation

###### 2020 Mathematics Subject Classification:

14A10, 14E99

Supported by the DFG-Graduiertenkolleg GK1821 “Cohomological Methods in
Geometry” at the University of Freiburg.

##  1\. introduction

The _stable_ and _augmented_ base locus, denoted $\mathbb{B}(\cdot)$ and
$\mathbb{B}_{+}(\cdot)$, are fundamental invariants in Birational Geometry and
in Complex Geometry, particularly when positivity and asymptotic behaviour of
divisors, line bundles, or vector bundles are discussed [Nak00, BDPP13,
ELM+06].

In this article, we study the behaviour of the stable and augmented base locus
under the pullback via a finite surjective morphism. Previous results in this
direction, [DCL21] work in the setting of complex manifolds. They, however,
rely on powerful analytic methods such as currents and Lelong numbers [Bou04].
If one is willing to impose strong positivity assumptions on the divisors,
using [DCDC19] and [Bir17], these results can be extended to projective
schemes. Our main result, Theorem 1.1, generalises these results to normal
varieties over a perfect field.

######  Theorem 1.1.

Let $X$ and $Y$ be normal projective varieties over a perfect field $k$. Let
$f:X\longrightarrow Y$ be a finite surjective morphism over $k$. Then the
following hold:

  1. (1.1.1)

For every $\mathbb{Q}$-divisor $D$ on $Y$, we have an equality of reduced
schemes $\mathbb{B}(f^{*}D)=f^{-1}\mathbb{B}(D)$.

  2. (1.1.2)

For every $\mathbb{R}$-divisor $\Delta$ on $Y$, we have an equality of reduced
schemes $\mathbb{B}_{+}(f^{*}\Delta)=f^{-1}\mathbb{B}_{+}(\Delta)$.

In Section 2, we collect the definitions of $\mathbb{Q}$-divisors,
$\mathbb{R}$-divisors, the stable and augmented base locus. In Section 3, we
prove Theorem 1.1 in the separable case by passing to the Galois closure.
There, we use generalised Reynolds operators to produce Galois-invariant
sections to study the stable base locus. In Section 4, we use foliations to
tackle the inseparable case.

###  1.1. Acknowledgements

The author would like to thank _Johan Commelin, Fabio Bernasconi, Andreas
Demleitner, Luca Di Cerbo, Erwan Rousseau and Roberto Svaldi_ for taking their
time to read earlier drafts of this paper and giving the author useful
comments and insights.

Special thanks to _Calum Spicer_ for the insight into tackling the inseparable
case; and to the author’s supervisor _Stefan Kebekus_ for making this article
even remotely readable!

##  2\. Preliminaries

This section contains the necessary background on the stable and augmented
base locus and foliations.

######  Convention 2.1.

Throughout we work with a perfect field $k$. A _$k$ -variety_ is an
irreducible, separated and integral scheme of finite type over $k$. We will,
in Section 4, come across morphisms between $k$-varieties that are not
morphism over $k$. To emphasis this, we will say _$k$ -morphism_ to mean a
morphism between $k$-varieties that is a morphism of schemes over the base
field $k$, a _morphism_ is a morphism of schemes that is not necessarily over
$k$.

######  Remark 2.2.

The perfect assumption on $k$ is crucial for Section 4.

###  2.1. Stable and augmented base locus

We will follow [Bir17] for the definition of augmented base locus for
$\mathbb{R}$-divisors on a projective scheme. We will then prove a couple of
small but key lemmas that will be used throughout the later sections.

######  Definition 2.3 ($\mathbb{Q}$-divisors, $\mathbb{R}$-divisors and
$\mathbb{R}$-linear equivalence).

Let $Y$ be a $k$-variety and denote the group of Cartier divisors on $Y$ as
$\operatorname{Div}(Y)$.

  1. (2.3.1)

A $\mathbb{Q}$-_divisor_ on $Y$ is an element in
$\operatorname{Div}(Y)\otimes\mathbb{Q}$.

  2. (2.3.2)

An $\mathbb{R}$-_divisor_ on $Y$ is an element in
$\operatorname{Div}(Y)\otimes\mathbb{R}$.

  3. (2.3.3)

Two $\mathbb{R}$-divisors $\Delta_{1}$ and $\Delta_{2}$ are
$\mathbb{R}$_-linearly equivalent_ if $\Delta_{1}-\Delta_{2}=\sum a_{i}L_{i}$
where $a_{i}\in\mathbb{R}$ and $L_{i}$ are Cartier divisors linearly
equivalent to 0. We denote $\mathbb{R}$-linear equivalence as
$\Delta_{1}\sim_{\mathbb{R}}\Delta_{2}$.

######  Definition 2.4 (Stable base locus).

Let $Y$ be a $k$-variety.

  1. (2.4.1)

For a Cartier divisor $L$, the stable base locus is defined as

$\mathbb{B}(L):=\Bigg{(}\bigcap_{m\in\mathbb{N}}\operatorname{Bs}(mL)\Bigg{)}_{\operatorname{red}},$

where $\operatorname{Bs}(mL)$ is the base locus of the linear system $|mL|$.

