# On Releasing Annotator-Level Labels and Information in Datasets

Vinodkumar Prabhakaran
Google Research
<EMAIL_ADDRESS>
&Aida Mostafazadeh Davani11footnotemark: 1
University of Southern California
<EMAIL_ADDRESS>
Mark Díaz
Google Research
<EMAIL_ADDRESS>
Authors contributed equally Work done while at Google Research

###### Abstract

A common practice in building NLP datasets, especially using crowd-sourced
annotations, involves obtaining multiple annotator judgements on the same data
instances, which are then flattened to produce a single “ground truth” label
or score, through majority voting, averaging, or adjudication. While these
approaches may be appropriate in certain annotation tasks, such aggregations
overlook the socially constructed nature of human perceptions that annotations
for relatively more subjective tasks are meant to capture. In particular,
systematic disagreements between annotators owing to their socio-cultural
backgrounds and/or lived experiences are often obfuscated through such
aggregations. In this paper, we empirically demonstrate that label aggregation
may introduce representational biases of individual and group perspectives.
Based on this finding, we propose a set of recommendations for increased
utility and transparency of datasets for downstream use cases.

##  1 Introduction

Obtaining multiple annotator judgements on the same data instances is a common
practice in NLP in order to improve the quality of final labels Snow et al.
(2008); Nowak and Rüger (2010). Cases of disagreement between annotations are
often resolved through majority voting, averaging, or adjudication in order to
derive a single “ground truth”, often with the aim of training supervised
machine learning models. However, in relatively subjective tasks such as
sentiment analysis or offensiveness detection, there often exists no single
“right” answer (Alm, 2011). Enforcing such a single ground truth in such tasks
will sacrifice valuable nuances about the task that are embedded in
annotators’ assessments of the stimuli, especially their disagreements Aroyo
and Welty (2013).

Annotators’ socio-demographic factors, moral values, and lived experiences
often influence their interpretations of language, especially in subjective
tasks such as identifying political stances (Luo et al., 2020), sentiment Díaz
et al. (2018), and online abuse Waseem (2016); Patton et al. (2019). For
instance, feminist and anti-racist activists systematically disagree with
crowd workers in their hate speech annotations Waseem (2016). Similarly,
annotators’ political affiliation is shown to correlate with how they annotate
the neutrality of political stances (Luo et al., 2020). A potential adverse
effect of majority voting in such cases is that it may sideline minority
perspectives in data.

In this paper, we analyze annotated data for eight different tasks across
three different datasets to study the impact of majority voting as an
aggregation approach. We answer two questions:

  * •

Q1: Does aggregated data uniformly capture all annotators’ perspectives, when
available?

  * •

Q2: Does aggregated data reflect certain socio-demographic groups’
perspectives more so than others?

Our analysis demonstrates that in the annotations for many tasks, the
aggregated majority vote does not uniformly reflect the perspectives of all
annotators in the annotator pool. For many tasks in our analysis, a
significant proportion of the annotators had very low agreement scores (0 to
0.4) with the majority vote label. While certain individual annotator’s labels
may have low agreement with the majority label due to valid/expected reasons
(e.g., if they produced noisy labels), we further show that these agreement
scores may vary significantly across different socio-demographic groups that
annotators identify with. This finding has important fairness implications, as
it demonstrates how the aggregation step can sometimes cause the final dataset
to under-represent certain groups’ perspectives.

Meaningfully addressing such issues in multiply-annotated datasets requires
understanding and accounting for systematic disagreements between annotators.
However, most annotated datasets often only release the aggregated labels,
without any annotator-level information. We argue that dataset developers
should consider including annotator-level labels as well as annotators’ socio-
demographic information (when viable to do so responsibly) when releasing
datasets, especially those capturing relatively subjective tasks. Inclusion of
this information will enable more research on how to account for systematic
disagreements between annotators in training tasks.

##  2 Background

NLP has a long history of developing techniques to interpret subjective
language Wiebe et al. (2004); Alm (2011). While all human judgments embed some
degree of subjectivity, some tasks such as sentiment analysis Liu et al.
(2010), affect modeling Alm (2008); Liu et al. (2003), emotion detection
Hirschberg et al. (2003), and hate speech detection Warner and Hirschberg
(2012) are agreed upon as relatively more subjective in nature. As Alm (2011)
points out, achieving a single real ‘ground truth’ is not possible, nor
essential in case of such subjective tasks. Instead, we should investigate how
to model the subjective interpretations of the annotators, and how to account
for them in application scenarios.

However, the current practice in the NLP community continues to be applying
different aggregation strategies to arrive at a single score or label that
makes it amenable to train and evaluate supervised machine learning models.
Oftentimes, datasets are released with only the final scores/labels,
essentially obfuscating important nuances in the task. The information
released about the annotations can be at one of the following four levels of
information-richness.

Firstly, the most common approach is one in which multiple annotations
obtained for a data instance are aggregated to derive a single “ground truth”
label, and these labels are the only annotations included in the released
dataset (e.g., Founta et al. (2018)). The aggregation strategy most commonly
used, especially in large datasets, is majority voting, although smaller
datasets sometimes use adjudication by an ‘expert’ (often one of the study
authors themselves) to arrive at a single label (e.g., in Waseem and Hovy
(2016)) when there are substantial disagreements between annotators. These
aggregation approaches rely on the assumption that there always exist a single
correct label, and that either the majority label or the ‘expert’ label is
more likely to be that correct label. What it fails to account for is the fact
that in many subjective tasks, e.g., detecting hate speech, the perceptions of
individual annotators may be as valuable as an ‘expert’ perspective.

