# UniTSA: A Universal Reinforcement Learning Framework for V2X Traffic Signal
Control

Maonan Wang , Xi Xiong , Yuheng Kan, Chengcheng Xu , Man-On Pun School of
Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China
and Shanghai AI Laboratory, Shanghai, China.Key Laboratory of Road and Traffic
Engineering, Ministry of Education, Tongji University, Shanghai,
China.SenseTime Group Limited and Shanghai AI Laboratory, Shanghai,
China.School of Science and Engineering, The Chinese University of Hong Kong,
Shenzhen, Shenzhen 518172, China.Corresponding author<EMAIL_ADDRESS>
###### Abstract

Traffic congestion is a persistent problem in urban areas, which calls for the
development of effective traffic signal control (TSC) systems. While existing
Reinforcement Learning (RL)-based methods have shown promising performance in
optimizing TSC, it is challenging to generalize these methods across
intersections of different structures. In this work, a universal RL-based TSC
framework is proposed for Vehicle-to-Everything (V2X) environments. The
proposed framework introduces a novel agent design that incorporates a
junction matrix to characterize intersection states, making the proposed model
applicable to diverse intersections. To equip the proposed RL-based framework
with enhanced capability of handling various intersection structures, novel
traffic state augmentation methods are tailor-made for signal light control
systems. Finally, extensive experimental results derived from multiple
intersection configurations confirm the effectiveness of the proposed
framework. The source code in this work is available at
https://github.com/wmn7/Universal_Light

_K_ eywords Traffic signal control $\cdot$ Universal models $\cdot$
Reinforcement learning $\cdot$ Traffic state augmentation

##  1 Introduction

Traffic congestion presents a critical challenge in urban areas worldwide,
leading to wasted time for individuals, excessive fuel consumption, and
increased greenhouse gas emissions [1]. To alleviate congestion, conventional
traffic signal control (TSC) methods such as fixed-cycle traffic control [2],
the Webster method [3], and Self-Organizing Traffic Light Control (SOTL) [4]
have been developed. However, as cities continue to grow, these traditional
traffic management approaches often prove insufficient to handle increasing
traffic volumes and dynamic road conditions [5]. The emergence of V2X
technologies in transportation systems offers a promising solution by
facilitating communication and data exchange among vehicles, infrastructure,
and other road users [6]. By leveraging real-time data derived from vehicles,
the traffic management infrastructure can efficiently regulate vehicle and
pedestrian movements at intersections by dynamically adapting to the changing
traffic conditions [7].

To control signal lights based on real-time traffic conditions, various RL-
based methods have been proposed. These methods employ three primary
approaches for adjusting traffic lights, namely “Choose the next phase", “Keep
or change the phase" and “Set the phase duration". More specifically, in
“Choose the next phase", the RL agent determines the next phase to activate,
allowing for a flexible phase sequence rather than a predetermined one [8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. While this approach
offers flexibility, it has the potential to confuse drivers since the phase
selection may appear random, which could incur increased risks of traffic
accidents. For the “Keep or change the phase" approach, the RL agent decides
whether to maintain or change the current phases [23, 24, 25]. Finally, the
“Set the phase duration" approach selects the optimal duration for the current
phase from a set of predefined options [26, 27, 28]. Through direct
interaction with the environment, the RL agent effectively learns to adapt to
traffic condition changes based on real-world experiences.

Despite the significant improvements achieved by the aforementioned RL-based
methods, a major limitation is that they are designed for interactions of
designated structures. In other words, these RL models have to be redesigned
and re-trained from scratch when dealing with intersections of different
structures including approaching roads, lanes, and phases, which incurs
substantial resources in terms of data collection, model development, and
testing [29]. Thus, it is crucial to develop universal models that can be
easily adapted and deployed across a wide range of intersections. As a result,
the implementation of V2X can be efficiently scaled up without requiring
extensive customization or redevelopment for each individual intersection
[30]. In the literature, several generalized models were proposed for
different junctions [11, 18, 10, 12]. Despite their good performance, these
generalized models are only applicable to those specific intersection
configurations considered in their design. Furthermore, these models exhibit
performance degradation when encountering intersections of unseen
configurations.

Motivated the discussions above, we present a Universal Traffic State
Augmentation RL-based framework (UniTSA). The proposed framework enables the
training of a universal agent using augmented data for TSC. In order to handle
intersections of diverse configurations, a junction matrix is developed to
describe the intersection state. By using this newly proposed matrix,
intersections of different structures can be characterized by matrices of the
same size. In addition, the “Keep or change the current phase” approach is
employed as the action design in this work, ensuring consistent model
structures across intersections of different configurations. To cope with
unseen intersection structures, five traffic state augmentation methods are
developed to enrich the agent’s data collection during the training process.
These augmentation methods provide more comprehensive training, which
ultimately leads to improved performance in intersections of configurations
not available in the training set. Furthermore, the Low-Rank Adaptation (LoRA)
[31] is employed to further fine-tune the model for crucial intersections.
Finally, extensive experiments using the Simulation of Urban MObility (SUMO)
[32] platform are conducted by taking into account intersections of various
approaching roads, lanes, and phases. The experimental results demonstrate
that the proposed UniTSA model achieves excellent performance even in unseen
intersections.

Our contributions can be summarized as follows:

  * •

An adaptive TSC framework called UniTSA is proposed for V2X by leveraging a
universal RL model built upon novel agent designs that can handle diverse
intersection structures. In addition, a fine-tuning mechanism is devised to
further enhance the performance in key intersections;

  * •

Traffic state augmentation methods are developed for the proposed TSC
framework, enhancing the agent’s understanding of diverse intersections with
improved performance in both training and testing sets;

  * •

Extensive experiments on $12$ intersections of diverse structures confirm that
the proposed UniTSA model substantially outperforms conventional universal
models. Moreover, given a new intersection, UniTSA can fine-tune its pre-
trained model to achieve comparable or even better performance with
significantly reduced training time as compared to training a new model from
scratch.

The remainder of the paper is structured as follows: Section 2 provides a
summary on the TSC-related work whereas Section 3 introduces the terminology
related to road and traffic signals. After that, Section 4 presents the
proposed UniTSA framework and the five traffic state augmentation methods
before Section 5 elaborates on the experimental setup and results on UniTSA.
Finally, Section 6 concludes the paper.

##  2 Related Work

Extensive research has been conducted in the field of transportation to study
TSC. Conventionally, fixed-time signal control is one of the earliest and
widely used approaches in TSC [33]. These methods rely on pre-determined
signal timings based on historical traffic patterns or engineering guidelines.
Various optimization techniques have been proposed to determine optimal fixed-
time plans for specific intersection configurations, where the Webster [3]
method is one of the most successful TSC methods for the single intersection
case. It can calculate the cycle length and phase split for a single
intersection according to the traffic volume during a certain period (i.e.,
past $15$ minutes or $30$ minutes). However, such fixed-time control methods
often incur suboptimal performance due to their lack of adaptability to
dynamically changing traffic conditions. To cope with this problem, actuated
control methods such as Sydney Coordinated Adaptive Traffic System (SCATS)
[34], Max-pressure control [35] and Self-Organizing Traffic Light Control
(SOTL) [4] were designed to adaptively adjust signal timings based on real-
time traffic demand. Despite their many advantages, these actuated control
methods are handicapped by the necessity that expert settings are required for
each intersection. Furthermore, the performance of these actuated control
methods degrades in complex scenarios.

Recently, RL-based TSC methods have attracted substantial attention due to
their outstanding adaptability to real-time traffic conditions and impressive
capability of learning the optimal control policies in complex scenarios [36].
Generally speaking, these RL-based TSC methods can be categorized into three
categories, namely the valued-based methods [8, 23, 25, 9, 10, 11, 12, 13,
14], the policy-based methods [15, 16, 17, 18, 19] and the actor-critic
methods [20, 28, 21, 22]. Despite their good performance, most existing RL-
based TSC methods focus on training models for specific intersection
configurations or scenarios. A few attempts on training generalized TSC models
have been made in the literature. For instance, MetaLight [11] trains a more
universal model by incorporating the meta-learning strategy proposed in [37].
However, MetaLight requires re-training of its model parameters for each new
intersection encountered. To overcome this shortcoming, [18, 10, 12]
established universal models through parameter sharing. However, these methods
do not preserve the original phase structure of traffic lights. In contrast,
our proposed method can maintain the original signal light structure while
fine-tuning crucial intersections to achieve significantly improved
performance.

Figure 1: A standard $4$-way intersection illustration.

##  3 Preliminary

In this section, some concepts used in this work are defined using a standard
four-way intersection as depicted in Fig. 1 as an example. These concepts can
be easily extended to the intersections of different structures.

  * •

Lane: A lane refers to a roadway that provides a defined path for vehicles to
travel in a specific direction. At a typical intersection, there are two types
of lanes: incoming lanes $l_{in}$ (where vehicles enter) and outgoing lanes
$l_{out}$ (where vehicle exit);

  * •

Traffic Movement: A traffic movement refers to the connection between an
incoming lane $l_{in}$ to an outgoing lane $l_{out}$. For the common $4$-way
intersection in the left side of Fig. 1, there are $12$ movements in total,
including right turns, left turns, and through movements in each of the four
directions;

  * •

Movement Signal: A movement signal is defined on the traffic movement. The
green signal means the corresponding movement is allowed whereas the red
signal is prohibited. As the right-turn traffic can move regardless of the
signal, only eight movement signals out of the $12$ possible movements in a
$4$-way intersection are used. More specifically, these eight movements
denoted by $M_{1},M_{2},\cdots,M_{8}$ are Northbound (N), Northbound Left-turn
(NL), Eastbound (E), Eastbound Left-turn (EL), Westbound(W), Westbound Left-
turn (WL), Southbound (S), Southbound Left-turn (SL) movements, respectively.
For instance, $m_{8}$ indicates that the vehicles can travel from west to
north;

  * •

Phase: A phase is a combination of movement signals. Each phase allows a
specific group of traffic movements to occur while restricting others. The
top-right portion of Fig. 1 illustrates the four phases of a $4$-way
intersection. For instance, Phase-1 involves $m_{1}$ and $m_{5}$, enabling
vehicles traveling north-south to proceed while simultaneously prohibiting
other movements;

  * •

Signal Plan: A signal plan represents a prearranged sequence and duration of
phases used to control the traffic signals at an intersection. Mathematically,
a signal plan is denoted by
$\\{(p_{1},t_{1}),(p_{2},t_{2}),\cdots,(p_{i},t_{i}),\cdots\\}$, where $p_{i}$
and $t_{i}$ represent a phase and its duration, respectively. Usually, the
phase sequence is in a cyclic order. For instance, the bottom-right portion of
Fig. 1 shows a cycle-based signal plan and the duration of each phase is
$t_{1}=50$, $t_{2}=20$, $t_{3}=50$ and $t_{4}=20$ as an example.

Figure 2: The overall framework of UniTSA.

##  4 Methodology

###  4.1 Framework

As shown in Fig. 2, the proposed UniTSA framework consists of two modules, one
designed to train a universal agent through different intersections and one
for fine-tuning the model for key intersections. More specifically, the first
module capitalizes on a novel RL agent design to allow the model to
characterize intersections of various topologies and signal schemes using the
same structure by incorporating features on movements and actions with next or
not while exploiting five novel traffic state augmentation methods. After
that, the second module fine-tunes the model derived by the first module for
some specific key intersections. More details about each step above will be
elaborated in the following sections.