  2. (2.4.2)

For a $\mathbb{Q}$-divisor $D$ and a positive integer $b$ such that $bD$ is
Cartier, the stable locus is defined as

$\mathbb{B}^{b}(D):=\mathbb{B}(bD).$

######  Remark 2.5.

In the setting of Definition (2.4.2), the stable base locus for $D$ does not
depend on the choice of the integer $b$. This follows from the observation,
for any Cartier divisor $L$ and any positive integer $m$, that
$\operatorname{Bs}(L)\supseteq\operatorname{Bs}(mL)$. This implies that
$\mathbb{B}(L)\supseteq\mathbb{B}(mL)$ and by Definition 2.4, we find
$\mathbb{B}(L)\subseteq\mathbb{B}(mL)$. Hence we will simply denote the stable
base locus as $\mathbb{B}(D):=\mathbb{B}^{b}(D)$.

We will use the definition of the augmented base locus for
$\mathbb{R}$-divisors from [Bir17, Definition 1.2].

######  Definition 2.6 (Augmented base locus).

Let $\Delta$ be an $\mathbb{R}$-divisor on a projective $k$-variety $Y$ and
$A$ an ample Cartier divisor on $Y$. Choose a representation
$\Delta\sim_{\mathbb{R}}\sum t_{i}A_{i}$ where $t_{i}\in\mathbb{R}$ and
$A_{i}$ are very ample Cartier divisors. Then, the _augmented base locus_ is

$\mathbb{B}^{A,\sum
t_{i}A_{i}}_{+}(\Delta):=\Bigg{(}\bigcap_{m\in\mathbb{N}}\mathbb{B}\bigl{(}\langle
m\Delta\rangle-A\bigr{)}\Bigg{)}_{\operatorname{red}}\leavevmode\nobreak\ ,$

where $\langle m\Delta\rangle=\sum\lfloor mt_{i}\rfloor A_{i}$.

######  Remark 2.7.

The augmented base locus does not depend on the choice of an ample Cartier
divisor $A$, or representation $\Delta\sim_{\mathbb{R}}\sum t_{i}A_{i}$
[Bir17, Lemma 3.1]. For this reason we drop the superfluous superscripts to
simply denote the augmented base locus as

$\mathbb{B}_{+}(\Delta):=\mathbb{B}^{A,\sum t_{i}A_{i}}_{+}(\Delta).$

By [Bir17, Lemma 3.1] this definition is consistent with the definitions found
in [Laz04b, ELM+06, ELM+09].

######  Notation 2.8 (Pullback of the stable and augmented base locus).

In the setting of Definition 2.4, let $g:Z\longrightarrow Y$ be any surjective
finite morphism that is not necessarily a $k$-morphism. We abuse notation by
denoting $g^{-1}\mathbb{B}(D)$ as the reduced scheme associated to the scheme
theoretic pullback of $\mathbb{B}(D)$. We do the same with the augmented base
locus.

######  Remark 2.9.

The following Lemma 2.10 will allow us to reduce statement (1.1.2) to
statement (1.1.1) of Theorem 1.1.

######  Lemma 2.10.

Let $X$ and $Y$ be projective $k$-varieties and $f:X\longrightarrow Y$ be a
finite morphism111not necessarily a $k$-morphism. Suppose that for all Cartier
divisors $L$ we have $\mathbb{B}(f^{*}L)=f^{-1}\mathbb{B}(L)$. Then for all
$\mathbb{R}$-divisors $\Delta$, we have
$\mathbb{B}_{+}(f^{*}\Delta)=f^{-1}\mathbb{B}_{+}(\Delta)$.

###### Proof.

To set up the proof, let $A$ be an ample Cartier divisor on $Y$. Let $\Delta$
be an $\mathbb{R}$-divisor and choose a representation
$\Delta\sim_{\mathbb{R}}\sum t_{i}A_{i}$ where $t_{i}\in\mathbb{R}$ and the
$A_{i}$ are very ample Cartier divisors such that $f^{*}A_{i}$ are very ample.
This is possible, as the pullback of an ample divisor under a finite morphism
is ample [Laz04a, Proposition 1.2.13]. Observe then that
$f^{*}\Delta\sim_{\mathbb{R}}\sum t_{i}f^{*}A_{i}$. By this choice of
representation, we find that for all $m$, we have $f^{*}\langle
m\Delta\rangle=\langle mf^{*}\Delta\rangle$.

Finally we compute that

$\displaystyle f^{-1}\mathbb{B}_{+}(\Delta)$
$\displaystyle=\bigcap_{m\in\mathbb{N}}f^{-1}\mathbb{B}\bigl{(}\langle
m\Delta\rangle-A\bigr{)}$
$\displaystyle=\bigcap_{m\in\mathbb{N}}\mathbb{B}\bigl{(}f^{*}\langle
m\Delta\rangle-f^{*}A\bigr{)}\ \ \ \ \textmd{by hypothesis}$
$\displaystyle=\bigcap_{m\in\mathbb{N}}\mathbb{B}\bigl{(}\langle
mf^{*}\Delta\rangle-f^{*}A\bigr{)}=\mathbb{B}_{+}(f^{*}\Delta).\qed$

The following is a trivial statement. As we will use it several times, we
formulate it as a lemma.

