# TiRE-GAN: Task-Incentivized Generative Learning Models for Radiomap
Estimation with Radio Propagation Model

Yueling Zhou*, Achintha Wijesinghe†, Songyang Zhang*, and Zhi Ding†, Fellow,
IEEE
*Department of Electrical and Computer Engineering, University of Louisiana at Lafayette, Lafayette, LA, USA, 70504
†Department of Electrical and Computer Engineering, University of California
at Davis, Davis, CA, USA, 95616
E-mail: {yueling.zhou1<EMAIL_ADDRESS>{achwijesinghe,
<EMAIL_ADDRESS>
###### Abstract

Enriching geometric information on radio frequency (RF) signal power
distribution in wireless communication systems, the radiomap has become an
essential tool for resource allocation and network management. Usually, a
dense radiomap is reconstructed from sparse observations collected by deployed
sensors or mobile devices, which makes the radiomap estimation an urgent
challenge. To leverage both physical principles of radio propagation models
and data statistics from sparse observations, this work introduces a novel
task-incentivized generative learning model, namely TiRE-GAN, for radiomap
estimation. Specifically, we first introduce a radio depth map as input to
capture the overall pattern of radio propagation and shadowing effects,
following which a task-driven incentive network is proposed to provide
feedback for radiomap compensation depending on downstream tasks. Our
experimental results demonstrate the power of the radio depth map to capture
radio propagation information, together with the efficiency of the proposed
TiRE-GAN for radiomap estimation.

###### Index Terms:

radiomap estimation, generative adversarial networks, radio propagation model

††publicationid: pubid:  978-1-6654-3540-6/22 © 2022 IEEE

##  I Introduction

The expansion of sensor networks and edge computing systems has stimulated
many new technologies in the next generation of wireless communications, such
as vehicle-to-everything (v2x) communications [1], unmanned aerial vehicle
(UAV)-assisted networking [2] and fifth-generation (5G) terrestrial and non-
terrestrial network (NTN) services [3]. All these applications require an
accurate description of radio frequency (RF) spectrum coverage and efficient
assessment of the radio environment, which lead to the new concept of
radiomaps [4]. Describing the RF power spectral density (PSD) over different
geometric locations, frequencies, and time, radiomaps can enrich the
information of spectrum coverage and assist the aforementioned spectrum
management applications [5]. Usually, dense radiomaps are reconstructed from
sparse observations collected by mobile sensors and user devices, as shown in
Fig. 1. To fully leverage the power of radiomaps in spectrum resource
allocation and network optimization, efficient radiomap estimation (RME) from
sparse observations has emerged as an urgent challenge.

Existing methods of radiomap estimation can be categorized into model-based or
learning-based approaches, each bearing its distinct constraints. Model-based
methods usually assume a specific radio propagation model, with which unknown
parameters are estimated by minimizing the error between observed samples and
predictions. Typical examples include the log-distance path loss (LDPL) models
[6], inverse distance weighted interpolation [7] and thin-plate splines
kernels [8]. However, the performances of model-based RME highly rely on the
selection of physical radio propagation models, which are usually unknown in
practical applications. Moreover, the PSD distribution sometimes is more
sensitive to surrounding obstacles rather than following a strict empirical
model, especially in complicated environments. The efficient extraction of
radio model information from realistic data samples remains an open question.

With the development of artificial intelligence (AI), recent interests of RME
focus on learning-based methods [9]. In [10], a fast radiomap estimation is
proposed based on Unet. Another work utilizing deep convolutional neural
networks is [11], where an auto-encoder structure is presented for radiomap
estimation. Other learning-based methods also include the conditional
generative adversarial nets (cGAN) [3] and transformers [12]. Despite many
successes, efficient learning-based RME requires sufficient data samples with
satisfying quality, which are usually inaccessible in realistic scenarios.
Although some works [13, 14] consider integrating the radio propagation models
with learning machines to address the data scarcity, the efficient model
selection remains challenging. Moreover, how to satisfy the special
requirements of downstream tasks, such as outage detection and network
planning, is another critical concern.

To address the aforementioned problems, this work proposed a Task-incentivized
Radiomap Estimation based on Generative Adversarial Networks (TiRE-GAN) for
radiomap estimation, leveraging the synergies of radio propagation model and
generative AI. Specifically, a radio depth map is introduced as an additional
input channel of the generator to capture the overall radio propagation
information and shadowing effects. Then, a task-incentivized block is proposed
to provide feedback for radiomap generation to compensate for the estimated
PSD based on downstream tasks. Our contributions can be summarized as follows:

  * •

We introduce a novel radio depth map (RDM) as input conditions to the cGAN,
serving as a proxy for the shadowing effects and radio propagation behavior,
thereby infusing the learning process with additional insights into the
physical principles of radio propagation.

  * •

We integrate the physics-conditioned cGAN with a novel task-incentivized
network to provide compensation to the estimated radiomaps depending on
downstream tasks, which significantly enhances the RME performance.

  * •

Our experimental results demonstrate the power of the proposed radio depth map
in characterizing radio propagation behavior, as well as the efficiency of the
proposed TiRE-GAN in radiomap estimation.

##  II Problem Description

Figure 1: Problem Description of RME: Given the distribution of buildings,
transmitter locations (in read), and the sparse observations collected by
mobile sensors and devices, we aim to predict the dense radiomaps. Figure 2:
Overall Structure of TiRE-GAN consisting of 3 parts: 1) The model-based data
preprocessing module utilizes the input observed information (e.i. building
map and transmitter location) to generate the radio depth map and then stacks
the four layers together as input; 2) The radiomap (RM) generation module is
in a cGAN generator structure, which learn and produce the prediction of
radiomap; 3) The feedback module consists of 2 parts: a cGAN discriminator and
a task-incentivized network. Both modules produce a loss term as feedback to
improve the generator.

In this section, we first introduce our problem formulation. As shown in Fig.
1, the radiomap estimation is formulated as a sparse sensing problem, where a
dense radiomap $\hat{M}_{R}$ is reconstructed given the landscape information
of the surrounding environment, sparse observations collected from sensors and
mobile devices, and locations of transmitters, formulated by the following
equation:

$\hat{M}_{R}=g(M_{U},M_{T},M_{S},\mathcal{X}),$ (1)

where $M_{U}$, $M_{T}$, $M_{S}$ and $\mathcal{X}$ represent the landscape map,
the encoded transmitter location, the sampled radiomap, and additional
extracted features, respectively.

Different from the interpolation formulation in [6] and [7], where the
unobserved PSD values are estimated only using the observed samples in the
same radiomap, we follow the classic prediction formulation and assume a set
of training pairs of $(M_{R},\mathcal{F})$ with
$\mathcal{F}=\\{M_{U},M_{T},M_{S},\mathcal{X}\\}$ to train the mapping
function $g(\cdot)$ by

$\min_{g(\cdot)}\quad L(M_{R},\hat{M}_{R}),$ (2)

where $L(\cdot)$ refers to the designed loss function.

In this work, our objectives can be summarized as two parts: 1) design an
effective mapping function $g(\cdot)$ by integrating the cGAN with a novel
task-incentivized network; and 2) design a suitable loss function $L(\cdot)$
to embed the physical principles of radio propagation. We shall introduce the
details of our proposed TiRE-GAN in Section III.

##  III Methods

###  III-A Overall Structure of TiRE-GAN Framework

We now introduce the overall structure of our proposed TiRE-GAN. As shown in
Fig. 2, the architecture of the proposed framework can be structured into
three components: 1) model-based data preprocessing; 2) radiomap generation
based on cGAN; and 3) task-incentivized feedback.

The overall operational sequence of the training process is delineated as
follows. To embed the radio propagation model, the urban landscape map and the
transmitter location data are processed to fabricate a physics-embedded Radio
Depth Map (RDM), which will be further discussed in Section III-B2.
Subsequently, this RDM, alongside the sampled radiomap, urban map, and
transmitter location data, are amalgamated into a quadruple-channel input for
the cGAN. Throughout the training phase, a pre-trained task-incentivized
network is incorporated as a contributory factor within the loss function for
the cGAN’s generator, enhancing the model’s ability to compensate for the
radiomap estimations by embedding physics-based constraints and the rewards
from downstream tasks. We will then introduce each module as follows.