###  4.2 Agent Design and Junction Matrix

State: Intersections may possess different numbers of lanes, which results in
state space of different dimensions when recording features in lane units. As
discussed in Section 3, it is observed that there are only eight valid
movement signals in a $4$-way intersection, regardless of the number of lanes.
Inspired by this observation, we propose to represent the information of an
intersection at time $t$ as a junction matrix denoted as

${\bm{J}}_{t}=\left[{\bm{m}}_{1}^{t},{\bm{m}}_{2}^{t},\cdots,{\bm{m}}_{8}^{t}\right]^{T},$
(1)

where $\left[\cdot\right]^{T}$ stands for the transpose of the enclosed matrix
while vector ${\bm{m}}_{i}^{t}$ of length eight represents information
extracted from the $i$-th movement at time $t$ for $i=1,2,\cdots,8$. More
specifically, ${\bm{m}}_{i}^{t}$ encompasses three components: traffic
characteristics, movement characteristics, and traffic signal characteristics.
The traffic characteristics quantify the congestion level of the movement
using parameters such as the average traffic flow $F^{i,t}$, the maximum
occupancy $O^{i,t}_{max}$, and the average occupancy $O^{i,t}_{mean}$ within
two consecutive actions. In addition, the movement characteristics provide
specific details about the movement itself, including the direction
($I^{i}_{s}$) indicating whether it is a straight movement or not and the
number of lanes ($L_{i}$) it occupies. Finally, the traffic signal
characteristics comprise three binary parameters, namely $I^{i,t}_{cg}$,
$I^{i,t}_{ng}$ and $I^{i,t}_{mg}$. These three parameters indicate whether the
movement signal is currently green or not, whether it will be green in the
next signal phase or not, and whether the current green duration has reached
the minimum duration or not, respectively. These eight features can be readily
obtained, making this design suitable for practical deployment. Therefore,
vector ${\bm{m}}^{t}_{i}$ is defined as follows:

${\bm{m}}_{i}^{t}=\left[F^{i,t},O^{i,t}_{max},O^{i,t}_{mean},I^{i}_{s},L_{i},I^{i,t}_{cg},I^{i,t}_{ng},I^{i,t}_{mg}\right]^{T}.$
(2)

(a)

(b)

Figure 3:  (a) A $3$-way intersection with (b) its junction matrix with zero
padding.

It has been observed that ${\bm{J}}_{t}$ at one time instance is not
sufficient to provide a comprehensive understanding of the traffic dynamics
for RL-based signal light controllers. To circumvent this obstacle, we propose
to incorporate temporal traffic information by exploiting the $K$ latest
observations, which enables the agent to better capture traffic patterns and
trends in traffic behaviors. As a result, the agent can more effectively adapt
its decision-making process in response to evolving traffic conditions.
Mathematically, we define the state ${\bm{S}}_{t}\in\mathbb{R}^{K\times
8\times 8}$ of the proposed agent at time $t$ as shown in Eq. (3):

${\bm{S}}_{t}=\left[{\bm{J}}_{t-K+1},{\bm{J}}_{t-K+2},\cdots,{\bm{J}}_{t}\right].$
(3)

Finally, it is worth pointing out that zero padding is employed when the
number of movements at an intersection is less than $8$, e.g. a $3$-way
intersection. For instance, Fig. 3a shows a common $3$-way intersection in
which only E, EL, W, and SL movement signals are in use. As shown in Fig. 3b,
zero padding is applied on the rows corresponding to those $4$ unused movement
signals, maintaining the matrix size identical to those of the $4$-way
intersection.

Action: A realistic and implementable action design has to take into account
the safety of all traffic participants. While the action design choose next
phase [10, 18, 12] can significantly improve the intersection efficiency, it
disrupts the original cycle sequence of signal lights, thereby compromising
driver safety. In sharp contrast, this work adopts the action design of "Keep
or change" [23, 25, 9]. This design adheres to the concept of cycles,
executing each phase sequentially (e.g., Phase 1, Phase 2, Phase 3, Phase 4,
Phase 1, Phase 2, and so on). The agent determines whether to keep the current
phase or change to the next phase based on the state ${\bm{S}}_{t}$. Due to
the availability of current phase information $I_{cg}$ and next phase
information $I_{ng}$ in the junction matrix, it is later shown that this
action design exhibits excellent scalability for intersections with different
signal plans.

Reward: The negative of the average queue length in each movement $q_{i}$ is
adopted as the reward. Metrics such as waiting time, travel time, and delay
are not used since it is impractical to obtain these metrics from real-world
traffic detection devices. Consequently, the proposed reward function is
defined as follows:

$r_{t}=\frac{\left(-\displaystyle\sum_{i=1}^{8}{q_{i}}\right)-\mu}{\sigma+\epsilon},$
(4)

where $\epsilon$ is a small number to prevent division by zero. Furthermore,
$\mu$ and $\sigma$ represent the mean and standard deviation of the first $R$
rewards, respectively. Mathematically, $\mu$ and $\sigma$ take the following
form:

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{1}{R-1}\displaystyle\sum_{j=1}^{R-1}{r_{j}},$ (5)
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{R-1}\displaystyle\sum_{j=1}^{R-1}{(r_{j}-\mu)^{2}}}.$
(6)

The reward is normalized to facilitate a faster training process.

Figure 4: Illustration of three traffic state augmentation methods applied to
both $4$-way and $3$-way intersections. Figure 5: Detailed steps in traffic
state augmentation block.

###  4.3 Traffic State Augmentation

In recent studies, data augmentation techniques have demonstrated their
effectiveness in enhancing the generalization capabilities of RL models [38,
39, 40]. By training on a more diverse set of augmented samples, RL agents can
improve their capability of handling unseen tasks. The most common data
augmentation techniques reported in the literature are Gaussian noise
addition, and masking. In this work, we propose three additional novel traffic
state augmentation methods specifically designed for RL-based TSC tasks,
namely movement shuffling, change of lane numbers and traffic flow scaling as
illustrated in Fig. 4.

Fig. 5 provides a detailed illustration of the step-by-step process involving
the traffic state ${\bm{S}}_{t}$ as it undergoes the five data augmentation
methods, leading to the final output $\tilde{\bm{S}}_{t}$. During training, a
minibatch of data is randomly sampled from the replay buffer or recently
augmented trajectories. While augmentation across the minibatch is stochastic,
it is consistent across the stacked frames. It is also worth noting that these
five traffic state augmentation methods can be directly applied to the
junction matrix ${\bm{J}}$, enabling the agent to learn and adapt to different
traffic scenarios and intersection structures.

Movement Shuffling: This method involves shuffling the rows of the junction
matrix, simulating different rotations, and flipping at the same intersection.
Intuitively, shuffling can be interpreted as an effective rotation of the
original intersection as depicted in Fig. 4. The assumption behind shuffling
is that the action taken by the agent should not change after the rotation of
the intersection. Mathematically, the movement shuffling operation can be
modeled as follows:

$\tilde{\bm{J}}_{t}^{\textrm{ms}}={\bm{P}}\cdot{\bm{J}}_{t},$ (7)

where ${\bm{J}}_{t}$ and $\tilde{\bm{J}}_{t}^{\textrm{ms}}$ represent the
original and augmented junction matrices, respectively. Furthermore,
${\bm{P}}$ is a permutation matrix designed to exchange rows in
${\bm{J}}_{t}$. After “movement shuffling” is applied to all junction matrices
in ${\bm{S}}_{t}$ in Eq. (3), the new state can be represented as
$\tilde{\bm{S}}_{t}^{\textrm{ms}}$:

$\tilde{\bm{S}}_{t}^{\textrm{ms}}=\left[\tilde{\bm{J}}_{t-K+1}^{\textrm{ms}},\tilde{\bm{J}}_{t-K+2}^{\textrm{ms}},\cdots,\tilde{\bm{J}}_{t}^{\textrm{ms}}\right].$
(8)

Change of Lane Numbers: To expose the agent to a wider range of road structure
combinations, we propose to randomly modify the number of lanes $L_{i}$ in
each movement vector
$\tilde{\bm{m}}_{i}^{\textrm{ms}}\in\tilde{\bm{J}}_{t}^{\textrm{ms}}$. This
augmentation method allows the agent to encounter various lane configurations
during training, enhancing its capability of handling diverse intersection
layouts. In addition, the traffic characteristics (e.g., traffic flow and
occupancy) are multiplied by the corresponding coefficients to maintain
relative values.

An example of this traffic state augmentation method is shown in Fig. 4. In
this example, a $4$-way intersection was modified to two in all directions by
reducing the number of lanes. Similarly, for the $3$-way intersection, we
increase the number of west and east-approaching lanes to $4$. While modifying
the number of lanes, we maintain the relative number of vehicles in each
direction. Thus, it is reasonable to assume that the actions taken by the
agent before and after such modifications should be identical. Mathematically,
the operation to change the lane number can be expressed as:

$\tilde{\bm{m}}_{i}^{\textrm{cln}}=f(\tilde{\bm{m}}_{i}^{\textrm{ms}},\tilde{L}_{i}),\
i=1,2,\cdots,8,$ (9)

where $L_{i}$ and $\tilde{L}_{i}$ are uniformly distributed random variables
representing the original and modified numbers of lanes, respectively.
Furthermore, $\tilde{\bm{m}}_{i}^{\textrm{ms}}$ stands for an individual
movement vector within the junction matrix after the “movement shuffling”
method whereas $f\left(\cdot,\cdot\right)$ denotes the function that adjusts
the traffic characteristics. Specifically, $f\left(\cdot,\cdot\right)$ can
take the following form:

$f(\tilde{\bm{m}}_{i}^{\textrm{ms}},L^{\prime})=\left\\{\begin{aligned}
&\left[\tilde{\bm{m}}_{i}^{\textrm{ms}}\right]_{k}\times\left(\frac{\tilde{L}_{i}}{L_{i}}\right),&\text{if}\
k=1,2,3,5\\\
&\left[\tilde{\bm{m}}_{i}^{\textrm{ms}}\right]_{k},&\text{otherwise}\end{aligned}\right.$
(10)

where $\left[\tilde{\bm{m}}_{i}^{\textrm{ms}}\right]_{k}$ represents the
$k$-th entry of $\tilde{\bm{m}}_{i}^{\textrm{ms}}$. Note that $f$ is designed
to ensure that the traffic characteristics $F$, $O_{max}$, and $O_{mean}$
maintain relative values regardless of the variations in the lane
configuration. After applying the “change of lane numbers” method in all
junction matrices in $\tilde{\bm{S}}_{t}^{\textrm{ms}}$, the new state can be
represented as $\tilde{\bm{S}}_{t}^{\textrm{cln}}$:

$\tilde{\bm{S}}_{t}^{\textrm{cln}}=\left[\tilde{\bm{J}}_{t-K+1}^{\textrm{cln}},\tilde{\bm{J}}_{t-K+2}^{\textrm{cln}},\cdots,\tilde{\bm{J}}_{t}^{\textrm{cln}}\right].$
(11)

where

$\tilde{\bm{J}}_{t}^{\textrm{cln}}=\left[\tilde{\bm{m}}_{1}^{\textrm{cln}},\tilde{\bm{m}}_{2}^{\textrm{cln}},\cdots,\tilde{\bm{m}}_{8}^{\textrm{cln}}\right]^{T}.$
(12)

Traffic Flow Scaling: To shift the agent’s focus from absolute car numbers to
relative car distributions, we introduce a flow scaling factor. By multiplying
the flow and occupancy values in the junction matrix with a uniformly
distributed random number $\alpha$, we can create variations in the relative
traffic volume for each movement. Notably, $\alpha$ remains consistent across
all movements within the same traffic state, ensuring that the relative
vehicle proportions between movements remain unchanged. This augmentation
method facilitates the agent to prioritize the relative significance of
different movements, thereby reducing its reliance on absolute values. It is
worth noting that traffic flow scaling does not alter the number of lanes in
each movement.