######  Lemma 2.11.

Let $X$ and $Y$ be normal $k$-varieties and $D$ a $\mathbb{Q}$-divisor on $Y$.
For any finite surjective morphism††footnotemark:  $f:X\longrightarrow Y$ we
find that $\mathbb{B}(f^{*}D)\subseteq f^{-1}\mathbb{B}(D)$. ∎

######  Corollary 2.12.

Let $X,Y$ and $Z$ be normal $k$-varieties, let
$Z\xrightarrow{g}X\xrightarrow{f}Y$ be a composition of finite surjective
morphisms††footnotemark: , and let $D$ be a $\mathbb{Q}$-divisor on $Y$. If we
have $\mathbb{B}\bigl{(}(f\circ g)^{*}L\bigr{)}=(f\circ g)^{-1}\mathbb{B}(L)$
then $\mathbb{B}(f^{*}L)=f^{-1}\mathbb{B}(L)$ holds.

###### Proof.

There is a chain of inclusions $\mathbb{B}\bigl{(}(f\circ g)^{*}D)\subseteq
g^{-1}\mathbb{B}(f^{*}D)\subseteq g^{-1}\bigl{(}f^{-1}\mathbb{B}(L)\bigr{)}$.
By hypothesis $\mathbb{B}\bigl{(}(f\circ g)^{*}D\bigr{)}=(f\circ
g)^{-1}\mathbb{B}(D)$, hence the chain of inclusions are indeed equalities.
The result follows from applying $g$ to the second inclusion. ∎

###  2.2. Foliation

We recall the definition of a foliation.

######  Definition 2.13 (Foliation).

Let $X$ be normal $k$-variety and denote the sheaf of Kähler differentials on
$X$ by $\Omega^{1}_{X}$. A _foliation_ on $X$ is a saturated subsheaf of the
tangent sheaf $\scr{T}_{X}:=\bigl{(}\Omega^{1}_{X}\bigr{)}^{\vee}$ that is
closed under the Lie bracket.

######  Definition 2.14 ($p$-closed Foliation).

Let $\scr{F}$ be a foliation on a normal $k$-variety $X$. Suppose that the
characteristic of the field $k$ is $p>0$. We say that $\scr{F}$ is _$p$
-closed_ if for all $U\subseteq X$ open and $\forall\partial\in\scr{F}(U)$,
the composition of $\partial$ with itself $p$-times is in $\scr{F}(U)$.

##  3\. The case of separable morphisms

This section is devoted to proving the following result:

######  Theorem 3.1.

Let $f:X\longrightarrow Y$ be a finite, surjective and separable $k$-morphism
of normal projective $k$-varieties. Then for every $\mathbb{Q}$-divisor $D$ on
$Y$, we have $\mathbb{B}(f^{*}D)=f^{-1}\mathbb{B}(D)$.

###  3.1. Symmetric Sections

To prove Theorem 3.1 we will first study the partial case when
$f:X\longrightarrow Y$ is a finite _Galois_ morphism, that is when $f$ is
isomorphic to a quotient of $X$ by a finite group $G$. In this case we make
use of the group action $G$ to produce a plethora of sections by generalising
the classical Reynolds operator.

Suppose that $L$ is a Cartier divisor on $Y$ and $\sigma\in
H^{0}\bigl{(}X,\scr{O}_{X}(f^{*}L)\bigr{)}$, we can apply the classical
Reynolds operator on $\sigma$ to obtain a section, which we denote by,
$R_{1}(\sigma)\in H^{0}\bigl{(}Y,\scr{O}_{Y}(L)\bigr{)}$. Even if
$\sigma(x)\neq 0$ for some $x\in X$, there is no guarantee that
$R_{1}\bigl{(}\sigma\bigr{)}(f(x))\neq 0$. Hence it is hard to control the
stable base locus with the Reynolds operator. The following Construction 3.2
remedies this issue.

######  Construction 3.2.

Let $X$ be a projective $k$-variety and $G<\operatorname{Aut}_{k}(X)$ be a non
trivial finite group. Denote by $f:X\longrightarrow Y$ the quotient of $X$ by
$G$. Take a Cartier divisor $L$ on $Y$. Write $G$ as

$G=\\{g_{1},g_{2},\ldots,g_{N}\\},$

such that $g_{1}$ is the identity element. For each $1\leq i\leq N$ there is a
linear map