###  III-B Model-Based Data Preprocessing

The data preprocessing module consists of two parts: 1) process the observed
features $\\{M_{U},M_{T},M_{S}\\}$ into aligned inputs of cGAN; and 2)
generate a radio depth map embedding the radio propagation behavior.

####  III-B1 Observed Data Preprocessing

Suppose that the radiomap is formatted in a $N\times N$ grid, i.e.,
$M_{R}\in\mathbb{R}^{N\times N}$. The landscape map of buildings is encoded as
a binary segmented image $M_{U}\in\mathbb{R}^{N\times N}$, where building
areas are marked by 1 and open areas are annotated by 0. Similar to the
building map, we encode the transmitter locations into a binary image
$M_{T}\in\mathbb{R}^{N\times N}$, where the transmitter locations are marked
as 1; otherwise, the pixel values are zero. The sparse observations can be
viewed as the corresponding pixels in gray-scale images
$M_{S}\in\mathbb{R}^{N\times N}$. For convenience, we first use a threshold to
eliminate the extremely low-value observations, and then normalize the PSD
value into $0\sim 1$ for the observed locations. Since linear normalization is
applied, we can easily re-scale the estimated radiomap back to the original
data domain. For the missing values in $M_{S}$, we set them as zeros. Since a
convolutional neural network is applied for the cGAN structure, zero-padding
does not impact the final radiomap estimation.

####  III-B2 Radio Depth Map

To embed the radio propagation information, we introduce the Radio Depth Map
(RDM), $\mathcal{X}=M_{D}\in\mathbb{R}^{N\times N}$ as a novel model-based
attribute, derived from the distribution of buildings and the spatial
locations of transmitters. Similar to a depth map in computer vision which
provides information on the distances of surfaces from a specific viewpoint,
the RDM is formulated as:

$M_{D}(x,y)=n\left(\sum_{t=1}^{N_{T}}P_{t}(x,y)\cdot B_{t}(x,y)\right),$ (3)

where $(x,y)$ denotes the two-dimensional coordinates of the target location,
$P_{t}(x,y)$ denotes the distance-based path loss gain from the $t$-th
transmitter, and the $B_{t}(x,y)$ capture the impact of buildings from the
viewpoint of $t$-th transmitter.

Suppose that $C(l,l_{t})$ identifies the set of pixels in the direct line of
sight between the target location $l=(x,y)$ and location $l_{t}=(x_{t},y_{t})$
of the $t$-th transmitter. The shadowing term impacted by buildings is
calculated by

$B_{t}(x,y)=\frac{\sum_{i\in C(l,l_{t})}(1-M_{U}(i))}{\sum_{i\in
C(l,l_{t})}1},$ (4)

which characterizes the ratio of non-building pixels between $l$ and $l_{t}$.
$B_{t}(x,y)$ is larger and the shadowing effect is smaller if the radio
propagates through fewer buildings.

To capture the path loss $P_{t}(x,y)$, we employ the Inverse Distance
Weighting (IDW) model, calculated by

$P_{t}(x,y)=d(l,l_{t})^{-\lambda},\ \lambda>0,$ (5)

where $d(l,l_{t})$ denotes the Euclidean distance between the target point $l$
and the $t$-th the transmitter. The parameter $\lambda$ governs the rate, at
which signal strength diminishes with increasing distance, necessitating a
positive value to ensure physical plausibility in the depth map. An exemplary
radio depth map is shown in the orange block of Fig. 2.

###  III-C cGAN Network

In this work, we employ a conditional Generative Adversarial Network (cGAN)
[15] as the foundational architecture, which consists of two principal
components: a generator $G$ and a discriminator $D$. Different from the
traditional GAN which generates data from random noise, the generator in cGAN
can generate data from given features $\mathcal{F}$ and incorporate additional
information $\mathbf{I}$ in the discriminator. In our radiomap estimation, the
conditional features consist of four channels as processed in the Data
Preprocessing Module, i.e., $\mathcal{F}=\\{M_{U},M_{T},M_{S},M_{D}\\}$ as
shown in Fig. 2. The additional information $\mathbf{I}$ can be designed as a
function of the region features, i.e., $\mathbf{I}=h(\mathcal{F})$ or some
determined labels, such as fake or real [16]. Then, the basic cGAN can be
formulated as

$\displaystyle\min_{G}\max_{D}\quad V(D,G)=\mathbb{E}_{y,I\sim p_{\rm
data}(\bf{y},\bf{I})}[\log D(\bf{y,I})]+$ $\displaystyle\mathbb{E}_{f\sim
p_{\rm data}(\bf{\mathcal{F}})}[\log(1-D(G(\bf{\mathcal{F}}),\mathbf{I}))],$
(6)

where $p_{\rm data}(\bf{y,I})$ is the joint distribution with the prior
knowledge $\mathbf{I}$, and $\mathbf{y}$ is the generated data. Generally, the
optimization of a min-max problem in cGAN can be split into two sub-problems,
and train $D$ and $G$ iteratively. Specifically, the loss function to optimize
$D$ is

$L_{D}=-V(D,G),$ (7)

and the general loss function to optimize $G$ is

$L_{G}=\mathbb{E}_{x\sim p_{\rm
data}(\bf{\mathcal{F}})}[\log(1-D(G(\bf{\mathcal{F}}),\mathbf{I}))].$ (8)

To leverage both radio propagation model information and data statistics from
observed samples, in this work, we will customize the loss function of the
generator as

$L=L_{G}+\alpha L_{MSE}+\beta L_{R}.$ (9)

Here, $L_{MSE}$ describes the error of RME by Mean Squared Error (MSE), i.e.,

$L_{MSE}(\mathbf{y},\hat{\mathbf{y}})=\frac{1}{K}\sum_{i=1}^{K}(y_{i}-\hat{y}_{i})^{2},$
(10)

where $K$ is the total number of pixels, $\mathbf{y}$ and $\hat{\mathbf{y}}$
are the ground truth and the predicted radiomap, while $y_{i}$ and
$\hat{y}_{i}$ are the true and predicted pixel value of the $i$th pixel.
$L_{R}$ is a task-incentivized regularization term from the feedback module,
which will be introduced in Section III-D.

###  III-D Feedback Design

The feedback module consists of two components: 1) a traditional cGAN
discriminator; and 2) a task-incentivized network (TIN) based on downstream
tasks.

####  III-D1 cGAN Discriminator

Conversely to the cGAN generator, the discriminator aims to detect whether the
generated data is fake or not. Here, we consider three-channels as the input
of the discriminator, where two are expanded image-sized one-hot encodings
representing the binary truthfulness of the input, and the third channel is
the generated radiomap. The discriminator outputs a scalar value, indicating
the likelihood of whether the provided radiomap is genuine. The discriminator
is trained by the binary cross entropy (BCE) loss as

$\vspace{-5mm}L_{D}(\mathbf{y},\hat{\mathbf{y}})=-\frac{1}{K}\sum_{i=1}^{K}\left[y_{i}\log(\hat{y}_{i})+(1-y_{i})\log(1-\hat{y}_{i})\right].$
(11)

Figure 3: Task-incentivized network (TIN) structure: We design the TIN based
on Unet, which contains 4 encoding blocks and 4 decoding blocks. Each encoding
block consists of 3 convolution layers and a pooling layer. Each decoding
block consists of one concatenation layer, a transpose convolution layer, and
3 convolution layers.

####  III-D2 Task-Incentivized Network

In traditional RME, the objective transcends the mere accuracy of radio signal
strength at discrete locations within a dense radiomap. However, in practical
scenarios, the final objective of RME is to assist spectrum planning and
network management. For example, one can easily identify the outage of regions
based on their radiomap by setting a certain threshold of service failure. To
compensate for the radiomap estimations based on downstream tasks, we
introduce a TIN module as shown in Fig. 2.

As an exemplary TIN, we adopt outage detection as a downstream task. More
specifically, the proposed TIN utilizes a UNet architecture for semantic
segmentation of radio power outage as Fig. 3, where the input constitutes a
radiomap and the output is an outage map indicating signal coverage or
absence. The training of the TIN is conducted using ground-truth radio
coverage and outage maps, thereby ensuring that the network learns to
accurately predict outage areas based on real-world data. Once trained, the
TIN operates within the cGAN framework as a pre-trained component. During
inference, the output generated by the cGAN’s generator serves as the input to
the TIN, enabling it to contribute an additional term $L_{R}$ to the loss
function of the cGAN. This term is specifically designed to encapsulate
aspects of human-centric interest, such as the distribution and
characteristics of signal outages, thereby guiding the cGAN toward generating
radiomaps that are not only precise but also aligned with the real-world
considerations and requirements of downstream tasks. This integration of the
TIN within the cGAN architecture exemplifies a strategic convergence of
machine learning precision with domain-specific insights, enhancing the
utility and applicability of the generated radiomaps.