Fig. 4 illustrates two plausible scaling methods by proportionally increasing
or decreasing the number of vehicles on each approach or adding small changes
to the traffic volume in two scenarios. Mathematically, the flow scaling
operation can be defined as:

$\tilde{\bm{m}}_{i}^{\textrm{tfs}}=g(\tilde{\bm{m}}_{i}^{\textrm{cln}},\alpha),\
i=1,2,\cdots,8,$ (13)

where $\alpha$ is the flow scaling factor. Furthermore,
$g\left(\cdot,\cdot\right)$ denotes a function that scales the flow and
occupancy values and takes the following form:

$g(\tilde{\bm{m}}_{i}^{\textrm{cln}},\alpha)=\left\\{\begin{aligned}
&\left[\tilde{\bm{m}}_{i}^{\textrm{cln}}\right]_{k}\times\alpha,&\text{if}\
k=1,2,3\\\
&\left[\tilde{\bm{m}}_{i}^{\textrm{cln}}\right]_{k}.&\text{otherwise}\end{aligned}\right.$
(14)

After replacing $\tilde{\bm{m}}_{1}^{\textrm{cln}}$ in Eq. (12) with
$\tilde{\bm{m}}_{i}^{\textrm{tfs}}$, the resulting state from the “traffic
flow scaling” method is denoted as $\tilde{\bm{S}}_{t}^{\textrm{tfs}}$.

Gaussian Noise Addition: Gaussian noise is added directly to the junction
matrix to introduce randomness into the training data. The additive noise can
affect all components of the junction matrix, including traffic
characteristics, movement characteristics, and traffic signal characteristics.
This augmentation method allows the agent to adapt to noisy and uncertain
traffic conditions, improving its robustness during inference. Mathematically,
the Gaussian noise addition operation can be modeled as:

$\tilde{\bm{J}}_{t}^{\textrm{gna}}=\tilde{\bm{J}}_{t}^{\textrm{tfs}}+{\bm{\Psi}}_{t},$
(15)

where ${\bm{\Psi}}_{t}\sim\mathcal{N}({\bm{0}},{\bm{I}})$ denotes the Gaussian
noise matrix, and
$\tilde{\bm{J}}_{t}^{\textrm{tfs}}\in\tilde{\bm{S}}_{t}^{\textrm{tfs}}$
corresponds to the junction matrix after applying the “traffic flow scaling”
method. After applying “Gaussian noise addition”, the new state can be
represented as $\tilde{\bm{S}}_{t}^{\textrm{gna}}$.

Masking: To encourage the agent to learn the traffic flow changes, we randomly
set the values in the junction matrix to zero at a specific time instance. By
masking certain components of the junction matrix, we create situations where
the agent must rely on the information before and after the masked period to
infer traffic dynamics. This augmentation method promotes the agent’s
capability of understanding and responding to traffic fluctuations.

Finally, a novel state denoted as $\tilde{\bm{S}}_{t}$ is obtained after the
application of the five data augmentation methods discussed above.

Figure 6: Three kinds of intersection feature extraction block (a) CNN-based
Structure; (b) RNN-based Structure; (c) Transformer-based Structure.

###  4.4 Intersection Feature Extraction

In this section, three neural network structures are utilized to extract
intersection information from the augmented traffic states, namely
Convolutional Neural Network (CNN), Recurrent Neural Network (RNN) and
transformers.

CNN-based Structure: As shown in Fig. 6(a), a CNN-based structure equipped
with two 2D convolutional layers is utilized to extract time series
information within a road junction, resulting in a hidden representation
${\bm{C}}_{t}$ given as:

${\bm{C}}_{t}=\text{ReLU}\left(\mathbf{W}^{2d}_{2}\mathbf{W}^{2d}_{1}{\tilde{\bm{S}}_{t}}\right),$
(16)

where $\mathbf{W}^{2d}_{1}$ and $\mathbf{W}^{2d}_{2}$ represent the learnable
parameters of the two 2D convolutional blocks, respectively. More
specifically, the first convolutional block, $\mathbf{W}^{2d}_{1}$, extracts
information regarding the traffic movement, and subsequently, the second
convolutional block, $\mathbf{W}^{2d}_{2}$, captures information specific to
the junction based on the movement information. Furthermore,
$\text{ReLU}(\cdot)$ denotes the ReLU function.

Next, the resulting ${\bm{C}}_{t}$ is passed through a multilayer perceptron
(MLP) layers, producing a feature vector ${\bm{O}}_{t}$ as follows:

${\bm{O}}_{t}=\mathbf{W}_{u}{\bm{C}}_{t}+\mathbf{b}_{u},$ (17)

where $\mathbf{W}_{u}$ and $\mathbf{b}_{u}$ are learnable parameters of the
MLP layer.

RNN-based Structure: In contrast to the CNN-based approach, the RNN-based
structure illustrated in Fig. 6(b) employs a parameter-sharing 1D
convolutional layer to extract information from each junction matrix. The 1D
convolutional layer operates on each junction matrix $\tilde{\bm{J}}_{t-K+i}$
for $i=1,2,\cdots,K$, which is the augmented state at a particular time step
within the history window. The resulting outputs denoted by
$\bm{c}_{1},\bm{c}_{2},\ldots,\bm{c}_{k}$, capture the information from the
intersection at $K$ different time instances and can be expressed as:

$\bm{c}_{i}=\text{ReLU}\left(\mathbf{W}^{1d}_{2}\mathbf{W}^{1d}_{1}\tilde{\bm{J}}_{t-K+i}\right),\
i=1,2,\cdots,K,$ (18)

where $\mathbf{W}^{1d}_{1}$ and $\mathbf{W}^{1d}_{2}$ represent the learnable
parameters of the two 1D convolutional layers. These outputs are then fed into
the RNN module:

$\bm{h}_{i}=\tanh(\mathbf{W}_{x}\bm{c}_{i}+\bm{h}_{i-1}\mathbf{W}_{h}+\mathbf{b}_{h}),\
i=1,2,\cdots,K,$ (19)

where $\mathbf{W}_{x}$, $\mathbf{W}_{h}$, and $\mathbf{b}_{h}$ represent the
weights for the hidden layer in the RNN structure. The final hidden state
$\bm{h}_{K}$ derived from the last RNN output is used to calculate the
features of the entire intersection over a period of time, denoted as
${\bm{O}}_{t}$:

${\bm{O}}_{t}=\mathbf{W}_{v}\bm{h}_{K}+\mathbf{b}_{v},$ (20)

where $\mathbf{W}_{v}$ and $\mathbf{b}_{v}$ are the weights for the output
layer in the RNN structure.

Transformer-based Structure: Fig. 6(c) shows the transformer-based approach.
we employ a weight-shared CNN network to extract features from the junction
matrix at each time step, as mentioned in Eq. (18). However, instead of using
an RNN block, we utilize a transformer encoder to capture temporal
dependencies in the sequence of features
$\tilde{\bm{C}}_{t}=[\bm{c}_{1},\bm{c}_{2},\ldots,\bm{c}_{K}]$. To incorporate
timing information, we insert a learnable embedding denoted as
$\bm{c}_{\text{class}}$ to $\tilde{\bm{C}}_{t}$ as follows:

$\bm{C}_{t}=\left[\bm{c}_{\text{class}},\tilde{\bm{C}}_{t}\right]=\left[\bm{c}_{\text{class}},\bm{c}_{1},\bm{c}_{2},\ldots,\bm{c}_{K}\right].$
(21)

The output state of the transformer encoder, based on this modified input
sequence, serves as the traffic state representation $\bm{O}_{t}$. In the
transformer encoder block, self-attention is calculated as follows:

$\bm{Q}_{C}=\bm{C}_{t}\bm{W}_{Q},\bm{K}_{C}=\bm{C}_{t}\bm{W}_{K},\bm{V}_{C}=\bm{C}_{t}\bm{W}_{V},$
(22)

and

$\bm{Z}_{t}=\phi\left(\frac{\bm{Q}_{C}\bm{K}_{C}^{T}}{\sqrt{d}}\right)\bm{V}_{C},$
(23)

where $\bm{Q}_{C}$, $\bm{K}_{C}$, and $\bm{V}_{V}$ represent the projected
query, key and value features, respectively, while $\bm{W}_{Q}$, $\bm{W}_{K}$,
and $\bm{W}_{V}$ the corresponding parameter metrics. In addition,
$\bm{Z}_{t}$ represents the output of self-attention, and $\phi(\cdot)$
denotes the softmax function. The final output $\bm{O}_{t}$ is the first value
of $\bm{Z}_{t}$.

###  4.5 RL Training and Fine-tuning

The Proximal Policy Optimization (PPO) algorithm [41] is adopted to train our
UniTSA model. As depicted in Fig. 2, the agent gathers trajectories consisting
of observations, actions, and rewards during the interactions with diverse
traffic scenarios of different structures. These trajectories serve as the
basis for computing the policy loss and value loss utilized to update the
weights of the Actor and Critic networks.

The policy loss quantifies the difference between the current policy and the
updated policy derived from the collected trajectories. It encourages the
agent to increase the probabilities of actions that lead to higher rewards
while reducing the probabilities of actions that lead to lower rewards.
Mathematically, the policy loss can be formulated as:

$\mathcal{L}_{\text{pf}}(\theta)=\hat{\mathbb{E}}_{t}\left[\min\left(r_{t}A_{t},\text{clip}\left(r_{t},1-\epsilon,1+\epsilon\right)A_{t}\right)\right],$
(24)

where $\hat{\mathbb{E}}_{t}$ stands for the expectation operator and $r_{t}$
represents the ratio between the new policy $\pi_{\theta}(a_{t}|s_{t})$ and
the old policy $\pi_{\theta_{\text{old}}}(a_{t}|s_{t})$ :

$r_{t}=\frac{\pi_{\theta}(a_{t}|s_{t})}{\pi_{\theta_{\text{old}}}(a_{t}|s_{t})}.$
(25)

Furthermore, $A_{t}=r_{t}+\gamma V(s_{t+1})-V(s_{t})$ denotes the advantage
function whereas the clip function can ensure stable policy updates.

The value loss measures the discrepancy between the estimated value function
and the actual rewards obtained during the interaction. It makes the value
function to better approximate the expected cumulative rewards. The value loss
can be defined as:

$\mathcal{L}_{\text{vf}}(\theta)=\hat{\mathbb{E}}_{t}\left[\left(V_{\theta}(s_{t})-\hat{R}_{t}\right)^{2}\right],$
(26)

where $V_{\theta}(s_{t})$ is the estimated value function under policy
$\theta$, and $\hat{R}_{t}=\sum_{k=0}^{\infty}{\gamma^{k}r_{t+k}}$ denotes the
reward-to-go. Once the policy loss and the value loss have been calculated,
the final objective function is expressed as Eq. (27):

$\mathcal{L}(\theta)=-\mathcal{L}_{\text{pf}}(\theta)+\lambda\mathcal{L}_{\text{vf}}(\theta),$
(27)

where $\lambda$ is the coefficient of value loss.