$\displaystyle R_{i}:H^{0}\bigl{(}X,\scr{O}_{X}(f^{*}L)\bigr{)}$
$\displaystyle\longrightarrow H^{0}\bigl{(}X,\scr{O}_{X}(i\cdot
f^{*}L)\bigr{)}=H^{0}\bigl{(}X,\operatorname{Sym}^{i}\scr{O}_{X}(f^{*}L)\bigr{)}$
$\displaystyle\sigma$ $\displaystyle\longmapsto\sum_{1\leq
j_{1}<j_{2}<\cdots<j_{i}\leq N}g_{j_{1}}^{*}\sigma\otimes
g_{j_{2}}^{*}\sigma\otimes\cdots\otimes g_{j_{i}}^{*}\sigma.$

It should be clear that $R_{i}(\sigma)$ are $G$-invariant by construction. It
will also be useful to define the following linear maps for $i=1,\ldots,N-1$:

$\displaystyle\tau_{i}:H^{0}\bigl{(}X,\scr{O}_{X}(f^{*}L)\bigr{)}$
$\displaystyle\longrightarrow H^{0}\bigl{(}X,\scr{O}_{X}(i\cdot
f^{*}L)\bigr{)}$ $\displaystyle\sigma$
$\displaystyle\longmapsto\sum_{1<j_{1}<j_{2}<\cdots<j_{i}\leq
N}g_{j_{1}}^{*}\sigma\otimes g_{j_{2}}^{*}\sigma\otimes\cdots\otimes
g_{j_{i}}^{*}\sigma$

######  Remark 3.3.

In Construction 3.2, the definitions of $R_{i}(\sigma)$ and $\tau_{i}(\sigma)$
do not depend on the choice of ordering of $G$.

######  Lemma 3.4.

In the setting of Construction 3.2, given a section $\sigma\in
H^{0}\bigl{(}X,\scr{O}_{X}(f^{*}L)\bigr{)}$ there are relations:

$R_{i}(\sigma)=\begin{cases}\sigma+\tau_{1}(\sigma)&\textmd{ for $i=1$}\\\
\sigma\otimes\tau_{i-1}(\sigma)+\tau_{i}(\sigma)&\textmd{ for $1<i<N$}\\\
\sigma\otimes\tau_{N-1}(\sigma)&\textmd{ for $i=N$}.\end{cases}$

###### Proof.

The cases $i=1$ and $i=N$ are clear. For $1<i<N$, we can split $R_{i}(\sigma)$
as follows

$\displaystyle R_{i}(\sigma)$ $\displaystyle=\sum_{1\leq
j_{1}<j_{2}<\cdots<j_{i}\leq N}g_{j_{1}}^{*}\sigma\otimes
g_{j_{2}}^{*}\sigma\otimes\cdots\otimes g_{j_{i}}^{*}\sigma$
$\displaystyle=\sum_{1=j_{1}<j_{2}<\cdots<j_{i}\leq
N}g_{j_{1}}^{*}\sigma\otimes g_{j_{2}}^{*}\sigma\otimes\cdots\otimes
g_{j_{i}}^{*}\sigma$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\sum_{1<j_{1}<\cdots<j_{i}\leq N}g_{j_{1}}^{*}\sigma\otimes
g_{j_{2}}^{*}\sigma\otimes\cdots\otimes g_{j_{i}}^{*}\sigma$
$\displaystyle=\sigma\otimes\tau_{i-1}(\sigma)+\tau_{i}(\sigma).\qed$

######  Lemma 3.5.

In the setting of Construction 3.2, let $\sigma\in
H^{0}\bigl{(}X,\scr{O}_{X}(f^{*}L)\bigr{)}$ and $x\in X$ be a scheme theoretic
point such that $\sigma(x)\neq 0$. Then, there exists an $1\leq i\leq N$ such
that $R_{i}(\sigma)(x)\neq 0$.

###### Proof.

Suppose that $R_{i}(\sigma)(x)=0$ for all $i=1,\ldots,N-1$. Then by the
relations of Lemma 3.4, one finds that

$R_{N}(\sigma)(x)=(-1)^{(N-1)}\sigma^{\otimes N}(x).$

This implies that $R_{N}(\sigma)(x)$ is non-zero as it is assumed that
$\sigma(x)\neq 0$. Hence $R_{i}(\sigma)(x)$ cannot simultaneously vanish for
all $1\leq i\leq N$. ∎

###  3.2. Proof of Theorem 3.1

Lemma 2.12 allows us to pass to the Galois closure and, without loss of
generality, assume that $f:X\longrightarrow Y$ is the quotient of $X$ by a
finite group $G$.

The inclusions $\mathbb{B}(f^{*}L)\subseteq f^{-1}\mathbb{B}(L)$ follows from
Lemma 2.11.