The TIN provides the task-incentivized regularization term $L_{R}$ in the loss
function of the generator, which is calculated by the pre-trained TIN using
the MSE loss to measure the error between the predicted and the ground truth
outage maps:

$L_{R}(z,\hat{z})=\frac{1}{K}\sum_{i=1}^{K}(z_{i}-\hat{z}_{i})^{2},$ (12)

where the $z$ and $\hat{z}$ are the ground truth and the predicted outage map,
while $z_{i}$ and $\hat{z}_{i}$ are the true and predicted pixel values of the
$i$th pixel. $K$ is the number of pixels.

Note that, our TIN can be integrated with any RM-assisted applications. Some
other typical examples also include ray tracing or wireless localization.

Figure 4: Performance comparison (sr=1) between Kriging, matrix
completion(MC), AutoEncoder (AE), RadioUNet (RU), conditional GAN (cGAN), our
TiRE-GAN, and the ground truth (GT) on 3 radiomap examples. TiRE-GAN achieves
the best radio propagation and the shadowing effect prediction.

##  IV Experiments

###  IV-A Experiment Setup

####  IV-A1 Data description

In this work, we utilize the RadiomapSeer dataset [10] for evaluation, which
comprises an extensive set of 700 maps from a variety of metropolitan areas,
including Ankara, Berlin, Glasgow, Ljubljana, London, and Tel Aviv. Each map
delineates a quadrant of $256\times 256$ square meters, depicted with a
granularity of $256\times 256$ pixels. The edifices are encoded by assigning a
pixel intensity of one, whereas open spaces are represented by a zero value.
Furthermore, the precise locations of transmitters are encoded with a pixel
intensity of one superimposed on a null background, facilitating accurate
localization.

####  IV-A2 Method Comparison

To evaluate the performance of our TiRE-GAN, we compare with both state-of-
the-art (SOTA) model-based methods, including Kriging [17], and data-driven
approaches, including Matrix completion (MC) [18], AutoEncoder (AE) [11],
RadioUnet (RU) [10] and conventional cGAN [15]. For the interpolation-based
method, we reconstruct the radiomap with observations in the same radiomap.
For the prediction-based method, we split the dataset as 4000/800/800
(70%/15%/15%) for training/validation/testing sets. In our TiRE-GAN, we set
$\alpha=1$ and $\beta=0.1$ in Eq. (9).

###  IV-B Overall performance

We first present the overall performance of different radiomap estimations
under different sampling ratios (sr). In this work, we evaluate the
performance based on the normalized radiomaps via MSE denoted by

$\text{MSE}(\mathbf{y},\hat{\mathbf{y}})=\frac{1}{K}\sum_{i=1}^{K}(y_{i}-\hat{y}_{i})^{2},$
(13)

and normalized MSE (NMSE) calculated by

$\text{NMSE}(\mathbf{y},\hat{\mathbf{y}})=\frac{1}{K}\sum_{i=1}^{K}\frac{(y_{i}-\hat{y}_{i})^{2}}{y_{i}^{2}},$
(14)

where $K=256^{2}$. Note that, since the linear normalization is applied, we
can easily re-scale the estimated radiomap back to the original data domain.
The results are presented in Table I and II. From the results, our proposed
TiRE-GAN always achieves superior performance against all other SOTA
approaches. Specifically, with the assistance of our proposed radio depth map,
our method can achieve a very high reconstruction accuracy even with a very
small sampling ratio and insufficient training samples, which validates the
effectiveness of the proposed method. The visualization results of different
methods in 3 radiomap examples are shown in Fig. 4, where the sample ratio is
1%. From the visualization results, the radioUnet, cGAN, and TiRE-GAN have
better performances than Kringing, MC, and AE in capturing the overall radio
propagation patterns. Compared to radioUnet and cGAN, our TiRE-GAN has more
explicit boundaries for the shadowing effects, resulting from the compensation
from the TIN feedback. The numerical results, together with the visualization
results demonstrate the efficiency of our proposed TiRE-GAN for radiomap
estimation.

TABLE I: Test NMSE Comparison ($\times 10^{-2}$) sr (%) | 0.03 | 0.1 | 0.5 | 1 | 3 | 5 | 10 | Average
---|---|---|---|---|---|---|---|---
Krigin | 24.5696 | 18.9280 | 14.2451 | 12.9608 | 14.7171 | 11.3540 | 10.9647 | 15.3913
RU | 6.5421 | 5.1111 | 2.6374 | 2.2206 | 1.8192 | 1.5191 | 1.2704 | 2.8744
AE | 54.7656 | 53.8440 | 45.6640 | 43.3302 | 26.0041 | 20.7167 | 14.3275 | 36.9503
AE deep | 50.6597 | 48.8197 | 21.4272 | 16.2537 | 6.8252 | 6.5503 | 5.8179 | 156.3537
cGAN | 11.5754 | 5.4805 | 2.1345 | 1.8301 | 1.3009 | 1.1894 | 1.0295 | 3.5057
MC | 83.0969 | 74.7810 | 69.0133 | 66.9996 | 62.8215 | 56.3645 | 55.4314 | 66.9297
TiRE-GAN | 1.9866 | 1.2608 | 0.7282 | 0.6084 | 0.5228 | 0.5119 | 0.4615 | 0.8686
TABLE II: Test MSE Comparison ($\times 10^{-4}$) sr (%) | 0.03 | 0.1 | 0.5 | 1 | 3 | 5 | 10 | Average
---|---|---|---|---|---|---|---|---
Krigin | 188.7554 | 141.3767 | 110.7954 | 101.5476 | 91.9940 | 89.0443 | 86.0208 | 115.5340
RU | 24.6458 | 19.3893 | 10.0221 | 8.4693 | 7.0856 | 5.9640 | 4.9978 | 11.5105
AE | 230.6564 | 226.6141 | 194.4702 | 186.5724 | 113.0439 | 87.7205 | 65.0134 | 157.7272
AE deep | 209.2621 | 201.4013 | 91.9589 | 70.0012 | 26.8177 | 26.4321 | 23.4719 | 92.7636
cGAN | 45.8797 | 22.0908 | 8.4572 | 7.2467 | 5.0700 | 4.6442 | 4.0541 | 13.9203
MC | 342.1892 | 307.9448 | 284.1937 | 275.9010 | 258.6960 | 232.1065 | 228.2640 | 275.6136
TiRE-GAN | 7.0604 | 4.6453 | 2.7678 | 2.3472 | 2.0377 | 2.0015 | 1.8137 | 3.2390

###  IV-C Robustness Against Noise

To evaluate the robustness of TiRE-GAN, we conduct the experiment, where a
noisy input pattern is substituted for the original sparse observation channel
by introducing additive Gaussian noise to each sampled value in the radiomap.
To showcase the effectiveness of the proposed radio depth map and TIN blocks,
we compare our TiRE-GAN with Kriging, matrix completion(MC), AutoEncoder (AE),
RadioUNet (RU), and conditional GAN (cGAN) under different signal-to-noise
ratios (SNR) as Fig. 5. Here, we consider a sampling ratio of 1% for all the
test cases. From the results, our TiRE-GAN always has better performance.
Moreover, the MSE of TiRE-GAN increases slower than those of other methods,
especially compared to the high performance methods (RadioUNet, cGAN),
demonstrating the robustness of the TiRE-GAN against noise.

Figure 5: The MSE ($\times 10^{-4}$) of Kriging, matrix completion(MC),
AutoEncoder (AE), RadioUNet (RU), conditional GAN (cGAN), and TiRE-GAN of
different SNRs under a sample ratio=1%.