This work proposes an effective universal model to control the traffic signals
by optimizing this objective function through PPO. Furthermore, since certain
intersections are more critical than others in practice, LoRA [31] is adopted
in the proposed model to fine-tune the performance on these important
intersections. More specifically, the LoRA modules are added to the weights of
dense layers in the Actor and Critic networks as shown in Fig. 2. During the
fine-tuning process, the original pretrained weights are kept constant while
the LoRA modules is being updated. This design allows the model to adapt to
the intersection-specific features without significantly increasing the number
of parameters, while maintaining training efficiency.

Figure 7:  All intersection topologies with their available phases which used
for the training and testing.

Let us consider a pretrained weight matrix ${\bm{W}}\in\mathbb{R}^{n\times m}$
in the network accompanied by a LoRA module
$\Delta{\bm{W}}={\bm{W}}_{A}{\bm{W}}_{B}^{T}$, where
${\bm{W}}_{A}\in\mathbb{R}^{n\times d}$, ${\bm{W}}_{B}\in\mathbb{R}^{m\times
d}$ with $d\ll n$. The output of this layer can be obtained as:

$z={\bm{W}}x+\Delta{\bm{W}}x={\bm{W}}x+\frac{\alpha}{r}{\bm{W}}_{A}{\bm{W}}_{B}^{T}x,$
(28)

where ${\bm{W}}_{A}$ and ${\bm{W}}_{B}$ are initialized as a zero matrix and a
zero-mean Gaussian distribution matrix, respectively. Furthermore, $\alpha$ is
a constant scale hyperparameter whereas $r$ is the rank of the LoRA module.
Through the combination of RL training using PPO and fine-tuning with LoRA,
the proposed universal model can effectively address the challenges posed by
important intersections of varying structures.

Table 1: Hyperparameter Setting Hyper-parameter | Value
---|---
Learning rate | $0.0001$
Trajectory memory size | $3000$
Clipping range $\epsilon$ | $0.2$
Discount factor $\gamma$ | $0.99$
Value function coefficient $\lambda$ | $0.9$
Scale hyperparameter $\alpha$ | $1$
Rank of LoRA module | $8$

##  5 Experiments

###  5.1 Experiment Settings

Extensive experiments were conducted to validate the proposed model using the
SUMO software package [32] in this section. SUMO is an open-source microscopic
traffic simulation tool designed for handling large networks. It provides the
Traffic Control Interface (TraCI) for controlling traffic lights and
retrieving traffic condition information for intersections. We calculate the
flow and occupancy of each movement by analyzing the positions and
trajectories of vehicles on the road. It is important to note that, in order
to simulate real-world conditions, we consider only vehicles within proximity
of $150$ m to the intersection, in lieu of all vehicles along the entire road.
Moreover, a green light was followed by a yellow light of $3$ s before
transitioning to a red light to ensure driver safety. The waiting time per
vehicle was used as a performance metric to evaluate the effectiveness of the
different methods. A low waiting time indicates that vehicles spent less time
passing through the intersection.

We utilized the Proximal Policy Optimization (PPO) implementation provided by
the Stable Baselines3 library [42]. To accelerate training, we employed $30$
parallel processes, and the total number of training environment steps was set
to $10$M. The state representation included the previous $K=8$ snapshots of
the junction matrix. The interval between two consecutive actions was $5$ s.

The hyper-parameters are configured as shown in Table 1. Furthermore, the
actor and critic networks were designed as two-layer fully connected networks.
The input sizes were $\\{64,32\\}$, and the output sizes were $\\{32,2\\}$ and
$\\{32,1\\}$ respectively for the actor and critic networks.

Table 2: All intersection configurations

| Training Dataset |  | Test Dataset
---|---|---|---
Intersection ID | INT-1 | INT-2 | INT-3 | INT-4 | INT-5 | INT-6 | INT-7 | INT-8 |  | INT-9 | INT-10 | INT-11 | INT-12
roads | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 3 |  | 4 | 4 | 3 | 3
lanes per road | (3,3,3,3) | (3,3,3,3) | (3,3,3,3) | (3,4,4,5) | (3,4,4,5) | (3,4,4,5) | (3,3,3) | (3,3,3) |  | (3,4,3,4) | (3,3,3,3) | (4,3,3) | (2,3,2)
phases | 4 | 4 | 2 | 4 | 4 | 6 | 3 | 3 |  | 4 | 5 | 3 | 3

###  5.2 Datasets

This study considers intersections with diverse structures, aiming to employ
one single universal model to predict actions for all intersections.
Specifically, $12$ intersections of varying numbers of phases, lanes on each
road, and approaching roads (i.e., $3$-way or $4$-way intersections) are
constructed and used for experiments. Among these $12$ intersections, eight
are used for training, while the remaining four are reserved for testing. The
topologies and phases of the $12$ intersections are depicted in Fig. 7, and
Table 2 provides a summary on their configurations. For instance, INT-4 shown
in Fig. 7 consists of four bi-directional approaching roads with three lanes
in the north-south direction, five lanes in the west-east direction, and four
lanes in each of the other two directions. INT-4 includes four phases, each
combining two different movement signals. Consequently, the configurations for
INT-4 in Table 2 indicate the presence of four roads, lanes per road specified
as $(3,4,4,5)$ in clockwise order, and four phases.

There are three different intersection topologies in the training dataset.
INT-1, INT-2 and INT-3 represent a regular $4$-way intersection scenario, with
each road consisting of three lanes; INT-4, INT-5 and INT-6 represent a large
$4$-way intersection scenario, featuring more than four lanes per road; INT-7,
INT-8 and INT-9 depict the $3$-way intersection scenario. For the
intersections with the same topology, we generate new intersections by
altering the sequence of phases or the number of phases. For instance, INT-1
and INT-2 have identical configurations, but the sequence of phases differs.
Similarly, INT-1 and INT-3 vary in terms of the number of phases.

To assess the performance of the proposed model on unseen intersections, four
testing scenarios are formed, namely INT-9, INT-10, INT-11 and INT-12. For
instance, INT-9 and INT-10 also represent $4$-way intersections, but INT-9
features different lane configurations compared to the training set, while
INT-10 differs in the number and combination of phases from the intersections
in the training set. INT-11 modifies the lane and phase of each road based on
INT-8. Finally, INT-12 simulates real-world traffic within the city of
Ingolstadt, Germany [43].

In addition to considering intersections with different structures, we
generate $100$ unique pieces of the route for each intersection. Three-
quarters of these routes were utilized for training, while the remaining
quarter was reserved for evaluation. Each route has a duration of $30,000$
seconds, equivalent to approximately $8$ hours.

###  5.3 Compared Methods

To evaluate the performance of the proposed UniTSA, we compare the resulting
universal model with several classic and state-of-the-art RL-based methods for
TSC.

FixTime [2]: Fixed-time control utilizes a predetermined cycle and phase
duration plan, which is widely used in situations with steady traffic flow. We
consider two versions of FixTime, FixTime-30 (Fix-30) and FixTime-40 (Fix-40).
These variants correspond to fixed-time signal control plans where each phase
has a duration of $30$ seconds and $40$ seconds, respectively.

Webster [3]: The Webster method determines the cycle length and phase split
based on traffic volume during a specific period. It has been proven that when
the traffic is uniform, the Webster method minimizes the travel time of all
vehicles passing the intersection or maximizes the intersection capacity [36].
Additionally, the Webster method can be adapted for real-time applications.
For fairness, we employ the Webster method to adjust the traffic lights based
on real-time traffic in this study.

SOTL [4]: Self-Organizing Traffic Light Control (SOTL) is an actuated signal
control method that dynamically adjusts signal durations based on a manually
determined threshold for the number of waiting vehicles. In this experiment,
we set the threshold based on [44].

MPLight [10]: MPLight incorporates the FRAP structure proposed in [9] and a
pressure-based reward mechanism inspired by [45]. Furthermore, MPLight employs
a sharing-parameter multilayer perceptron (MLP) to enhance adaptability across
different intersection configurations.

AttendLight [18]: This method adopts the attention mechanism to train a
universal model capable of handling intersections with diverse structures and
traffic flow distributions. It employs two attention models: the first
attention model addresses variations in the number of roads and lanes, while
the second attention model enables decision-making across intersections with
different numbers of phases.

In addition to the baseline methods, we also consider several variations of
our method:

UniTSA (Single): This method focuses on training the model within a single
environment. It utilizes the agent design described in Section 4.2 and employs
an RNN-based structure for extracting information from the traffic state.

UniTSA (Multi): In contrast to UniTSA (Single), this method trains the model
across multiple environments simultaneously. We explore the performance of
UniTSA (Multi) using various neural network designs, as discussed in Section
4.4, including UniTSA (Multi+CNN), UniTSA (Multi+RNN), and UniTSA
(Multi+Trans).

UniTSA (Multi+TSA): This variant of UniTSA enhances the training process by
incorporating five traffic state augmentation methods into UniTSA (Multi). It
results in UniTSA (Multi+CNN+TSA), UniTSA (Multi+RNN+TSA), and UniTSA
(Multi+Trans+TSA).

Table 3: Quantitative results (average waiting time per vehicle) of training intersections for universal models. A lower value indicates better performance and the lowest values are highlighted in bold. | INT-1 | INT-2 | INT-3 | INT-4 | INT-5 | INT-6 | INT-7 | INT-8
---|---|---|---|---|---|---|---|---
Fix-30 [2] | 39.458 | 38.862 | 8.323 | 35.123 | 35.031 | 51.923 | 18.920 | 18.595
Fix-40 [2] | 50.696 | 52.108 | 10.667 | 44.888 | 45.540 | 61.743 | 23.411 | 24.759
Webster [3] | 26.466 | 26.889 | 5.751 | 25.413 | 24.541 | 40.128 | 12.755 | 13.574
SOTL [4] | 16.048 | 16.561 | 2.764 | 28.169 | 27.902 | 27.404 | 8.639 | 7.925
MPLight [10] | 19.111 | 14.659 | 4.469 | 16.067 | 19.925 | 19.115 | 6.654 | 7.523
AttendLight [18] | 16.483 | 13.893 | 3.860 | 16.903 | 18.915 | 20.795 | 6.532 | 8.104
UniTSA (Multi+CNN) | 13.776 | 13.580 | 3.265 | 14.790 | 15.437 | 18.751 | 6.592 | 6.894
UniTSA (Multi+RNN) | 13.692 | 13.437 | 3.495 | 15.198 | 14.896 | 18.612 | 6.393 | 6.625
UniTSA (Multi+Trans) | 14.976 | 20.459 | 2.811 | 16.489 | 16.481 | 23.793 | 7.552 | 7.175
UniTSA (Multi+CNN+TSA) | 14.071 | 14.150 | 3.036 | 15.990 | 16.472 | 24.383 | 6.329 | 6.328
UniTSA (Multi+RNN+TSA) | 13.450 | 13.578 | 3.007 | 14.470 | 14.462 | 18.689 | 6.242 | 6.164
UniTSA (Multi+Trans+TSA) | 13.335 | 13.314 | 3.311 | 14.648 | 14.566 | 18.579 | 6.643 | 6.571

###  5.4 Results of the training intersections

In this section, the performance of the proposed UniTSA is compared against
that derived from several existing approaches, including the FixTime approach,
Webster model, SOTL, MPLight, and AttendLight, on the training intersections.
In particular, UniTSA models of different network structures were examined.