Conversely, take $x\notin\mathbb{B}(f^{*}D)$. This means that there exists an
$m\in\mathbb{N}$ and a section $\sigma\in H^{0}\bigl{(}X,\scr{O}_{X}(m\cdot
f^{*}D)\bigr{)}$ such that $\sigma(x)\neq 0$. To show that $x\notin
f^{-1}\mathbb{B}(D)$, we need to produce a section in
$H^{0}\bigl{(}Y,\scr{O}_{Y}(n\cdot D)\bigr{)}$, for some $n>0$, that does not
vanish at $f(x)$. By Lemma 3.5, we find that there exists an $i$, such that
$R_{i}(\sigma)\in f^{*}H^{0}\bigl{(}Y,\scr{O}_{Y}(mi\cdot D)\bigr{)}$
satisfying $R_{i}(\sigma)(x)\neq 0$. ∎

##  4\. The Case of inseparable morphisms

The goal of this section is to remove the separability hypothesis of Theorem
3.1, and hence proving Theorem 1.1. In characteristic $0$, every finite
morphism is automatically separable. In positive characteristic, a finite
field extension splits into a separable extension and a purely inseparable
extension. Hence it enough to study purely inseparable morphisms. In this
case, the idea is to use Ekedahl’s theory of foliations and purely inseparable
morphisms to factorise a purely inseparable $k$-morphism into purely
inseparable morphisms of height one [Eke87]. We then study the pullback of the
stable and augmented base locus via a purely inseparable morphism of height
one.

######  Setting 4.1.

Assume that the perfect field $k$ has characteristic $p>0$.

###  4.1. Ekedahl’s theory of foliations and purely inseparable morphisms

The original paper [Eke87] covers the setting of smooth $k$-varieties.
However, it is crucial to work in the setting of normal $k$-varieties. The
excellent [PW22] gives a nice exposition of this more general setting in
[PW22, Section 2.4], which we follow closely.

######  Definition 4.2 (Absolute Frobenius).

In Setting 4.1, let $X$ be a normal $k$-variety. The _absolute Frobenius_
morphism $F_{X}:X\longrightarrow X$ is the endomorphism of schemes induced by
the identity on the underlying topological space and the morphism between the
sheaves of rings, $F^{\\#}_{X}:\scr{O}_{X}\longrightarrow\scr{O}_{X}$, is
given by sending $s\mapsto s^{p}$.

######  Remark 4.3.

In general, the absolute Frobenius morphism is not a $k$-morphism. The
pushforward $\bigl{(}F_{X}\bigr{)}_{*}\scr{O}_{X}$ is isomorphic to
$\scr{O}_{X}$ as sheaves. However the $\scr{O}_{X}$-module structure on
$\bigl{(}F_{X}\bigr{)}_{*}\scr{O}_{X}$ is given by the rule

$\displaystyle\scr{O}_{X}\times\bigl{(}F_{X}\bigr{)}_{*}\scr{O}_{X}$
$\displaystyle\longrightarrow\bigl{(}F_{X}\bigr{)}_{*}\scr{O}_{X}$
$\displaystyle(r,s)\quad\quad\ \ $ $\displaystyle\longmapsto\ \ r^{p}\cdot s.$

We will label $\bigl{(}F_{X}\bigr{)}_{*}\scr{O}_{X}$ with this
$\scr{O}_{X}$-module structure by $\scr{O}_{F_{X}}$.

######  Definition 4.4 (Purely inseparable morphism).

In Setting 4.1, a finite surjective morphism222not necessarily a $k$-morphisms
$g:X\longrightarrow Z$ between normal $k$-varieties is called _purely
inseparable_ if the field extension $k(Z)\subseteq k(X)$ is a purely
inseparable extension.

######  Definition 4.5 (Purely inseparable morphism of height one).

In Setting 4.1, a finite surjective morphism††footnotemark:
$g:X\longrightarrow Z$ between normal $k$-varieties is called _purely
inseparable of height one_ if it is purely inseparable and there exists a
morphism $h:Z\longrightarrow X$ such that $h\circ g=F_{X}$, where $F_{X}$ is
the absolute Frobenius of X.

####  4.1.1. Foliations and finite morphisms

Here we highlight how to obtain a foliation from a finite morphism and vice-
versa.

######  Construction 4.6 (Quotient by a foliation).

In Setting 4.1, let $X$ be a normal $k$-variety and $\scr{F}$ a foliation on
$X$. The functions of $X$ that are annihilated by $\scr{F}$ define a sheaf
$\scr{A}_{\scr{F}}$. More precisely, the assignment sending each open set
$U\subseteq X$ to

$\scr{A}{F}(U):=\\{s\in\scr{O}_{X}(U)\mid\textmd{$\forall V\subseteq U$ open,
$\forall\partial\in\scr{F}(V):\partial s=0$}\\},$

defines $\scr{A}_{\scr{F}}$. The $\scr{O}_{X}$-module structure of
$\scr{A}_{\scr{F}}$ is defined by the multiplication rule

$\displaystyle\scr{O}_{X}\times\scr{A}{F}$ $\displaystyle\longrightarrow\
\scr{A}{F}$ $\displaystyle(r,s)\quad$ $\displaystyle\longmapsto r^{p}\cdot s$

There is an inclusion of $\scr{O}_{X}$-modules
$\scr{A}{F}\hookrightarrow\scr{O}_{F_{X}}$. Taking the relative spectrum of
this inclusion we obtain

${X}$${\operatorname{Spec}_{X}\scr{A}{F}}$${X}$${\operatorname{Spec}k}$${\operatorname{Spec}k}$${\operatorname{Spec}k.}$$\scriptstyle{F_{X}}$$\scriptstyle{F_{\operatorname{Spec}k}}$

The $k$-morphism $X\longrightarrow\operatorname{Spec}_{X}\scr{A}_{\scr{F}}$ is
called the _quotient of $X$ by $\scr{F}$_.