###  IV-D Outage Evaluation of Generated radiomap

To further evaluate the performance of reconstructed radiomaps in downstream
tasks, we compare the performance of RM-assisted outage detection. Here, we
input the generated radiomaps from different methods into the pre-trained
outage detection networks. The numerical results of the whole datasets with
sr=1% and the exemplary visualization results are shown in Table III and Fig.
6, respectively. From the results, the traditional model-based methods and
conventional auto-encoder perform poorly in the outage detection. Our TiRE-GAN
always has superior performance, which demonstrates the effectiveness of TIN
in radiomap compensation for downstream tasks.

TABLE III: Outage Map MSE comparison Method | Kriging | MC | AE
---|---|---|---
MSE | 0.078134 | 0.075765 | 0.049377
Method | RU | cGAN | TiRE-GAN
MSE | 0.028439 | 0.030352 | 0.026682
Figure 6: Illustration of Outage Map Prediction under sr=1%. We compared the
outage map predicted by the generated radiomap from Kriging, matrix
completion(MC), AutoEncoder (AE), RadioUNet (RU), conditional GAN (cGAN), and
our TiRE-GAN. The corresponding radiomap and the ground truth outage map(GT)
are shown on the right for reference. The result shows that TiRE-GAN has the
best capability to predict the outage map with a clear boundary and range.

##  V Conclusion

In this work, we introduce a task-incentivized generative learning model,
namely TiRE-GAN for radiomap estimation. More specifically, to characterize
the overall radio propagation behavior, we introduce a radio depth map as
additional physics-embedded features as the condition to guide the data-driven
generative models. To compensate for the radiomap estimation for downstream
tasks, we introduce a task-incentivized network to provide feedback to predict
the detailed features in the radiomap. From our experimental results, the
proposed TiRE-GAN always achieves superior performance, even with insufficient
training samples, which addresses the barriers of conventional learning
approaches. Our numerical and visualization results demonstrate the
effectiveness of the proposed radio depth map and TiRE-GAN. Future research
endeavors will delve into the optimization of the generator’s reward
mechanism. Another promising direction is the multi-band radiomap estimation
for different frequencies.

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# Generically nonreduced components of Hilbert schemes on fourfolds

Joachim Jelisiejew  Faculty of Mathematics, Informatics and Mechanics,
University of Warsaw, Banacha 2, 02-097 Warsaw<EMAIL_ADDRESS>To
Gianfranco

###### Abstract.

We exhibit generically nonreduced components of the Hilbert scheme of at least
$21$ points on a smooth variety of dimension at least four. The result was
announced in [Jel23] and answers a question [Ame10, Problem 3.8]. The method
is similar to the one of [JŠ22, §6].

University of Warsaw. Partially supported by National Science Centre grant
2020/39/D/ST1/00132. The article is accepted in the special volume of
_Rendiconti del seminario matematico di Torino_ dedicated to Prof. Casnati.

##  1\. Introduction

Let $X$ be a smooth quasi-projective variety over a field $\Bbbk$. The Hilbert
scheme of $d$ points $\operatorname{Hilb}_{d}(X)$ is a moduli space of central
importance, with applications to combinatorics [Hai01, Hai03], algebra,
enumerative geometry [Ric22], and classical algebraic geometry [Bea83]. Many
of the applications are limited to the case when $\dim X\leq 2$ as in this
case $\operatorname{Hilb}_{d}(X)$ is smooth [Fog68]. For a good and gentle
introduction to Hilbert schemes, see [Ber12] or [MS05, Chapter 18]. See also
[Jel23] for a list of open problems.

The possible singularities of $\bigsqcup_{d}\operatorname{Hilb}_{d}(X)$ for
$\dim X\geq 3$ are only partially understood. A point
$[Z]\in\operatorname{Hilb}_{d}(X)$ is smooth for every $Z\subseteq X$ which
can be embedded into a smooth surface. As a very particular case, this implies
that $\operatorname{Hilb}_{d}(X)$ is smooth for $d\leq 3$. In contrast, the
Hilbert scheme $\operatorname{Hilb}_{d}(X)$ is singular for every $d\geq 4$
and $\dim X\geq 3$, in fact for every $x\in X$, any degree $d$ subscheme
$Z\subseteq V(\mathfrak{m}_{x}^{2})$ gives a singular point [MS05, Cor 18.30].

The singularities in the case $\dim X=3$ are constrained as the Hilbert scheme
is a critical locus [DS09]. Understanding the singularities is a very active
research area, see for example [GGGL23, RS23, KJS23, Rez23].

The singularities in the case $\dim X\geq 16$ can be almost arbitrary: the
Hilbert scheme satisfies Murphy’s Law up to retraction, see [Jel20]. For
important singularity types, such as nonreduced ones, sharper bounds on $\dim
X$ are known. Szachniewicz [Sza21] proved that
$\operatorname{Hilb}_{d}(\mathbb{A}^{6})$ is nonreduced for every $d\geq 13$;
it has an embedded component. See [Erm12, Sch23] for some results in similar
direction on fixed loci.

One instance where _up to retraction_ cannot be ignored is when we consider
generic nonreducedness. In particular, the results above do not prove that the
Hilbert scheme has any generically nonreduced components. Proving that such
components do exist and already in codimension four is the main aim of the
current article.

###  1.1. Generic nonreducedness

We work over a field of characteristic zero, in particular over a perfect
field. An irreducible component of a finite type $\Bbbk$-scheme is either
_generically smooth_ , that is, its general point is smooth, or _generically
nonreduced_ which means that every point is nonreduced.

The problem is that generic nonreducedness does not propagate along
retractions. For example, consider

$\frac{\Bbbk[\\![y]\\!]}{(y^{2})}\hookrightarrow\frac{\Bbbk[\\![x,y]\\!]}{(xy,y^{2})}.$

The source of this map is generically nonreduced, while the target is
generically reduced. Geometrically speaking, the above map comes from a
retraction of $V(xy,y^{2})\subseteq\mathbb{A}^{2}$ onto
$V(y^{2})\subseteq\mathbb{A}^{1}$ by contracting the $x$ axis:

Therefore, from [Jel20] it does not follow that
$\operatorname{Hilb}_{d}(\mathbb{A}^{16})$ admits generically nonreduced
components. Neither it follows from subsequent paper of Szachniewicz [Sza21].
In contrast, in the paper [JŠ22] the authors show that
$\operatorname{Quot}_{8}(\mathbb{A}^{4})$ has a generically nonreduced
component.

The aim of the present note is to apply the method of Jelisiejew-Šivic to the
case of the Hilbert scheme and show the following theorem, which resolves
[Ame10, Problem 3.8]. Let
$\mathcal{L}_{H}\subseteq\operatorname{Hilb}_{\mathrm{pts}}(\mathbb{A}^{n})$
denote the locus of $[Z]$ such that $Z=\operatorname{Spec}(A)$ is an
irreducible scheme corresponding to the local algebra $A$ with Hilbert
function $H_{A}=H$.

######  Theorem 1.1.

Let $\Bbbk$ be a field of characteristic zero and let $H=(1,4,10,s)$ for
$s\in\\{6,7,8,9\\}$. Then
$\mathcal{L}_{H}\subseteq\operatorname{Hilb}_{15+s}(\mathbb{A}^{4})$ is an
irreducible component, and this component, with the scheme structure inherited
from the Hilbert scheme, is generically nonreduced. Therefore, the Hilbert
scheme $\operatorname{Hilb}_{d}(\mathbb{A}^{4})$ admits generically nonreduced
components for all $d\geq 21$.

Prior to Theorem 1.1 it was not known whether
$\operatorname{Hilb}_{d}(\mathbb{A}^{4})$ or
$\operatorname{Hilb}_{d}(\mathbb{A}^{5})$ are _reduced_ for all $d$. It
remains an open question whether $\operatorname{Hilb}_{d}(\mathbb{A}^{3})$ is
nonreduced for $d$ high enough and whether this scheme has generically
nonreduced components, see [Jel23, Problem XIV].

There are three main steps of the argument. First, the locus $\mathcal{L}_{H}$
is closed for the functions $H$ as in theorem. Moreover, it is contained in a
dominant Białynicki-Birula cell, which implies that on an open subset
$U\subseteq\operatorname{Hilb}_{15+s}(\mathbb{A}^{4})$ containing
$\mathcal{L}_{H}$, the Hilbert scheme admits a retraction $\pi\colon U\to
U^{\mathbb{G}_{m}}$ which maps any point $[\operatorname{Spec}(S/I)]\in U$ to
$\operatorname{Spec}(S/\operatorname{in}(I))$, where
$S=\Bbbk[x_{1},\ldots,x_{4}]$ and $\operatorname{in}(I)$ is the ideal of top
degree forms.