Table 3 shows the average waiting time per vehicle achieved by UniTSA and
other baseline methods on the training intersections. It is clear that the RL-
based approaches demonstrated superior performance as compared to the
conventional approaches in most intersection scenarios. Despite that SOTL
achieved the shortest waiting time in the INT-2, it required manually defined
thresholds for different environments, limiting its generalization in large-
scale scenarios.

Among the RL-based universal methods, namely MPLight, AttendLigh and UniTSA,
UniTSA demonstrated significant performance improvements as compared to other
RL-based methods. On average, UniTSA achieved $15\%$ and $12\%$ performance
improvement over MPLight and AttendLight, respectively, across the eight
intersections evaluated. When compared to MPLight, the proposed method
incorporates not only parameter sharing techniques before replacing the MLP
with RNN or Transformer block to capture the temporal information of the
traffic state. In contrast to AttendLight, UniTSA simplifies the state design
by representing the traffic state at a specific time instance using the
junction matrix. Furthermore, five methods of traffic state enhancement are
introduced, allowing the agent to observe a wider variety of traffic
intersection states during training, which contributes to the improved
performance of our model.

Finally, we explored the impact of different network structures and traffic
state augmentation. Inspection of Table 3 reveals that an RNN-based structure
yielded superior results compared to a CNN-based structure. Furthermore, in
the absence of traffic state augmentation, UniTSA (Multi+CNN) and UniTSA
(Multi+RNN) outperformed UniTSA (Multi+Trans) across most intersections, which
is rather surprising. This can be attributed to the fact that the Transformer-
based approach necessitates a larger volume of training data. However, in the
presence of traffic state augmentation, the model can interact with a broader
range of intersections with varying structures. As a result, UniTSA
(Multi+Trans+TSA) outperformed UniTSA (Multi+RNN) and UniTSA (Multi+RNN+TSA)
in many scenarios.

Table 4: Quantitative results of test intersections for universal models. | INT-9 | INT-10 | INT-11 | INT-12
---|---|---|---|---
Fix-30 | 40.341 | 38.360 | 17.136 | 17.440
Fix-40 | 60.043 | 50.580 | 23.504 | 20.570
Webster | 26.191 | 27.766 | 11.981 | 13.280
SOTL | 23.031 | 28.066 | 7.452 | 8.070
MPLight | 23.674 | 21.475 | 8.447 | 15.047
AttendLight | 18.080 | 18.501 | 7.393 | 12.982
UniTSA (Multi+CNN) | 17.877 | 18.244 | 7.063 | 12.820
UniTSA (Multi+RNN) | 16.771 | 15.450 | 6.807 | 10.140
UniTSA (Multi+Trans) | 17.640 | 16.269 | 6.598 | 9.630
UniTSA (Multi+CNN+TSA) | 13.075 | 14.245 | 6.794 | 7.550
UniTSA (Multi+RNN+TSA) | 12.677 | 14.208 | 5.914 | 6.730
UniTSA (Multi+Trans+TSA) | 13.054 | 16.186 | 5.983 | 6.190

###  5.5 Results of the test intersections

Next, we evaluate the key feature of UniTSA to see whether it can be utilized
for unseen intersections. Four intersections, namely INT-9, INT-10, INT-11 and
INT-12, are specifically used for the testing purposes. As discussed in
Section 5.2, these intersections differ from the scenarios in the training set
in terms of the number of lanes or the number of phases. Table 4 summarizes
the performance achieved by different methods, including baseline approaches
and various variants of our proposed UniTSA method. Consistent with the
results obtained on the training set, the RL-based algorithms significantly
outperformed the traditional traffic control algorithms.

Among the RL-based universal methods, UniTSA models demonstrated superior
performance across all test intersections as shown in Table 4. Among all the
methods, UniTSA (Multi+RNN+TSA) and UniTSA (Multi+Trans+TSA) excelled in terms
of reducing average travel time for vehicles passing through the
intersections. UniTSA (Multi+RNN+TSA) showcases an average travel time
reduction of approximately $32.9\%$ and $41.3\%$ as compared to MPLight and
AttendLight, respectively. These results confirm the effectiveness of our
approach in optimizing TSC and enhancing traffic flow efficiency.

Notably, incorporating traffic state augmentation techniques leads to improved
performance among the different UniTSA variants. For instance, UniTSA
(Multi+RNN+TSA), UniTSA (Multi+RNN+TSA), and UniTSA (Multi+RNN+TSA) exhibit
enhancements of $23.4\%$, $19.8\%$, and $17.9\%$, respectively, when compared
to UniTSA (Multi+RNN), UniTSA (Multi+RNN), and UniTSA (Multi+RNN). This
improvement can be attributed to the inclusion of a greater variety of
intersection scenarios within the training data through traffic state
augmentation techniques, such as the “Lane number change" method, which
enables the generation of diverse combinations of lane configurations.

(a)

(b)

(c)

(d)

Figure 8: The environment steps of different methods at the four test intersections. (a) INT-9. (b) INT-10. (c) INT-11. (d) INT-12. Table 5: Quantitative results of fine-tuning in test intersections. | INT-9 | INT-10 | INT-11 | INT-12
---|---|---|---|---
UniTSA (Multi+RNN) | 16.771 | 15.450 | 6.807 | 10.140
UniTSA (Multi+RNN+TSA) | 12.677 | 14.208 | 5.914 | 6.730
1M Environment Steps
UniTSA (Single) | 23.683 | 14.863 | 8.417 | 5.024
UniTSA (Multi+RNN+TSA) + FT | 10.995 | 12.553 | 4.475 | 3.534
10M Environment Steps
UniTSA (Single) | 10.353 | 12.254 | 4.359 | 3.089
UniTSA (Multi+RNN+TSA) + FT | 10.292 | 11.081 | 4.153 | 3.110

###  5.6 Results of fine-tuning in test intersections

In practical scenarios, certain intersections require special attention due to
their significance. To address this, we begin with the universal model trained
by UniTSA (Multi+RNN). The RNN-based UniTSA is chosen because it demonstrates
superior performance across most intersections compared with both CNN-based
and Transformer-based structures. In comparison to the UniTSA (Single), which
is trained on a single scenario, the resulting model can quickly reach or even
surpass the performance of the single-environment model after only a few
training steps.

Fig. 8 shows the change of cumulative rewards over training steps for
different models in the test intersections. The green dashed line represents
the result of applying the universal model directly to new intersections
without any fine-tuning. Notably, the model already exhibits promising results
without any additional fine-tuning or transfer learning. The blue line
represents the model trained from scratch whereas the orange line the fine-
tuned model based on the universal model. It is observed that the single-
environment model converges at approximately $3$M training steps. In sharp
contrast, the fine-tuning model achieves comparable performance with only
around $1$M training steps, resulting in approximately $66\%$ reduction in
computation time while maintaining similar performance.

Table 5 provides a detailed analysis of the model’s performance after fine-
tuning. At $1$M training steps, the fine-tuning model demonstrated an average
performance improvement of $36\%$ as compared to UniTSA (Single) across the
four test intersections. Even after $10$M training steps, the fine-tuning
models continued to outperform the models trained from scratch by $3\%$. This
aspect is particularly appealing for real-time applications. For instance, in
a road network with over $1000$ junctions, it is possible to significantly
reduce the number of interactions with the environment while maintaining
comparable performance, thereby greatly enhancing training efficiency.

(a)

(b)

(c)

(d)

Figure 9: Comparative analysis of traffic state augmentation methods on the
selected training and test intersections (a) INT-1. (b) INT-7. (c) INT-9. (d)
INT-11.

###  5.7 Comparative analysis on traffic state augmentations

In this section, we analyze the effectiveness of traffic state augmentation
methods in improving the performance of the UniTSA (Multi+RNN) model. We
conducted experiments by applying different combinations of two traffic state
augmentation techniques on both the training and test intersections. To
evaluate the impact of these methods, we propose the Average Waiting Time
(AWT) ratio between the models with and without traffic state augmentations
with an AWT ratio of less than $1$ indicating an improvement achieved by the
corresponding traffic state augmentation methods.

Fig. 9 showcases the outcomes of the comparative analysis on INT-1, INT-7,
INT-9, and INT-11 intersections, which represent common intersection
structures encountered in real-world scenarios. Each bar in the figure
corresponds to the average AWT ratio for a specific combination of traffic
state augmentations, and the error bars represent the $95\%$ confidence
interval. The blue dashed line represents the AWT ratio of UniTSA
(Multi+RNN+TSA), which employs all available traffic state augmentation
methods.

Fig. 9a and Fig. 9b depict the results obtained from the training
intersections INT-1 and INT-7, respectively. Inspection of these figures
reveals that the average performance improvement achieved through traffic
state augmentations was around $2\%$. Furthermore, the inclusion of noise and
mask methods incurred performance degradation in INT-7, which can be
attributed to the fact that the model has already captured the underlying
patterns and characteristics of the training intersections to a large extent.
As a result, introducing additional variations through traffic state
augmentation may not provide substantial benefits in the training set.
However, traffic state augmentation methods demonstrated significant
improvements when confronted with unseen intersection structures.

Fig. 9c and Fig. 9d illustrate the AWT ratios for the test intersections INT-9
and INT-11, respectively. These results clearly demonstrate that most traffic
state augmentation methods enhanced the performance of the base policy in the
test intersections, which consists of unseen intersections. The diverse
training samples generated through traffic state augmentation contribute to
the improved performance. Among the traffic state augmentation techniques,
movement shuffle and traffic flow scale were particularly effective in
enhancing the model’s performance. These techniques enable the model to adapt
and learn from a wider range of scenarios, resulting in improved performance
on the test set.

##  6 Conclusion

In this paper, a universal RL-based TSC framework called UniTSA has been
proposed for diverse intersection structures in V2X environments. More
specifically, UniTSA offers the capability to train a universal RL agent by
incorporating a junction matrix to characterize intersection states. To handle
unseen intersections, new traffic state augmentation methods have been
proposed to enrich the agent’s data collection, resulting in improved
performance and generalization for unseen intersection configurations. As a
result, UniTSA eliminates the necessity of extensive customization and
redevelopment for each individual intersection while offering a simple,
efficient, and open-sourced implementation, which makes UniTSA a valuable
framework for future research in data-efficient and generalizable RL-based TSC
methods within the field of V2X. Extensive experimental results have
demonstrated that UniTSA achieved the shortest average waiting time across
various intersection configurations, surpassing the performance of the
existing methods and outperforming the models trained from scratch with fine-
tuning.

## Acknowledgments

This work was supported National Key Research and Development Program of China
under Grant No. 2020YFB1807700 and the Shanghai Pujiang Program under Grant
No. 21PJD092.

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# Double lines in the quintic del Pezzo fourfold

Kiryong Chung  Department of Mathematics Education, Kyungpook National
University, 80 Daehakro, Bukgu, Daegu 41566, Korea<EMAIL_ADDRESS>
###### Abstract.