######  Construction 4.7 (Foliation associated to a finite morphism).

In Setting 4.1, let $f:X\longrightarrow Y$ be a finite surjective $k$-morphism
between normal $k$-varieties. The sections of $\scr{T}_{X}$ which annihilate
$f^{-1}\scr{O}_{Y}\subseteq\scr{O}_{X}$ define a subsheaf of $\scr{T}_{X}$.
More precisely, the assignment sending each open set $U\subseteq X$ to

$\scr{F}_{f}(U):=\\{\partial\in\scr{T}_{X}(U)\mid\textmd{$\forall V\subseteq
U$ open, $\forall s\in f^{-1}\scr{O}_{Y}(V):\partial s=0$}\\}$

defines a subsheaf of $\scr{O}_{X}$-modules of $\scr{T}_{X}$.

######  Lemma 4.8.

In Construction 4.7, the sheaf $\scr{F}_{f}$ is a $p$-closed foliation.

###### Proof.

To show that $\scr{F}_{f}$ is closed under the Lie bracket, fix open sets
$V\subseteq U\subseteq X$ and a function $s\in f^{-1}\scr{O}_{Y}(V)$. For all
$\partial,\partial^{\prime}\in\scr{F}_{f}(U)$, compute the Lie bracket to find

$[\partial,\partial^{\prime}](s)=\partial\bigl{(}\partial^{\prime}(s)\bigr{)}-\partial^{\prime}\bigl{(}\partial(s)\bigr{)}=0.$

A similar computation shows that $\scr{F}_{f}$ is $p$-closed.

Now we show that $\scr{F}_{f}$ is a saturated in $\scr{T}_{X}$. If
$\scr{F}_{f}=\scr{T}_{X}$ or $\scr{F}_{f}=0$, then there is nothing to prove,
so let us assume $\scr{F}_{f}$ is a proper subsheaf. Fix an open set
$U\subseteq X$. Suppose for a contradiction that ${\tau}$ is a _non-zero_
torsion section in $\left.\raise
2.0pt\hbox{$\scr{T}_{X}$}\right/\hskip-2.0pt\raise-2.0pt\hbox{$\scr{F}_{f}$}(U)$.
By shrinking $U$ to a smaller open set, if necessary, we can choose a lift
$\partial\in\scr{T}_{X}(U)$ of $\tau$. Since $\tau$ is non zero, this implies
that $\partial\notin\scr{F}_{f}(U)$. Since $\tau$ is a torsion section, there
exists a non zero function $r\in\scr{O}_{X}(U)$ such that $r\cdot\tau=0$. This
means that $r\cdot\partial\in\scr{F}_{f}(U)$.

To arrive at the contradiction, pick a section $t\in f^{-1}\scr{O}_{Y}(V)$ for
some $V\subseteq U$ open such that its evaluation $\partial(t)\neq 0$ however
then $\bigl{(}r\cdot\partial\bigr{)}(t)=0$. Since $X$ is integral, as it is
normal, this leads to contradiction as it suggests that $r$ is a zero divisor.
∎

####  4.1.2. Ekedahl’s Theorem

Restricting the class of morphisms in Construction 4.7, we obtain a
correspondence between $p$-closed foliations and purely inseparable
$k$-morphisms of height one.

######  Remark 4.9.

It is a necessary condition that $k$ is a perfect field in Theorem 4.10 below.

######  Theorem 4.10 ([PW22, Proposition 2.9]).

In Setting 4.1, let $X$ be a normal $k$-variety. There is a 1-1 correspondence
between

  1. (4.10.1)

$p$-closed foliations $\scr{F}\subset\scr{T}_{X}$, and

  2. (4.10.2)

purely inseparable $k$-morphisms $X\longrightarrow Z$ of height one.

The correspondence is given by:

  * •

sending the foliation $\scr{F}$ to the quotient of $X$ by $\scr{F}$, and

  * •

sending the morphism $X\xrightarrow{g}Z$ to the $p$-closed foliation
$\scr{F}_{g}$.

Moreover the correspondence satisfies
$[k(Z):k(X)]=p^{\operatorname{rank}\scr{F}}$. ∎

######  Remark 4.11.

Theorem 4.10 asserts that the quotient by a $p$-closed foliation is in
particular a normal variety, and the quotient morphism is finite and
surjective.

###  4.2. Factorising purely inseparable morphisms

Using Theorem 4.10, we will factorise a purely inseparable $k$-morphism into
height one pieces.

######  Proposition 4.12.

Let $f:X\longrightarrow Y$ be a purely inseparable $k$-morphism between normal
$k$-varieties. Then there exists a positive integer $l$ and normal
$k$-varieties $Z_{1},...,Z_{l}$ that factorise $f$

$X=Z_{1}\longrightarrow Z_{2}\longrightarrow\cdots\longrightarrow Z_{l}=Y.$

The morphisms $Z_{\bullet}\longrightarrow Z_{\bullet+1}$ are purely
inseparable $k$-morphisms of height one.