Second, primary obstruction yields quadratic equations for the fibre

$\pi^{-1}([Z])\subseteq\left(T_{\operatorname{Hilb}_{15+s}(\mathbb{A}^{4}),[Z]}\right)_{<0}.$

Third, for a chosen $[Z_{0}]\in\mathcal{L}_{H}$ we computer-check using
_Macaulay2_ that the quadrics alone cut out a $4$-dimensional scheme in the
affine space
$\left(T_{\operatorname{Hilb}_{15+s}(\mathbb{A}^{4}),[Z_{0}]}\right)_{<0}$. It
follows that $\dim\pi^{-1}([Z_{0}])\leq 4$. The fibre $\pi^{-1}([Z_{0}])$ is a
cone and has a translation action by $\mathbb{A}^{4}$, so the fibre is equal
to $\\{Z_{0}+v\ |\ v\in\mathbb{A}^{4}\\}$ as a set and hence $\mathcal{L}_{H}$
contains an open neighbourhood of $[Z_{0}]$, so this locus is a component. A
syzygetic argument shows that the containment
$T_{\mathcal{L}_{H},[Z]}\subseteq
T_{\operatorname{Hilb}_{15+s}(\mathbb{A}^{4}),[Z]}$ is strict for every
$[Z]\in\mathcal{L}_{H}$, hence $\mathcal{L}_{H}$ cannot be generically
reduced.

###  1.2. Open questions and possible generalizations

Consider now $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ and the unique very
compressed Hilbert function $H=H_{n,d}$ given by the condition that there
exists a $\delta$ such that

$H_{n,d}(i)=\begin{cases}\binom{n+i-1}{i}=\dim\Bbbk[x_{1},\ldots,x_{n}]_{i}&\mbox{
for }i<\delta\\\ 0&\mbox{ for }i>\delta\\\
d-\sum_{i=0}^{\delta-1}\binom{n+i-1}{i}&\mbox{ for }i=\delta\end{cases}$

The locus $\mathcal{L}_{H}\subseteq\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ is
irreducible and closed also in this more general case. We then have three
possibilities for a general $[Z]\in\mathcal{L}_{H}$:

  1. SMOOTH

the scheme $[Z]$ has only trivial negative tangents, so $\mathcal{L}_{H}$ is a
component and $[Z]$ is a smooth point of
$\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ on this component,

  2. DEFORMS

the scheme $[Z]$ has nontrivial negative tangents and some of them
“integrate”, that is, the fibre $\pi^{-1}([Z])$, which is a cone, contains
more points than just $\mathbb{A}^{n}$. In this case $\mathcal{L}_{H}$ is not
an irreducible component and without additional information we cannot say much
about whether its points are reduced in
$\operatorname{Hilb}_{d}(\mathbb{A}^{n})$.

  3. NONRED

the scheme $[Z]$ has nontrivial negative tangents and $\pi^{-1}([Z])$ is, as a
topological space, equal to $\mathbb{A}^{n}$. In this case $\mathcal{L}_{H}$
is a generically nonreduced component.

######  Example 1.2.

By [CEVV09] the case SMOOTH occurs for example for $H=(1,4,3)$. The case
DEFORMS occurs for example for $H=(1,4,4)$. The case NONRED occurs for $H$ as
in Theorem 1.1.

We stress that above we look at a general point of $\mathcal{L}_{H}$. This
makes a difference: for example for $H=(1,6,6)$ the case SMOOTH occurs, so
$\mathcal{L}_{H}$ is a generically smooth component, however Szachniewicz
[Sza21] found an embedded component of
$\operatorname{Hilb}_{13}(\mathbb{A}^{6})$ inside $\mathcal{L}_{H}$. It is a
completely open problem to understand whether having an embedded component is
typical or exceptional for $\mathcal{L}_{H}$ which fall into the SMOOTH case.

One motivation to discuss the more general situation is the case $n=3$, the
Hilbert scheme of $\mathbb{A}^{3}$. Taking $d=96$ and $H=H_{3,96}$ we get that
$\mathcal{L}_{H}$ is too big to fit in the smoothable component of
$\operatorname{Hilb}_{96}(\mathbb{A}^{3})$, see [Iar72]. A syzygetic argument,
see Lemma 3.2 below, also shows that the case SMOOTH cannot hold. Moreover, it
is known that the fibre $\pi^{-1}([Z])$ for a general $[Z]$ is cut out by
quadrics only. Actually, this holds whenever $(T^{2}_{[Z]})_{<-2}=0$, where
$T^{2}_{[Z]}\subseteq\operatorname{Ext}^{1}(I_{Z},\mathcal{O}_{Z})$ is the
Schessinger’s functor, see [Har10, Chapter 3]. It is possible to obtain the
quadrics explicitly using _Macaulay2_. However, the Gröbner basis computation
necessary for determining $\dim\pi^{-1}([Z])$ is out of reach, at least using
standard algorithms. We warn the reader that it is not clear, even
intuitively, whether we should expect NONRED or DEFORMS in this case, since it
may be that $\mathcal{L}_{H}$ lies in the closure of a compressed (not very
compressed) component similar to the ones discussed in [Iar84].

The question about $H=(1,4,10,s)$ in [Ame10] is also formulated for $s=10$. In
this case one could try the approach above, however there are $50$ negative
tangents (see Lemma 3.2 below) and the approach is infeasible on our hardware.
Of course, perhaps this is only a question of computational cost, however we
prefer to leave the case $s=10$ open, in the hope that it will stimulate
further progress on understanding the Yoneda multiplication in
$\operatorname{Ext}^{\bullet}(\mathcal{O}_{Z},\mathcal{O}_{Z})$ and in
particular the primary obstruction.

##  2\. Acknowledgements

We would like to thanks the Editors, in particular Ada Boralevi, for making
this volume possible. We also thank the organizers of the GC workshop: Ada,
Enrico, Paolo, and Roberto for this serene and productive time. The comments
of the anonymous referee were helpful as well.

##  3\. Preliminaries

We work over a field $\Bbbk$ of characteristic zero. The characteristic
assumption will be used mostly for justifying the computations (we believe
that the result holds for most characteristics). Let
$S=\Bbbk[x_{1},\ldots,x_{n}]$ be a polynomial ring and
$\mathbb{A}^{n}=\operatorname{Spec}(S)$. For a subscheme
$Z\subseteq\mathbb{A}^{n}$ we denote by $I_{Z}$ its ideal and by
$\mathcal{O}_{Z}=S/I_{Z}$ its coordinate ring.

######  Proposition 3.1.

The tangent space to $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ is given
by $\operatorname{Hom}_{S}(I_{Z},\mathcal{O}_{Z})$. This space is canonically
isomorphic to $\operatorname{Ext}^{1}_{S}(\mathcal{O}_{Z},\mathcal{O}_{Z})$.

###### Proof.

A self-contained proof for $\operatorname{Hom}_{S}(I_{Z},\mathcal{O}_{Z})$ can
be found in [Str96]; also the Ext functor naturally appears there. The
isomorphism

$\operatorname{Hom}_{S}(I_{Z},\mathcal{O}_{Z})\to\operatorname{Ext}^{1}_{S}(\mathcal{O}_{Z},\mathcal{O}_{Z})$

follows from the long exact sequence obtained by applying
$\operatorname{Hom}_{S}(-,\mathcal{O}_{Z})$ to $0\to I_{Z}\to
S\to\mathcal{O}_{Z}\to 0$. ∎

Further in the paper, when discussing $\operatorname{Hom}_{S}(-,-)$ and
$\operatorname{Ext}_{S}^{\bullet}(-,-)$, we drop the subscript $S$ from the
notation.

If $I_{Z}$ is presented as

$S^{\oplus d_{2}}\to S^{\oplus d_{1}}\to I_{Z}\to 0,$

then $\operatorname{Hom}(I_{Z},\mathcal{O}_{Z})$ is the kernel of the natural
map $\operatorname{Hom}(S^{\oplus
d_{1}},\mathcal{O}_{Z})\to\operatorname{Hom}(S^{\oplus
d_{2}},\mathcal{O}_{Z})$. Computing this kernel is best performed with a
computer.