Let $Y$ be the del Pezzo $4$-fold defined by the linear section
$\textrm{Gr}(2,5)$ by $\mathbb{P}^{7}$. In this paper, we classify the type of
normal bundles of lines in $Y$ and describe its parameter space. As a
corollary, we obtain the desigularized model of the moduli space of stable
maps in $Y$. Also we compute the intersection Poincaré polynomial of the
stable maps space.

###### Key words and phrases:

Rational curves, Fano variety, Desingularization, Intersection cohomology

###### 2020 Mathematics Subject Classification:

14E15, 14E08, 14M15, 32S60.

##  1\. Introduction

###  1.1. Motivation

In previous series of papers [CK11, CHK12, CM17], the authors completely
solved the comparison problem of different moduli spaces (i.e., the stable
maps space, Hilbert scheme of curves and the stable sheaves space) of rational
curves in a homogeneous variety $X$ when the degree of curves is $\leq 3$. As
a result, we obtain the moduli theoretic birational model (in the sense of log
minimal model program) and compute the cohomology group of the moduli spaces.
In this case, the convexity of $X$ provides the mild singularity of the moduli
space of stable maps and thus one can set it as the initial point of
comparison. But many of Fano varieties are not convex. As such a toy example
is the minimal compactification of $\mathbb{C}^{3}$: the quintic del Pezzo
$3$-fold $W_{5}$ and Mukai variety $W_{22}$. In the case of $W_{5}$, our
starting point of the comparison is the Hilbert scheme (which is isomorphic to
the moduli space of stable sheaves). In [Chu22], we obtain the desingularized
model of the moduli space of stable maps in $W_{5}$. In this paper, as well-
known example of the minimal compactification of $\mathbb{C}^{4}$, we study
the rational curves in quintic del Pezzo $4$-fold $Y$ which is unique up to
isomorphism. We deal with the first non-trivial case, that is, the degree two
rational curves in $Y$. We obtain the desingularized model of stable maps
space and thus its intersection cohomology group. Similar to the $3$-fold case
([Chu22]), the crucial part is to classify types of the normal bundle of a
line in $Y$. In general, the geometry of lines in Fano variety has been an
important role for determining the geometric properties of Fano variety
([PZ16, KPS18, PCS19]).

###  1.2. Results

Unless otherwise stated we define the quintic del Pezzo $4$-fold $Y$ by the
linear section $\mathrm{Gr}(2,5)$ by $\\{p_{12}-p_{03}=p_{13}-p_{24}=0\\}$
where $\\{p_{ij}\\}$ are the Plücker coordinate of $\mathbb{P}^{9}$ under the
Plücker embedding $\mathrm{Gr}(2,5)\subset\mathbb{P}^{9}$. It is known that
the normal bundle $N_{L/Y}$ of a line $L$ in $Y$ is one of the following types
([PZ16, Lemma 1.6])

$N_{L/Y}\cong\mathcal{O}_{L}(1)\oplus\mathcal{O}_{L}^{\oplus
2}\;\mathrm{or}\;\mathcal{O}_{L}(-1)\oplus\mathcal{O}_{L}(1)^{\oplus 2}.$

Let us call the line of the first case (resp. the second case) by free line
(resp. non-free line). Let $\mathbf{H}_{d}(Y)$ be the Hilbert scheme of curves
$C$ with Hilbert polynomial $\chi(\mathcal{O}_{C}(m))=dm+1$ in $Y$. Let us
define the _double line_ $L^{2}$ as the non-split extension sheaf $F$
($\cong\mathcal{O}_{L^{2}}$)

$0\rightarrow{\mathcal{O}_{L}(-1)}\rightarrow{F}\rightarrow{\mathcal{O}_{L}}\rightarrow
0,$

where $L$ is a line. A double line $L^{2}$ in $Y$ supported on $L$ is
classified by
$\mathrm{Ext}^{1}(\mathcal{O}_{L},\mathcal{O}_{L}(-1))\cong\mathrm{H}^{0}(N_{L/Y}(-1))$.
Hence the double line $L^{2}$ for a (resp. non-free) free line $L$ in $Y$ is
unique (resp. isomorphic to $\mathbb{P}^{1}$). The main result of this paper
is the following.

######  Theorem 1.1.

Let $\mathbf{D}(Y)$ be the locus of double lines in $\mathbf{H}_{2}(Y)$. Then
$\mathbf{D}(Y)$ is a $4$-dimensional smooth subvariety of $\mathbf{H}_{2}(Y)$.

By combining the result of [CHL18] and Theorem 1.1, it turns out that non-free
lines in $Y$ consist of lines meeting with the _dual conic_ $C_{v}^{\vee}$ at
a point (Corollary 3.2). Furthermore we obtain the designularized model (i.e.,
a subvariety of _complete conics_) of the moduli space of stable maps of
degree $2$ in $Y$, which enable to compute the intersection cohomology group
of the moduli space. For detail description, see Corollary 3.4.

######  Notation 1.2.

  * •

Let us denote by $\mathrm{Gr}(k,n)$ the Grassimannian variety parameterizing
$k$-dimensional subspaces in a fixed vector space $V$ with $\dim V=n$.

  * •

We sometimes do not distinguish the moduli point $[x]\in\mathcal{M}$ and the
object $x$ parameterized by $[x]$ when no confusion can arise.

  * •

Let us shortly denote the projectivized linear subspace
$\mathbb{P}(e_{i},\cdots,e_{j})=\mathbb{P}(\text{span}\\{e_{i},\cdots,e_{j}\\})$
in $\mathbb{P}(V_{5})$ where $\\{e_{0},e_{2},\cdots,e_{4}\\}$ is the standard
basis of the vector space $V_{5}$ of dimension $5$.

### Acknowledgements

The author gratefully acknowledges the many helpful suggestions of Sang-Bum
Yoo and Joonyeong Won during the preparation of the paper.

##  2\. Preliminaries

In this section we collect some facts about the quintic del Pezzo fourfold
which are mostly taken from [Pro94] and [CHL18].

One can define a Schburt variety relating with lines and planes in
$\mathrm{Gr}(2,5)$ as follow. For fixed a flag
$p\in\mathbb{P}^{1}\subset\mathbb{P}^{2}\subset\mathbb{P}^{3}\subset\mathbb{P}^{4}$,
let

  * •

$\sigma_{3,2}=\\{\ell\,|\,p\in\ell\subset\mathbb{P}^{2}\\}$,

  * •

$\sigma_{3,1}=\\{\ell\,|\,p\in\ell\subset\mathbb{P}^{3}\\}$,

  * •

$\sigma_{2,2}=\\{\ell\,|\,\ell\subset\mathbb{P}^{2}\\}$.

Clearly, $\sigma_{3,2}$ is a line in $\mathrm{Gr}(2,5)$ and thus it is
parameterized by $\mathrm{Gr}(1,3,5)$. Also, we note that the planes in
$\text{Gr}(2,5)$ with $\sigma_{3,1}$(resp. $\sigma_{2,2}$)-type is
parameterized by the _flag_ variety $\text{Gr}(1,4,5)$ (resp.
$\text{Gr}(3,5)$). The projection maps
$v_{1}:\mathrm{Gr}(1,3,5)\to\mathrm{Gr}(1,5)$ and
$v_{2}:\mathrm{Gr}(1,4,5)\to\mathrm{Gr}(1,5)$ are called by the _vertex map_.
In [Pro94], the Hilbert scheme of lines and planes in $Y$ are explicitly
described. For a projective variety $X$ with fixed embedding in
$\mathbb{P}^{N}$, let $\mathbf{H}_{1}(X)$ (resp. $\mathbf{F}_{2}(X)$) be the
Hilbert scheme of lines (resp. planes) in $X$

######  Proposition 2.1 ([Pro94, Proposition 2.7]).

Let
$i:\mathbf{H}_{1}(Y)\subset\mathbf{H}_{1}(\mathrm{Gr}(2,5))=\mathrm{Gr}(1,3,5)$
be the inclusion map and $v_{1}:\mathrm{Gr}(1,3,5)\rightarrow\mathrm{Gr}(1,5)$
be the vertex map. Then the composition map $v_{1}\circ
i:\mathbf{H}_{1}(Y)\to\mathrm{Gr}(1,5)$ is a smooth blow-up along the smooth
conic $C_{v}\subset\mathrm{Gr}(1,5)$.

Let us call $C_{v}$ by the _vertex conic_ in Proposition 2.1.

######  Proposition 2.2 ([Pro94, Proposition 2.2]).

The Hilbert scheme of planes in $Y$ is isomorphic to

$\mathbf{F}_{2}(Y)\cong C_{v}\sqcup\\{[S]\\}.$

Here each point $t\in C_{v}(\cong\mathbb{P}^{1})$ parameterizes the
$\sigma_{3,1}$-type planes $P_{t}$ such that the vertex of the plane $P_{t}$
is the point $\\{t\\}$ in $C_{v}$. Also the point $[S]$ parameterizes the
$\sigma_{2,2}$-type plane $S$ in $Y$ determined by the linear spanning
$\langle C_{v}\rangle\subset\mathrm{Gr}(1,5)\cong\mathbb{P}^{4}$ of $C_{v}$.

Let $\\{e_{0},e_{1},e_{2},e_{3},e_{4}\\}$ be the standard coordinate vectors
of the space $V_{5}(\cong\mathbb{C}^{5})$, which provides the original
projective space $\mathbb{P}(V_{5})(=\mathbb{P}^{4})$. Let
$\\{p_{ij}\\}_{0\leq i<j\leq 4}$ be the Plücker coordinates of
$\mathbb{P}^{9}$. Let $\mathbb{P}^{7}=H_{1}\cap H_{2}$ be the linear subspace
of $\mathbb{P}^{9}$ defined by $p_{12}-p_{03}=p_{13}-p_{24}=0$. The vertex
conic $C_{v}$ is given by ([CHL18, Lemma 6.3])

$C_{v}=\\{[a_{0}:a_{1}:a_{2}:a_{3}:a_{4}]\mid
a_{0}a_{4}+a_{1}^{2}=a_{2}=a_{3}=0\\}\subset\mathbb{P}(V_{5}).$

######  Remark 2.3.

From the proof of [CHL18, Lemma 6.3], we know that $\sigma_{3,1}$-type planes
$P_{t}$ in $Y$ are $P_{t}=\mathbb{P}(V_{1}\wedge V_{4})$ where
$V_{1}=\text{span}\\{e_{0}+te_{1}-t^{2}e_{4}\\}$ and
$V_{4}=\text{span}\\{e_{0},e_{1},e_{2}+te_{3},e_{4}\\}$. Also the unique plane
$S$ in $Y$ is given by $S=\mathbb{P}(\wedge^{2}V_{3})$ such that
$V_{3}=\text{span}\\{e_{0},e_{1},e_{4}\\}$.

The positional relation of planes in $Y$ are as follows.

######  Proposition 2.4 ([Pro94, Proposition 2.2]).

Let $P_{t}$ be a $\sigma_{3,1}$-type plane and $S$ be the unique
$\sigma_{2,2}$-type plane in $Y$. Then

  1. (1)

the intersection part $P_{t}\cap S$ is tangent line of the dual conic111That
is, the curve is generated by the tangent lines of $C_{v}$. $C_{v}^{\vee}$ in
$Y$.