To prove Proposition 4.12, we will need the following technical lemma.

######  Lemma 4.13.

In Construction 4.7, the foliation $\scr{F}_{f}$ is zero if and only if $f$ is
a separable morphism.

######  Remark 4.14.

The consequence of Lemma 4.13 is that, the quotient by $\scr{F}_{f}$ is the
identity morphism if and only if $f$ is a separable morphism.

We will prove Lemma 4.13 after we give a proof of Proposition 4.12.

###### Proof of Proposition 4.12.

If the morphism $f:X\longrightarrow Y$ is the identity, we are done. Supposing
otherwise, we will show that then it is possible to factorise
$f:X\longrightarrow Y$ into

(4.14.1) $X\xrightarrow{g}Z_{2}\xrightarrow{f_{2}}Y$

where $g$ is a purely inseparable $k$-morphism of height one and $f_{2}$ is a
purely inseparable $k$-morphism. The morphism $g$ is simply the quotient of
the foliation associated to the finite morphism $f:X\longrightarrow Y$, which
by Theorem 4.10 is purely inseparable of height one and by Lemma 4.13 is not
the identity morphism.

Using the fact that purely inseparable morphisms are universal homeomorphisms
[GW10, Exercise 12.32], we find that $f$ and $g$ are universal homeomorphisms
and that the underlying topological spaces of $Y$, $X$ and $Z_{2}$ are all
homeomorphic. Hence the morphism $f_{2}:|Z_{2}|\longrightarrow|Y|$, between
the underlying topological spaces is given by $g^{-1}\circ f$.

For the morphism of sheaves of rings, observe that there is an inclusion of
sheaves $f^{-1}\scr{O}_{Y}\subseteq
g^{-1}\scr{O}_{Z_{2}}\subseteq\scr{O}_{X}$. The sheaf $g^{-1}\scr{O}_{Z_{2}}$
is the sheaf of functions annihilated by the foliation $\scr{F}_{f}$, which
contains $f^{-1}\scr{O}_{Y}$, as in Construction 4.6. For an open set
$V\subseteq X$ we obtain a ring homomorphism

(4.14.2)
$\scr{O}_{Y}\bigl{(}f(V)\bigr{)}\hookrightarrow\scr{O}_{Z_{2}}\bigl{(}g(V)\bigr{)}.$

For any open set $U\subseteq Z_{2}$, by setting $V=g^{-1}(U)$ in Equation
(4.14.2), we obtain a well defined morphism of sheaves
$f_{2}^{\\#}:f^{-1}_{2}\scr{O}_{Y}\longrightarrow\scr{O}_{Z_{2}}$. Hence we
get a morphism of schemes $f_{2}:Z_{2}\longrightarrow Y$ factorising $f$ as
found in Equation (4.14.1).

We proceed by applying the same argument to $f_{2}:Z_{2}\longrightarrow Y$.
Since $f:X\longrightarrow Y$ a finite morphism, this process of factorising
must stop after a finite number of steps. This proves the proposition. ∎

In preparation to prove Lemma 4.13, it will be useful to give an alternative
description of $\scr{F}_{f}$ in terms of differentials, to give us a concrete
connection between $\scr{F}_{f}$ and the separability of $f$.

######  Construction 4.15.

In the Setting of Construction 4.7, Let $\scr{M}$ be the image of the map
$f^{*}\Omega^{1}_{Y}\longrightarrow\Omega^{1}_{X}$. Denote by
$\operatorname{Ann}{\scr{M}}$ the sections of $\scr{T}_{X}$ that are
annihilated by $\scr{M}$. This is described in the following exact sequence:

(4.15.1)
$0\longrightarrow\operatorname{Ann}{\scr{M}}\longrightarrow\scr{T}_{X}\longrightarrow\bigl{(}\scr{M}\bigr{)}^{\vee}.$

Moreover, since the quotient of $f^{*}\Omega^{1}_{Y}$ by $\scr{M}$ is
$\Omega^{1}_{X/Y}$, we find that
$\operatorname{Ann}{\scr{M}}=\bigl{(}\Omega^{1}_{X/Y}\bigr{)}^{\vee}$.

######  Lemma 4.16.

The sheaves $\scr{F}_{f}$ and $\operatorname{Ann}\scr{M}$ constructed in
Construction 4.7 and Construction 4.15 are isomorphic.

###### Proof.

Let $U\subseteq X$ be an open set and fix a section $s\in
f^{-1}\scr{O}_{Y}(U)$ and fix a derivation
$\partial\in\operatorname{Ann}\scr{M}(U)$. Observe that $ds\in\scr{M}(U)$. We
can compute that $\partial(ds)=\partial(s)=0$, which shows that
$\partial\in\scr{F}_{f}(U)$.