We propose one example, which is straightforward, but it will be important in
the following. Recall that when $I_{Z}$ is graded, also the tangent space
$\operatorname{Hom}_{S}(I_{Z},\mathcal{O}_{Z})$ is graded with

$\operatorname{Hom}_{S}(I_{Z},\mathcal{O}_{Z})_{i}=\left\\{\varphi\colon
I_{Z}\to\mathcal{O}_{Z}\ |\
\varphi((I_{Z})_{j})\subseteq(\mathcal{O}_{Z})_{i+j}\mbox{ for all
}j\right\\}.$

######  Lemma 3.2.

Suppose that $[Z]\in\operatorname{Hilb}_{15+s}(\mathbb{A}^{4})$ is given by a
_homogeneous_ ideal $I_{Z}$ and that $\mathcal{O}_{Z}$ is very compressed with
Hilbert function $(1,4,10,s)$. Suppose further that $I_{Z}$ is generated by
cubics. Then

$\dim\operatorname{Hom}(I_{Z},\mathcal{O}_{Z})_{0}=(20-s)s\quad\mbox{and}\quad\dim\operatorname{Hom}(I_{Z},\mathcal{O}_{Z})_{-1}\geq
4s^{2}-55s+200.$

###### Proof.

By assumption, the presentation of $I_{Z}$ is

$S(-5)^{\beta}\oplus S(-4)^{4\cdot(20-s)-35}\to S(-3)^{20-s}\to I_{Z}\to 0.$

Let us first look at degree zero. If we consider the full linear space
$\operatorname{Hom}_{\Bbbk}(I_{Z},\mathcal{O}_{Z})_{0}$, then any relation
between generators of $I_{Z}$ is mapped to $(\mathcal{O}_{Z})_{\geq 4}=0$, so
$\operatorname{Hom}_{\Bbbk}(I_{Z},\mathcal{O}_{Z})_{0}=\operatorname{Hom}(I_{Z},\mathcal{O}_{Z})_{0}$.

Let us now look at degree one. By similar considerations, for every linear map
$\varphi\in\operatorname{Hom}_{\Bbbk}(I_{Z},\mathcal{O}_{Z})_{-1}$, the image
of $S(-5)^{\beta}$ is zero and the image of $S(-4)^{4\cdot(20-s)-35}$ is
contained in the $s$-dimensional space $(\mathcal{O}_{Z})_{3}$. Thus, the
relations in the presentation yield at most
$s\cdot\left(4\cdot(20-s)-35\right)$ linear-algebraic conditions on the images
of minimal homogeneous generators and so

$\dim\operatorname{Hom}(I_{Z},\mathcal{O}_{Z})_{-1}\geq
10\cdot(20-s)-s\cdot\left(4\cdot(20-s)-35\right)=4s^{2}-55s+200,$

as claimed. ∎

######  Proposition 3.3 (Very compressed loci).

Let $H$ be any very compressed Hilbert function and $\delta$ be the largest
index such that $H(\delta)\neq 0$. Then the very compressed locus
$\mathcal{L}_{H}$ is closed in $\operatorname{Hilb}_{d}(\mathbb{A}^{n})$,
isomorphic to
$\mathbb{A}^{n}\times\operatorname{Gr}(H(\delta),\binom{n-1+\delta}{\delta})$
and has dimension

$n+\binom{n-1+\delta}{\delta}-H(\delta).$

###### Proof.

A point $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ lies in
$\mathcal{L}_{H}$ if and only if, first, the support of $[Z]$ is a single
point $z$ and, second, the ideal $I_{Z}$ is contained in
$\mathfrak{m}_{z}^{\delta}$. The first condition is closed and the second is
closed provided that the first one is satisfied. The description of
$\mathcal{L}_{H}$ as a product is immediate, see [Sza21, Proposition 2.27]. ∎

###  3.1. Białynicki-Birula decompositions

The general theory of Białynicki-Birula decompositions is beautiful but quite
complicated, see [JS19, JS21, Tha96]. We would like to apply it to the
standard scalar torus action on the Hilbert scheme. We will see below that in
this special case things simplify considerably. Therefore, we gather below
only the necessary facts and restrict to the affine case and to the positive
Białynicki-Birula decomposition, that is, when considering the limit at $t\to
0$.111Be aware that in some articles by the author, notably [Jel19], the sign
$X^{+}$ denotes the _negative_ Białynicki-Birula decomposition, that is, the
one coming from considering $\lim_{t\to\infty}$.

The following allows us to reduce to considering the affine case.

######  Proposition 3.4 ([Sum74], [JS19, Proposition 5.3(2)]).

Suppose that $X$ is a quasi-projective scheme. Then there is a open cover
$\\{U_{i}\\}$ by affine $\mathbb{G}_{m}$-stable schemes. Moreover, for every
such cover the Białynicki-Birula decomposition $X^{+}$ of $X$ is covered by
the Białynicki-Birula decompositions $U_{i}^{+}$ of $U_{i}$.

Next, a $\mathbb{G}_{m}$-action on an affine scheme $\operatorname{Spec}(A)$
is the same as a $\mathbb{Z}$-grading on the algebra $A$. In this case, the
Białynicki-Birula decomposition can be characterised explicitly as follows.

######  Proposition 3.5.

Let $X=\operatorname{Spec}(A)$ be an affine scheme with an action of
$\mathbb{G}_{m}$. Then, the positive Białynicki-Birula decomposition $X^{+}$
of $X$ is a closed subscheme $X^{+}$ given by the ideal generated by $A_{<0}$.
The fixed locus of $X$ is given by the ideal generated by
$\\{A_{<0}\\}\cup\\{A_{>0}\\}$. The composition

$\frac{A}{A_{<0}\cdot A+A_{>0}\cdot A}\simeq\frac{A_{0}}{(A_{<0}\cdot
A)_{0}}\hookrightarrow\frac{A_{\geq 0}}{(A_{<0}\cdot A)_{\geq
0}}\simeq\frac{A}{A_{<0}\cdot A}$

gives a morphism $\pi\colon X^{+}\to X^{\mathbb{G}_{m}}$. The canonical closed
embedding $s\colon X^{\mathbb{G}_{m}}\to X^{+}$ is a section of $\pi$. We
obtain the following diagram, where $\pi$ and $s$ are closed embeddings

${X^{+}}$${X}$${X^{\mathbb{G}_{m}}}$$\scriptstyle{\theta}$$\scriptstyle{\pi}$$\scriptstyle{s}$

Thus, for every $x\in X^{\mathbb{G}_{m}}$, the fibre $\pi^{-1}(x)$ is given by
spectrum of an $\mathbb{N}$-graded algebra $B\simeq\frac{A}{A_{<0}\cdot
A+(\mathfrak{m}_{x})_{0}\cdot A}$, which satisfies $B_{0}=\Bbbk$.

###### Proof.

See for example [JS19, Proposition 4.5, Example 4.6]. ∎

For a homogeneous maximal ideal $\mathfrak{m}$ in a $\mathbb{Z}$-graded ring
$A$, the cotangent space at $[\mathfrak{m}]\in\operatorname{Spec}(A)$ is the
subquotient $\mathfrak{m}/\mathfrak{m}^{2}$, so is also naturally graded. The
tangent space at $[\mathfrak{m}]$ is also graded, the weights are opposite.

######  Example 3.6.

In the setup of Proposition 3.5, take $x\in X^{\mathbb{G}_{m}}(\Bbbk)$. Then
the cotangent space at $x\in X^{+}$ is the non-negative part of the cotangent
space of $x\in X$. Dualising, we obtain that

$d\theta\colon T_{X^{+},x}\to T_{X,x}$

identifies $T_{X^{+},x}$ with $(T_{X,x})_{\leq 0}$, the non-positive part of
$T_{X,x}$.

The weights of the tangent space are crucial for comparing $X^{+}$ and $X$, as
the following proposition says.

######  Proposition 3.7 ([JS19, Proposition 1.6]).

Let $X$ be a separated scheme locally of finite type (for example, this holds
if $X$ is quasi-projective). Assume that $x\in X^{\mathbb{G}_{m}}(\Bbbk)$ is
such that $d\theta_{x}$ is surjective (that is, an isomorphism). Then up to
restricting to a $\mathbb{G}_{m}$-stable affine neighbourhood of $x$ we can
assume that $\theta$ is an isomorphism.