  2. (2)

the intersection part $P_{t}\cap P_{t^{\prime}}$ is a point in $S$ for any
$t\neq t^{\prime}\in C_{v}$.

The lines in $Y$ have a stratification relating with the plane’s type in $Y$.

######  Proposition 2.5 ([Pro94, Corollary 3.7]).

Let $L$ be a line in $Y$ and $R=\bigcup\limits_{t\in C_{v}}P_{t}$ be the union
of planes in $Y$. Then there are five types of lines in $Y$ such that the
automorphism group $\mathrm{Aut}(Y)$ of $Y$ transitively acts on each stratum.

  1. (a)

$L\nsubseteq R\cup S$.

  2. (b)

$L\subset R$, $L\cap S=\\{\mathrm{pt}.\\}$ and $L\cap C_{v}^{\vee}=\emptyset$.

  3. (c)

$L\subset R$, $L\cap S=L\cap C_{v}^{\vee}=\\{\mathrm{pt}.\\}$.

  4. (d)

$L\subset S$ and $L$ is a tangent line of $C_{v}^{\vee}$.

  5. (e)

$L\subset S$ and $L\cap C_{v}^{\vee}=\\{p_{1},p_{2}\\}$ for $p_{1}\neq p_{2}$.

In Section $6$ of [CHL18], the authors reproduce the results of Proposition
2.1, 2.2, and 2.4 by specifying the linear subspace
$\mathbb{P}^{7}\subset\mathbb{P}^{9}$.

######  Example 2.6.

Let $P_{t_{0}}$ be the plane determined by the vertex $\mathbb{P}(e_{0})$ and
the three dimensional space $\mathbb{P}(e_{0},e_{1},e_{2},e_{4})$. The
intersection point is $P_{t_{0}}\cap C_{v}^{\vee}=\mathbb{P}(e_{0}\wedge
e_{1})$ which is the tangent line of $C_{v}$ at $\mathbb{P}(e_{0})$.
Furthermore, the example of lines in Proposition 2.5 are given in Table 1.

Type | Vertex |  Plane
---|---|---
(a) | $\mathbb{P}(e_{2})$ |  $\;\;\;\;\;\;\mathbb{P}(e_{0},e_{2},e_{3})$
(b) | $\mathbb{P}(e_{0})$ |  $\;\;\;\;\;\;\mathbb{P}(e_{0},e_{2},e_{4})$
(c) | $\mathbb{P}(e_{0})$ |  $\;\;\;\;\;\;\mathbb{P}(e_{0},e_{1},e_{2})$
(d) | $\mathbb{P}(e_{0})$ |  $\;\;\;\;\;\;\mathbb{P}(e_{0},e_{1},e_{4})$
(e) | $\mathbb{P}(e_{1})$ |  $\;\;\;\;\;\;\mathbb{P}(e_{0},e_{1},e_{4})$
Table 1. Example of lines in $Y$

Let $\mathbf{H}_{2}(Y)$ be the Hilbert scheme of conics in $Y$. For a general
conic $C$ in $Y$, it determines linear spanning in two meanings: the linear
space $\mathbb{P}^{2}$ containing $C$ in
$\mathbb{P}(\wedge^{2}V_{5})=\mathbb{P}^{9}$ and the linear space
$\mathbb{P}^{3}$ containing two skew lines in
$\mathbb{P}(V_{5})=\mathbb{P}^{5}$. Motivated this observation, we have a
birational model of $\mathbf{H}_{2}(Y)$ as follow. Let $\mathcal{U}$ be the
universal subbundle on $\mathrm{Gr}(4,5)$ and

$\mathcal{K}:=\mathrm{ker}\\{\wedge^{2}\mathcal{U}\subset\wedge^{2}\mathcal{O}^{\oplus
5}\to\mathcal{O}^{\oplus 2}\\}$

be the kernel of the composition map where the arrow is given by
$\\{p_{12}-p_{03},p_{13}-p_{24}\\}$. Let
$\mathbf{S}(Y):=\mathrm{Gr}(3,\mathcal{K})$ be the relative Grassmannian over
$\mathrm{Gr}(4,5)$.

######  Proposition 2.7 ([CHL18, Proposition 6.7 and Remark 6.8]).

Under above definition and notation, $\mathbf{H}_{2}(Y)$ is obtained from
$\mathbf{S}(Y)$ by a blow-down followed by a blow-up

$\textstyle{\widetilde{\mathbf{S}}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{S}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\mathbf{H}_{2}(Y),}$

where

  1. (1)

the blow-up center in $\mathbf{S}(Y)$ (resp. $\mathbf{H}_{2}(Y)$) is a
disjoint union $\mathbb{P}^{1}\sqcup\mathbb{P}^{1}$ (resp. $\mathbb{P}^{5}$)
of $\mathbb{P}^{1}$’s and

  2. (2)

the space $\widetilde{\mathbf{S}}(Y)$ is a relative conics space over
$\mathrm{Gr}(4,5)$ such the fiber over $\mathrm{Gr}(4,5)$ is the Hilbert
scheme $\mathbf{H}_{2}(\mathrm{Gr}(2,4)\cap H_{1}\cap H_{2})$ of conics in the
quadric surface $\mathrm{Gr}(2,4)\cap H_{1}\cap H_{2}$.

In special $\mathbf{H}_{2}(Y)$ is an irreducible and smooth variety of
dimension $7$.

######  Remark 2.8.

The relative Grassimannian $\mathbf{S}(Y)=\mathrm{Gr}(3,\mathcal{K})$ in
Proposition 2.7 can be regarded as the incident variety

(1) $\mathbf{S}(Y)=\\{(U_{3},V_{4})\mid
U_{3}\subset\mathcal{K}_{[V_{4}]}\\}\subset\mathrm{Gr}(3,\wedge^{2}V_{5})\times\mathrm{Gr}(4,V_{5}),$

where
$\mathcal{K}_{[V_{4}]}=\text{ker}\\{\wedge^{2}V_{4}\subset\wedge^{2}V_{5}\stackrel{{\scriptstyle(p_{12}-p_{03})\oplus(p_{13}-p_{24})}}{{\rightarrow}}\mathbb{C}\oplus\mathbb{C}\\}$.
Also the birational correspondence
$\Psi:\mathbf{S}(Y)\dashrightarrow\mathbf{H}_{2}(Y)$ is
$\Psi([(U_{3},V_{4})])=\mathbb{P}(U_{3})\cap\mathrm{Gr}(2,V_{4})$. Note that
the map $\Psi$ is not defined at the two distinct points $[P_{t}]$ and $[S]$
over a linear subspace $\mathbb{P}^{1}(\cong C_{v})$ in $\mathrm{Gr}(4,5)$
([CHL18, Remark 6.8]).

##  3\. Results

In this section we prove Theorem 1.1. As corollaries, we have a description of
the locus of non-free lines in $Y$ (Corollary 3.2). Also we obtain the
desinguarized model of stable maps space in $Y$ and thus its intersection
cohomology (Corollary 3.4).

###  3.1. Proof of Theorem 1.1

Firstly, we describe the closure of the birational inverse $\Psi^{-1}$ of the
double line in $\mathbf{H}_{2}(Y)$ in Proposition 2.7. Then we find explicitly
the strict transform of the closure along the blow-up/down maps in Proposition
2.7.

Let $\bar{\mathbf{D}}(Y)$ be the locus of the pairs $(U_{3},V_{4})$ in
$\mathbf{S}(Y)$ such that the restriction $q_{G}|_{\mathbb{P}(U_{3})}$ to
$\mathbb{P}(U_{3})$ of the quadric form $q_{G}$ associated to
$\mathrm{Gr}(2,V_{4})$ is rank $\leq 1$. Let

$p=p_{2}\circ i:\bar{\mathbf{D}}(Y)\rightarrow\mathrm{Gr}(4,5)$

be the composition of the second projection map
$p_{2}:\mathbf{S}(Y)\rightarrow\mathrm{Gr}(4,5)$ in equation (1) and the
inclusion map $i:\bar{\mathbf{D}}(Y)\subset\mathbf{S}(Y)$.

######  Lemma 3.1.

Under above definition and notations, the image
$p(\bar{\mathbf{D}}(Y)):=Q_{3}$ is an irreducible quadric $3$-fold in
$\mathrm{Gr}(4,5)$ with the homogenous coordinates
$x_{0},x_{1},x_{2},x_{3},x_{4}$ such that $Q_{3}$ is defined by
$x_{1}^{2}+4x_{0}x_{2}=0$.

###### Proof.

For the chart $x_{3}\neq 0$, let

$[V_{4}]:=\left(\begin{matrix}1&0&0&a&0\\\ 0&1&0&b&0\\\ 0&0&1&c&0\\\
0&0&0&d&1\\\ \end{matrix}\right)$

be an affine chart of $\mathrm{Gr}(4,5)$ with
$a=x_{0}/x_{3},b=x_{1}/x_{3},c=x_{4}/x_{3},d=x_{2}/x_{3}$ and
$V_{4}=\text{span}\\{e_{0}+ae_{3},e_{1}+be_{3},e_{2}+ce_{3},e_{4}+de_{3}\\}$.
Then for an affine chart of $\mathrm{Gr}(2,V_{4})$

$[V_{2}]=\left(\begin{matrix}1&0&t_{1}&t_{3}\\\ 0&1&t_{2}&t_{4}\\\
\end{matrix}\right),$

the affine chart of $\mathrm{Gr}(2,V_{4})$ in $\mathrm{Gr}(2,5)$ is

$[V_{2}][V_{4}]=\left(\begin{matrix}1&0&t_{1}&a+ct_{1}+dt_{3}&t_{3}\\\
0&1&t_{2}&b+ct_{2}+dt_{4}&t_{4}\end{matrix}\right).$

After eliminating the variables $\\{t_{1},t_{2},t_{3},t_{4}\\}$ by the
computer program ([GS]), we have a defining equation of
$\mathrm{Gr}(2,V_{4})\cap H_{1}\cap H_{2}$

(2) $\begin{split}\langle
bcp_{01}^{2}+c^{2}p_{01}p_{02}&+cdp_{01}p_{04}-ap_{01}^{2}+bp_{01}p_{04}+cp_{02}p_{04}+dp_{04}^{2}+dp_{01}p_{14}-p_{02}p_{14},\\\
\;h_{1},\;h_{2},\;h_{3},&\;h_{4},\;h_{5},\;h_{6}\rangle\end{split}$

where

$\begin{split}h_{1}&=p_{03}-p_{12},\;h_{2}=p_{12}-bp_{01}-cp_{02}-dp_{04},\;h_{3}=p_{13}-p_{24},\\\
h_{4}&=p_{23}+ap_{02}+bp_{12}-dp_{24},\;h_{5}=p_{24}+ap_{01}-cp_{12}-dp_{14},\;h_{6}=p_{34}-ap_{04}-bp_{14}-cp_{24}\end{split}$

in $\mathbb{P}^{9}\times\mathbb{C}_{(a,b,c,d)}$.