Conversely, fix a derivation $\partial\in\scr{F}_{f}(U)$ for any open set
$U\subseteq X$. We need to show that for any $\omega\in\scr{M}(U)$, that
$\partial\omega=0$. Since $\scr{M}$ is locally generated by sections of the
form $f^{*}dy$, where $y$ is a local section of $\scr{O}_{Y}$; it is enough to
check that $\partial(f^{*}dy)=0$. This follows from the computation
$\partial(f^{*}dy)=f^{*}(\partial dy)=0$, since $f^{*}y$ is a local section of
$f^{-1}\scr{O}_{Y}$. ∎

###### Proof of Lemma 4.13.

By Lemma 4.16 and Construction 4.15, we get that
$\scr{F}_{f}=\operatorname{Ann}\scr{M}=\bigl{(}\Omega^{1}_{X/Y}\bigr{)}^{\vee}$.
This allows us to work with relative differentials.

Suppose that $f:X\longrightarrow Y$ is separable. Since $\Omega^{1}_{X/Y}$ is
supported on the ramification locus of $f$, it is torsion. Therefore the dual
$\bigl{(}\Omega^{1}_{X/Y}\bigr{)}^{\vee}=\scr{F}_{f}=0$.

Conversely, suppose that $f$ is not separable. Denote the generic point of $X$
by $\eta$. We compute that

$\displaystyle\bigl{(}\operatorname{Ann}{\scr{M}}\bigr{)}_{\eta}=\Bigl{(}\bigl{(}\Omega^{1}_{X/Y}\bigr{)}^{\vee}\Bigr{)}_{\eta}$
$\displaystyle=\Bigl{(}\bigl{(}\Omega^{1}_{X/Y}\bigr{)}_{\eta}\Bigr{)}^{\vee}$
[Har77, III Propostion 6.8]
$\displaystyle=\bigr{(}\Omega^{1}_{K(X)/K(Y)})^{\vee}\neq 0$
$\displaystyle\textmd{\cite[cite]{[\@@bibref{}{Hart}{}{}, II Theorem 8.6A]}}.$

Hence $\scr{F}_{f}\neq 0$. ∎

###  4.3. Stable base locus under purely inseparable morphisms

In this subsection, we make use of the Frobenius morphism to understand the
pullback of the stable base locus via a purely inseparable height one
morphism.

######  Lemma 4.17.

In Setting 4.1, let $F_{X}:X\longrightarrow X$ be the absolute Frobenius
morphism and let $D$ be a $\mathbb{Q}$-divisor on $X$. Then
$\mathbb{B}(F_{X}^{*}D)=F_{X}^{-1}\mathbb{B}(D)$.

###### Proof.

The left hand side of the expression is simply $\mathbb{B}(pD)$ since
$F_{X}^{*}D=pD$. The right hand side is $\mathbb{B}(D)$ as the stable base
locus is a closed subset of $X$ and the absolute Frobenius morphism leaves the
points of $X$ fixed. The expression immediately follows from the observation
that $\mathbb{B}(pD)=\mathbb{B}(D)$. ∎

######  Lemma 4.18.

In Setting 4.1, let $g:X\longrightarrow Z$ be a purely inseparable height one
morphism333not necessarily a $k$-morphism and $D$ a $\mathbb{Q}$-divisor on
$Z$. Then $\mathbb{B}(g^{*}D)=g^{-1}\mathbb{B}(D)$.

###### Proof.

Since $g$ is purely inseparable of height one, there exists a morphism
$Z\xrightarrow{i}X$ such that $g\circ i=F_{X}$. We claim that $i\circ
g=F_{Z}$, the Frobenius endomorphism of $Z$. Then by Lemma 4.17 and Lemma 2.12
the result follows.

To prove the claim, observe that $F_{X}\circ g=g\circ F_{Z}$ by [Sta18, Tag
0CC7]. Since $F_{X}=g\circ i$, if we can show that $g$ is an epimorphism, or
in other words left cancellative, then the claim follows. This follows from
the observation that $g$ is surjective and the map of sheaves of rings
$g^{\\#}:\scr{O}_{Z}\longrightarrow\scr{O}_{X}$ is injective. ∎

###  4.4. Proof of Theorem 1.1

We start with a series of reduction steps. It is enough to prove Statement
(1.1.1) due to Lemma 2.10.

In characteristic $0$, the morphism $f:X\longrightarrow Y$ is necessarily
separable, this case is covered by Theorem 3.1. We can then assume we are in
Setting 4.1, namely in positive characteristic.

Since any finite field extension can be split into a separable extension
followed by a purely inseparable extension [Sta18, Tag 030K], we can assume
without loss of generality that $f:X\longrightarrow Y$ is a purely inseparable
extension.

By Proposition 4.12, we obtain a factorisation of $f$

$X\longrightarrow Z_{1}\longrightarrow\cdots\longrightarrow
Z_{l}\longrightarrow Y.$

Since all the morphisms in the factorisation are purely inseparable of height
one, by Lemma 4.18 we get $\mathbb{B}(f^{*}D)=f^{-1}\mathbb{B}(D)$. ∎

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