###  3.2. Białynicki-Birula decomposition of the Hilbert scheme of points

Let $\mathbb{G}_{m}=\operatorname{Spec}(\Bbbk[t^{\pm 1}])$ be a one-
dimensional torus and consider its action

$\mathbb{G}_{m}\times\mathbb{A}^{n}\to\mathbb{A}^{n}$

by rescaling: $\lambda\cdot(x_{1},\ldots,x_{n})=(\lambda x_{1},\ldots,\lambda
x_{n})$ for every $\Bbbk$-point $(x_{1},\ldots,x_{n})\in\mathbb{A}^{n}(\Bbbk)$
and $\lambda\in\Bbbk^{\times}=\mathbb{G}_{m}(\Bbbk)$. For every closed
subscheme $Z\subseteq\mathbb{A}^{n}$ and $\lambda\in\mathbb{G}_{m}(\Bbbk)$ we
obtain a new closed subscheme $\lambda\cdot Z$ given by the closed embedding

(3.1)
${Z}$${\mathbb{A}^{n}}$${\mathbb{A}^{n}.}$$\scriptstyle{\lambda\cdot}$$\scriptstyle{\simeq}$

When we view a point as a closed subscheme, both definitions agree. A
subscheme $Z$ is a $\mathbb{G}_{m}$-fixed point if and only if its ideal
$I_{Z}$ is homogeneous.

Construction (3.1) generalizes readily to the case when $Z$ is closed in
$\mathbb{A}^{n}\times S$, for any scheme $S$. This yields an action
$\mathbb{G}_{m}\times\operatorname{Hilb}_{d}(\mathbb{A}^{n})\to\operatorname{Hilb}_{d}(\mathbb{A}^{n})$,
which on $\Bbbk$-points agrees with (3.1).

For any set $A$ of $d$ monomials in $S$, consider the locus
$U_{A}\subseteq\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ which consists of
$[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ such that $A$ spans
$\mathcal{O}_{Z}$. These loci are open and $\mathbb{G}_{m}$-stable, hence the
corresponding Białynicki-Birula cells $U_{A}^{+}$ cover
$(\operatorname{Hilb}_{d}(\mathbb{A}^{n}))^{+}$, see Proposition 3.4. The loci
above are important for the computational aspects, see for example [LR16].

We would like now to understand when Proposition 3.7 can be applied in the
case of Hilbert schemes, so we are interested in the weights on the tangent
space.

######  Lemma 3.8.

Let $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ be a
$\mathbb{G}_{m}$-fixed point. Then
$(T_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]})_{>0}$ vanishes if and only
if $\mathcal{O}_{Z}$ is very compressed.

###### Proof.

Take $S=\Bbbk[x_{1},\ldots,x_{n}]$. Suppose first that $\mathcal{O}_{Z}$ is
very compressed. Then there exists an $s$ such that $I_{Z}\subseteq S_{\geq
s}$ and $(\mathcal{O}_{Z})_{>s}=0$. A tangent at $Z$ of strictly positive
degree $i$ corresponds to a homomorphism $\varphi\colon
I_{Z}\to\mathcal{O}_{Z}$ such that
$\varphi((I_{Z})_{j})\subseteq(\mathcal{O}_{Z})_{j+i}$. The source is nonzero
only for $j\geq s$, but for such a $j$ we have $i+j>s$, so the target is zero.
It follows that $\varphi=0$.

Suppose now that $\mathcal{O}_{Z}$ is not very compressed. This implies that
there exists an $s$ such that $I_{s}\neq 0$ and $(\mathcal{O}_{Z})_{\geq
s+1}\neq 0$. Pick a set of minimal generators of $I_{Z}$ and let $g$ be an
element of lowest degree. Pick a socle element $h\in\mathcal{O}_{Z}$ of
highest degree. Then $\deg(h)\geq s+1>s\geq\deg(g)$. There exists a
homomorphism $\varphi\colon I_{Z}\to\mathcal{O}_{Z}$ which satisfies
$\varphi(g)=h$ and sends all other minimal generators to zero. It follows that
$\varphi$ is homogeneous of strictly positive degree, equal to
$\deg(h)-\deg(g)$. ∎

###  3.3. Primary obstructions

Primary obstructions govern the order two part of deformation theory and can
be computed explicitly. We discuss them below.

Consider two tangent vectors at a point
$[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$. They yield maps

$\varphi_{i}\colon\operatorname{Spec}\left(\frac{\Bbbk[\varepsilon_{i}]}{(\varepsilon_{i}^{2})}\right)\to\operatorname{Hilb}_{d}(\mathbb{A}^{n}),$

for $i=1,2$ and two elements

$\varphi_{1},\varphi_{2}\in
T_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]}\simeq\operatorname{Ext}^{1}\left(\mathcal{O}_{Z},\mathcal{O}_{Z}\right).$

The two tangent vectors span an at most $2$-dimensional space and the
corresponding morphism is

$\varphi_{12}\colon\operatorname{Spec}\left(\frac{\Bbbk[\varepsilon_{1},\varepsilon_{2}]}{(\varepsilon_{1},\varepsilon_{2})^{2}}\right)\to\operatorname{Hilb}_{d}(\mathbb{A}^{n}),$

which restricts to $\varphi_{1}$, $\varphi_{2}$ in the natural way. We may ask
when $\varphi_{12}$ does extend to a map $\widetilde{\varphi_{12}}$ from
$\operatorname{Spec}\left(\Bbbk[\varepsilon_{1},\varepsilon_{2}]/(\varepsilon_{1}^{2},\varepsilon_{2}^{2})\right)$.
Deformation theory [FGI+05, Chapter 5] implies that an extension exists if and
only if an obstruction

$ob_{\varphi_{12},\widetilde{\varphi}_{12}}\in\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})$

vanishes. The key observation is that we can describe the obstruction
explicitly.

######  Theorem 3.9 ([JŠ22, Theorem 4.18]).

The obstruction $ob_{\varphi_{12},\widetilde{\varphi}_{12}}$ is equal to

$\varphi_{1}\circ\varphi_{2}+\varphi_{2}\circ\varphi_{1},$

where $\circ$ denotes Yoneda’s multiplication in
$\operatorname{Ext}^{\bullet}(\mathcal{O}_{Z},\mathcal{O}_{Z})$ applied to
$\varphi_{1},\varphi_{2}\in\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})$.

Let
$\mu\colon\operatorname{Sym}^{2}\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})\to\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})$
be given by

$\mu(\varphi_{1}\cdot\varphi_{2}):=\varphi_{1}\circ\varphi_{2}+\varphi_{2}\circ\varphi_{1}.$

Consider its transpose

$\mu^{\vee}\colon\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\to\left(\operatorname{Sym}^{2}\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})\right)^{\vee}\simeq\operatorname{Sym}^{2}\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\right).$

As explained in [JŠ22, §4.2], Theorem 3.9 yields the following corollary.
Recall that $\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}$
is the cotangent space at $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$.

######  Corollary 3.10.

Consider the complete local ring
$(\hat{\mathcal{O}}_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]},\mathfrak{m}_{[Z]})$.
Its truncation to second order satisfies

$\frac{\hat{\mathcal{O}}_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]}}{\mathfrak{m}_{[Z]}^{3}}\simeq\frac{\operatorname{Sym}^{\bullet}\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}}{\operatorname{im}\left(\mu^{\vee}\colon\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\to\operatorname{Sym}^{2}\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\right)+\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\right)^{3}}.$

###  3.4. Primary obstructions and Białynicki-Birula decompositions

As written, Corollary 3.10 does not directly involve the dimension of the
local ring. Moreover, we would like to apply it for the fibre of the
Białynicki-Birula decomposition. Both subtleties “cancel out”: restriction to
the fibre gives us an $\mathbb{N}$-grading which allows to pass from the third
neighbourhood to the full complete local ring.