For the chart $x_{4}\neq 0$, let
$a=x_{0}/x_{4},b=x_{1}/x_{4},u=x_{3}/x_{4},d=x_{2}/x_{4}$. By doing the same
calculation as before, we obtain the local equation of
$\mathrm{Gr}(2,V_{4})\cap H_{1}\cap H_{2}$ as follows:

(3) $\begin{split}\langle
ap_{04}^{2}+ap_{01}p_{14}+bp_{04}p_{14}&-dp_{14}^{2}-p_{01}p_{34}-aup_{04}p_{14}-bup_{14}^{2}-up_{04}p_{34}+u^{2}p_{14}p_{34},\\\
\;k_{1},\;k_{2},\;k_{3},&\;k_{4},\;k_{5},\;k_{6}\rangle\end{split}$

where

$\begin{split}k_{1}=&p_{02}+bp_{01}+dp_{04}-up_{12},\;k_{2}=p_{03}-p_{12},\;k_{3}=p_{12}-ap_{01}+dp_{14}-up_{24},\\\
k_{4}=&p_{13}-p_{24},\;k_{5}=p_{23}+ap_{12}+bp_{24}-dp_{34},\;k_{6}=p_{24}+ap_{04}+bp_{14}-up_{34}\end{split}$

in $\mathbb{P}^{9}\times\mathbb{C}_{(a,b,u,d)}$.

Since the restricted form $q_{G}$ associated to $\mathrm{Gr}(2,V_{4})$ is of
$\text{rank}(q_{G}|_{\mathbb{P}(U_{3})})\leq 1$, the quadratics of the
defining equations (2) and (3) is of rank $\leq 3$. Therefore the defining
equation of the image $p(\bar{\mathbf{D}}(Y))$ is given by

$\langle b^{2}+4ad\rangle$

in both cases. ∎

Obviously, the singular locus of $\text{Sing}(Q_{3})(\cong\mathbb{P}^{1})$ is
defined by $I_{\mathrm{Sing}(Q_{3})}=\langle x_{0},x_{1},x_{2}\rangle$.

###### Proof of Theorem 1.1.

Step 1. For each $[V_{4}]\in Q_{3}\setminus\text{Sing}(Q_{3})$, the quadric
surface $\mathrm{Gr}(2,V_{4})\cap H_{1}\cap H_{2}$ is of rank $3$. Hence the
fiber $p^{-1}([V_{4}])$ is isomorphic to $\mathbb{P}^{1}$ which parameterizes
tangent planes (i.e., lines) of the quadric cone $\mathrm{Gr}(2,V_{4})\cap
H_{1}\cap H_{2}$. If $[V_{4}]\in\text{Sing}(Q_{3})$, the singular quadric
surface $\mathrm{Gr}(2,V_{4})\cap H_{1}\cap H_{2}$ is the union of the plane
$P_{t}$ and $S$. In fact, for the affine chart $x_{3}\neq 0$ (similarly,
$x_{4}\neq 0$), it is defined by the union of $\sigma_{3,1}$-type planes:

$\langle
c^{2}p_{01}+cp_{04}-p_{14},p_{23},cp_{24}-p_{34},cp_{02}-p_{12},-cp_{12}+p_{24},p_{13}-p_{24},p_{03}-p_{12}\rangle$

and the $\sigma_{2,2}$-plane $S$:

$\langle
p_{02},p_{23},cp_{24}-p_{34},cp_{02}-p_{12},-cp_{12}+p_{24},p_{13}-p_{24},p_{03}-p_{12}\rangle$

which matches with Remark 2.3 (by letting $c=t$). Hence the fiber
$p^{-1}([V_{4}])$ is isomorphic to $\mathbb{P}^{1}$ which parameterizes planes
containing the intersection line $P_{t}\cap S$. After all, $\bar{D}(Y)$ is a
$\mathbb{P}^{1}$-fiberation over $Q_{3}$.

Step 2. Note that the birational map $\Psi$ in Proposition 2.7 is not defined
for the two points: $\\{[P_{t}],[S]\\}$ over
$\text{Sing}(Q_{3})(\cong\mathbb{P}^{1})$. Hence the blow-up center of
$\eta:\widetilde{\mathbf{S}}(Y)\rightarrow\mathbf{S}(Y)$ is contained in
$\bar{\mathbf{D}}(Y)$ and thus the strict transform of $\bar{\mathbf{D}}(Y)$
by the blow-up map $\eta$ is nothing but the blow-up
$\widetilde{\mathbf{D}}(Y)$ of $\bar{\mathbf{D}}(Y)$ along the center
$\mathbb{P}^{1}\sqcup\mathbb{P}^{1}$. Since the blow-center of
$\bar{\mathbf{D}}(Y)$ is of $\mathbb{Z}_{2}$-quotient singularity, one can
easily check that $\widetilde{\mathbf{D}}(Y)$ is smooth and the exceptional
divisor $\mathbf{E}$ in $\widetilde{\mathbf{D}}(Y)$ is a
$\mathbb{P}(1,2,2)(\cong\mathbb{P}^{2})$-bundle over
$\mathbb{P}^{1}\sqcup\mathbb{P}^{1}$. Each fiber $\mathbb{P}^{2}$
parameterizes the double line in the plane because any flat family in
$\bar{\mathbf{D}}(Y)$ is obviously supported on lines by its construction.

Step 3. The restriction to each fiber $\mathbb{P}^{1}$ of the normal bundle
$\mathcal{N}_{\mathbf{E}/\widetilde{\mathbf{D}}(Y)}$ of the exceptional
divisor $\mathbf{E}$ is
$\mathcal{N}_{\mathbf{E}/\widetilde{\mathbf{D}}(Y)}|_{\mathbb{P}^{1}}\cong\mathcal{O}_{\mathbb{P}^{1}}(-1)$,
the image $\mathbf{D}(Y)$ of the restriction to $\widetilde{\mathbf{D}}(Y)$ of
the blow-down map $\widetilde{\mathbf{S}}(Y)\rightarrow\mathbf{H}_{2}(Y)$ is
smooth by the Fujiki-Nakano criterion ([FN72]). So we finish the proof. ∎

###  3.2. Non-free lines in $Y$ and the intersection cohomology of stable maps

######  Corollary 3.2.

Let $Z$ be the locus of non-free lines in the Hilbert scheme
$\mathbf{H}_{1}(Y)$ of lines in $Y$. Then $Z$ is isomorphic to a
$\mathbb{P}^{1}$-fiberation over the vertex conic
$C_{v}(\cong\mathbb{P}^{1})$.

###### Proof.

Geometrically, non-free lines are lines in $Y$ meeting with the dual conic
$C_{v}^{\vee}$ at a point uniquely. Since the automorphism of $Y$ transitively
acts on each stratum of Proposition 2.5, it is enough to check each case in
Table 1. For the case (d) and (e), $L$ is non-free line if and only if $L$ is
a tangent line of $C_{v}^{\vee}$ by [CHL18, Proposition 6.6]. Thus the lines
of the case (d) are only non-free. For the case (c), the line $L$ is defined
by $p_{03}=p_{04}=p_{12}=p_{13}=p_{14}=p_{23}=p_{24}=p_{34}=0$. Thus for the
affine chart $x_{3}\neq 0$, it lies on irreducible quadric cones defined by
$dp_{04}^{2}+(dp_{01}-p_{02})p_{14}=h_{1}=p_{12}-dp_{04}=h_{3}=p_{23}-dp_{24}=p_{24}-dp_{14}=p_{34}=0$
for $d\neq 0$. Hence there exists one parameter family of double lines
supported on $L$. That is, $L$ is non-free. For other cases (a) and (b), we
know that each line is free by a similar computation. ∎

Let $C$ be a projective connected reduced curve. A map $f:C\to Y$ is
considered _stable_ if $C$ has at worst nodal singularities and
$|\mathrm{Aut}(f)|<\infty$. Let $\mathcal{M}(Y,d)$ be the moduli space of
isomorphism classes of stable maps $f:C\to Y$ with genus $g(C)=0$ and
$\mathrm{deg}(f^{*}\mathcal{O}_{Y}(1))=d$. The moduli space $\mathcal{M}(Y,d)$
might be singular and reducible depending on the geometric property (for
example, convexity) of $Y$.

######  Remark 3.3.

Let $f:C\rightarrow L\subset Y$ be a stable map of the degree $\deg(f)=2$ such
that $L$ is non-free. From the tangent bundle sequence of $L\subset Y$,
$\mathrm{H}^{1}(f^{*}T_{Y})\cong\mathrm{H}^{1}(f^{*}N_{L/Y})\cong\mathrm{H}^{1}(f_{*}\mathcal{O}_{C}\otimes
N_{L/Y})\cong\mathbb{C}$. That is, $Y$ is not convex and thus
$\mathcal{M}(Y,2)$ is not smooth stack ([FP97]).

Let $X$ be a quasi-projective variety. For the (resp. intersection) Hodge-
Deligne polynomial $\mathrm{E}_{c}(X)(u,v)$ (resp. $\mathrm{IE}_{c}(X)(u,v)$)
for compactly supported (resp. intersection) cohomology of $X$, let

$\mathrm{P}(X)=\mathrm{E}_{c}(X)(-t,-t)\;(\mathrm{resp.}\;\mathrm{IP}(X)=\mathrm{IE}_{c}(X)(-t,-t))$

be the _virtual_ (resp. intersection) Poincaré polynomial of $X$. A map
$\pi:X\rightarrow Y$ is _small_ if for a locally closed stratification of
$Y=\bigsqcup_{i}Y_{i}$ such that the restriction map
$\pi|_{\pi^{-1}(Y_{i})}:\pi^{-1}(Y_{i})\rightarrow Y_{i}$ is _etale_ locally
trivial, the inequality

$\dim\pi^{-1}(y)<\frac{1}{2}\mathrm{codim}_{Y}(Y_{i})$

holds for each closed point $y\in Y_{i}$ except a dense open stratum of $Y$.
Let $\pi:X\rightarrow Y$ be a small map such that $X$ has at most finite group
quotient singularities. Then $\mathrm{P}(X)=\mathrm{IP}(Y)$ ([Max18,
Definition 6.6.1 and Theorem 6.6.3]).

######  Corollary 3.4.

The intersection cohomology of the moduli space $\mathcal{M}(Y,2)$ is given by

$\mathrm{IP}(\mathcal{M}(Y,2))=1+4t^{2}+10t^{4}+15t^{6}+15t^{8}+10t^{10}+4t^{12}+t^{14}.$

###### Proof.

By the same method of the proof of Theorem 1.2 in [Chu22], one can show that
the blow-up $\widetilde{\mathbf{H}}_{2}(Y)$ of $\mathbf{H}_{2}(Y)$ along
$\mathbf{D}(Y)$ is smooth one and thus we have a birational morphism

$\pi:\mathbf{H}_{2}(Y)\rightarrow\mathcal{M}(Y,2)$

such that the exceptional divisor (i.e., $\mathbb{P}^{2}$-bundle over
$\mathbf{D}(Y)$) contracts to a $\mathbb{P}^{2}$-bundle over
$\mathbf{H}_{1}(Y)$. From Corollary 3.2, the map $\pi$ is a small map and thus
$\mathrm{P}(\widetilde{\mathbf{H}}_{2}(Y))=\mathrm{IP}(\mathcal{M}(Y,2))$. By
the equality

$\mathrm{P}(\widetilde{\mathbf{H}}_{2}(Y))=\mathrm{P}(\mathbf{H}_{2}(Y))+(\mathrm{P}(\mathbb{P}^{2})-1)\cdot\mathrm{P}(\mathbf{D}(Y)),$

Proposition 2.1 and Corollary 3.2, we obtain the result. ∎

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