######  Proposition 3.11.

Let $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ be a
$\mathbb{G}_{m}$-fixed $\Bbbk$-point and consider its Białynicki-Birula fibre
$\operatorname{Spec}(A)$. Assume that the subspace

$\left(T_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]}\right)_{\leq-2}$

is zero. Then there is a surjection of graded algebras

$\frac{\operatorname{Sym}^{\bullet}\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}}{\operatorname{im}(\mu^{\vee}\colon\left(\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}\to\operatorname{Sym}^{2}\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee})}\twoheadrightarrow
A.$

###### Proof.

By Proposition 3.4 we already know that there is an open
$\mathbb{G}_{m}$-stable neighbourhood $[Z]\in
U\subseteq\operatorname{Hilb}_{d}(\mathbb{A}^{n})$ such that the Białynicki-
Birula fibre $\pi^{-1}([Z])$ is contained in $U^{+}$. Using Proposition 3.5 we
conclude that $\pi^{-1}([Z])$ is a spectrum of an $\mathbb{N}$-graded algebra
$B$ with $B_{0}=\Bbbk$.

We now employ Corollary 3.10. To make the notation lighter, we put
$E:=\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})$. The complete
local ring $\hat{B}$ of $[Z]\in\operatorname{Spec}(B)$ is a quotient of the
complete local ring of $[Z]\in\operatorname{Hilb}_{d}(\mathbb{A}^{n})$. Hence,
also the truncation $\hat{B}/\mathfrak{m}^{3}$ is a quotient of the truncation
of

$\frac{\hat{\mathcal{O}}_{\operatorname{Hilb}_{d}(\mathbb{A}^{n}),[Z]}}{\mathfrak{m}^{3}}\simeq\frac{\operatorname{Sym}^{\bullet}E^{\vee}}{\operatorname{im}\left(\mu^{\vee}\colon\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})^{\vee}\to\operatorname{Sym}^{2}E^{\vee}\right)+\left(E^{\vee}\right)^{3}}.$

The cotangent space of $[Z]\in\operatorname{Spec}(B)$ has no nonpositive
weights, so $\hat{B}/\mathfrak{m}^{3}$ is in fact a quotient of

$\frac{\operatorname{Sym}^{\bullet}\left(E_{<0}\right)^{\vee}}{\operatorname{im}\left(\mu^{\vee}\colon\left(\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}\to\operatorname{Sym}^{2}\left(E_{<0}\right)^{\vee}\right)+\left(\left(E_{<0}\right)^{\vee}\right)^{3}}$

We now lift this from infinitesimal second order to a more global situation
using the $\mathbb{N}$-grading. Consider the map of graded rings

$p\colon\operatorname{Sym}^{\bullet}E_{<0}\to B.$

By Example 3.6, this map is an isomorphism on cotangent spaces. Since $B$ is
$\mathbb{N}$-graded with $B_{0}=\Bbbk$, the map $p$ is a surjection, by
induction on the degree. Moreover, again thanks to the grading and to the fact
that all $E_{<0}=E_{-1}$, we obtain an isomorphism
$\hat{B}/\mathfrak{m}^{3}\simeq B/B_{\geq 3}$.

We summarize the situation on a commutative diagram

${\operatorname{Sym}^{\bullet}E_{<0}}$${B}$${\frac{\operatorname{Sym}^{\bullet}\left(E_{<0}\right)^{\vee}}{\operatorname{im}\left(\mu^{\vee}\colon\left(\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}\to\operatorname{Sym}^{2}\left(E_{<0}\right)^{\vee}\right)+\left(\left(E_{<0}\right)^{\vee}\right)^{3}}}$${\frac{\hat{B}}{\mathfrak{m}^{3}}}$${\frac{B}{B_{\geq
3}}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

This shows that the image $p(\operatorname{im}\mu^{\vee})$ in $B_{2}$ is zero.
The claim follows. ∎

###  3.5. Computational input

As mentioned in the introduction, currently there is not enough knowledge
about the Ext algebra to perform a conceptual analysis of the primary
obstruction. In this section we include a somewhat brute-force check of
specific examples.

The package _MatricesAndQuot_ is available as an auxiliary file for the arXiv
version of [JŠ22]. Needless to say, many alternatives exist, in particular the
computation can be performed using Ilten’s _VersalDeformations_ package
[Ilt12] or Lella’s _HilbertAndQuotSchemesOfPoints.m2_ package [Lel22] or the
framework [BCR22].

######  Proposition 3.12 (Key computational output).

Let $H=(1,4,10,s)$ For every $s=6,7,8,9$ there is an example of
$[Z]\in\mathcal{L}_{H}\subseteq\operatorname{Hilb}_{15+s}(\mathbb{A}^{4})$
such that the algebra

$\frac{\operatorname{Sym}^{\bullet}\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}}{\operatorname{im}(\mu^{\vee}\colon\left(\operatorname{Ext}^{2}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee}\to\operatorname{Sym}^{2}\left(\operatorname{Ext}^{1}(\mathcal{O}_{Z},\mathcal{O}_{Z})_{<0}\right)^{\vee})}$

is $4$-dimensional.

###### Proof.

This is an explicit _Macaulay2_ computation. See code below. We work over
characteristic $17$ for efficiency. By semicontinuity the same result (for the
ideals given by obvious lifts of generators) is true in characteristic zero.
See [KK22, (10.12)-(10.13)] for a detailed discussion of this method.

    S = (ZZ/17)[x_1 .. x_4];
    loadPackage("MatricesAndQuot", Reload=>true);
    I9 = ideal(x_2*x_3^2, x_2^2*x_3+x_1*x_4^2+x_4^3,
    x_1^2*x_2+x_1*x_3^2+x_2*x_4^2,
    x_1^3+x_2^2*x_4+x_2*x_4^2, x_3*x_4^2, x_1^2*x_4,
    x_1*x_2*x_4+x_3^2*x_4+x_1*x_4^2, x_1*x_2*x_3+x_3^3+x_1^2*x_4,
    x_2^3+x_2*x_3*x_4+x_4^3, x_1^3+x_2^3+x_3^2*x_4,
    x_1^2*x_3+x_1*x_2*x_4+x_2*x_4^2);
    assert(degree I9 == 24); -- case (1,4,10,9)
    assert(dim primaryObstruction(S^1/I9) == 4);
    I8 = I9 + ideal(x_1*x_2^2);
    assert(degree I8 == 23); -- case (1,4,10,8)
    assert(dim primaryObstruction(S^1/I8) == 4);
    I7 = I8 + ideal(x_1*x_3^2);
    assert(degree I7 == 22); -- case (1,4,10,7)
    assert(dim primaryObstruction(S^1/I7) == 4);
    I6 = I7 + ideal(x_1^2*x_3);
    assert(degree I6 == 21); -- case (1,4,10,6)
    assert(dim primaryObstruction(S^1/I6) == 4);

The total computation time is a few minutes, the case I9 takes most of it. ∎

###  3.6. Proof of Theorem 1.1

We proceed to the proof of our main theorem. Recall that we consider Hilbert
functions $H=(1,4,10,s)$ for $s\in\\{6,7,8,9\\}$.

###### Proof of Theorem 1.1.

We follow the strategy outlined in the introduction. Fix an $s$, let
$d=1+4+10+s$, and pick a point $[Z]\in\mathcal{L}_{H}$ as in Proposition 3.12.
The fibre $\pi^{-1}([Z])$ is connected as it is a cone. The group scheme
$(\mathbb{A}^{4},+)$ acts on the fibre $\pi^{-1}([Z])$ by translations. From
this and from $\dim\pi^{-1}([Z])=4$ it follows that the fibre is, as a set,
equal to the $(\mathbb{A}^{4},+)$-orbit of the cone point $[Z]$. By
semicontinuity of fibre dimensions, the same holds for fibres near $[Z]$. It
follows that, as a set, $\mathcal{L}_{H}$ contains an open neighbourhood of
$[Z]$.

From Proposition 3.3 and Lemma 3.2 it follows that for every point
$[Z^{\prime}]\in\mathcal{L}_{H}$, the tangent spaces to
$\operatorname{Hilb}_{d}(\mathbb{A}^{4})$ and $\mathcal{L}_{H}$ differ already
in degree $-1$. This can happen only if the component of the Hilbert scheme
that topologically is equal to $\mathcal{L}_{H}$ has no smooth points, so it
is generically nonreduced.

To obtain generically nonreduced components of
$\operatorname{Hilb}_{d}(\mathbb{A}^{4})$ for $d\geq 21$ consider a scheme $Z$
as above for $d=21$ and enlarge it to a scheme

$Z\sqcup\bigsqcup_{d-21}\operatorname{Spec}(\Bbbk)$

embedded (in any way) into $\mathbb{A}^{4}$. ∎

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