Secondly, some datasets (e.g., Jigsaw (2018); Davidson et al. (2017)) release
the distribution across labels rather than a single aggregated label. In
binary classification tasks, this corresponds to the percentage of annotators
who chose one of the labels. In multi-class classification, this may be the
distribution across labels obtained for an instance. While this provides more
information than a single aggregated label does (e.g., identifies the
instances with high disagreement), it fails to capture annotator-level
systematic differences.

Thirdly, some datasets release annotations made by each individual annotators
in an anonymous fashion (e.g., Kennedy et al. (2020); Jigsaw (2019)). Such
annotator-level labels allow downstream dataset users to investigate and
account for systematic differences between individual annotators’ perspectives
on the tasks, although they do not contain any information about each
annotators’ socio-cultural backgrounds. Finally, some recent datasets (e.g.,
Díaz et al. (2018)) also release such socio-demographic information about the
annotators in addition to annotator-level labels. This information may include
various identity subgroups the annotators self-identify with (e.g., gender,
race, age range, etc.), or survey responses from the annotators that capture
their value systems, lived experiences, or expertise, as they relate to the
specific task at hand. Such information, while tricky to share responsibly,
would help enable analysis around representation of marginalized perspectives
in datasets, as we demonstrate in the next section.

Figure 1: Histograms represent the frequency distribution of annotator
agreement with the aggregated label for eight tasks under three datasets for
Emotions, Sentiment and Hate Speech datasets. The lack of uniformity in the
distributions means that annotator perspectives are not equally captured in
the majority labels.

##  3 Impacts of Aggregation

In this section, we investigate how the aggregation of multiple annotations to
a single label impact representations of individual and group perspectives in
the resulting datasets. We analyze annotations for eight binary classification
tasks, across three different datasets: hate-speech Kennedy et al. (2020),
sentiment Díaz et al. (2018), and emotion Demszky et al. (2020). Table 1 shows
the number of instances, annotators and individual annotations present in the
datasets. For hate-speech and emotion datasets, we use the binary label in the
raw annotations, whereas for the sentiment dataset, we map the 5-point ordinal
labels (-2, -1, 0, +1, +2) in the raw data to a binary distinction denoting
whether the text was deemed positive or negative.111We do this mapping for the
purposes of this analysis, where we are focusing on binary tasks. Ideally, a
more nuanced 5-point labeling schema will be more useful. While the emotion
dataset contains annotations for 28 different emotions, in this work, for
brevity, we focused on the annotations for only the six standard Ekman
emotions Ekman (1992) — anger, disgust, fear, joy, sadness, and surprise. In
particular, we use the raw annotations for these six emotions, rather than the
mapping of all 28 emotions onto these six emotions that Demszky et al. (2020)
use in some of their experiments.

Dataset | #instances | #annotators | #annotations
---|---|---|---
Hate-speech | 27,665 | 18 | 86,529
Sentiment | 14,071 | 1,481 | 59,240
Emotion | 58,011 | 82 | 211,225
Table 1: Statistics on the three datasets we analyze: Hate-speech Kennedy et
al. (2020), Sentiment Díaz et al. (2018), and Emotions Demszky et al. (2020)

Figure 2: Average and standard deviation of annotator agreement with
aggregated labels, calculated for annotators grouped by their socio-
demographics under gender, race, and political affiliation.

###  3.1 Q1: Do Aggregated Labels Represent Individual Annotators Uniformly?

First, we investigate whether the aggregated labels obtained through majority
labels provide a more or less equal representation for all annotator
perspectives. For this analysis, we calculate the majority label for each
instance as the label that half or more annotators who annotated that instance
agreed on. We then measure Cohen’s Kappa agreement score for each individual
annotator’s labels and the majority labels on the subset of instances they
annotated. While lower agreement scores between some individual annotators and
the majority vote is expected (e.g., if the annotator produced noisy labels,
or they misunderstood the task), the assumption is that the majority label
captures the perspective of the ‘average human annotator’ within the annotator
pool.

Figure 1 represents the histogram of annotators’ agreement scores with
majority votes for all eight tasks. While the majority vote in some tasks such
as joy and sadness (to some extent) do represent most of the annotator pool
more or less uniformly (i.e., majority vote agrees with most annotators at
around the same rate), in most cases, the majority vote under-represents or
outright ignores the perspectives of a substantial number of annotators. For
instance, majority vote for disgust has very low agreement ($\kappa<0.3$) with
almost one-third (27 out of 82) of the annotator pool. Similarly, majority
vote for sentiment has very low agreement with around one-third (450+) of
their annotator pool.

###  3.2 Q2: Do Aggregated Labels Represent All Social Groups Uniformly?

While the analysis on Q1 reveals that certain annotator perspectives are more
likely to be ignored in the majority vote, it is especially problematic from a
fairness perspective, if these differences vary across different social
groups. Here, we investigate whether specific socio-demographic groups and
their perspectives are unevenly disregarded through annotation aggregation. To
this end, we analyze the sentiment analysis dataset (Díaz et al., 2018) since
it includes raw annotations as well as annotators’ self-identified socio-
demographic information. Furthermore, as observed in Figure 1, a large subset
of annotators in this dataset are in low agreement with the aggregated labels.

We study three demographic attributes, namely race, gender, and political
affiliation and compare the agreement scores between the aggregated labels and
the individual annotators’ labels within each group. Figure 2 shows the
average and standard deviation of annotators’ agreement scores with aggregated
labels for each demographic group: race (Asian, Black, and White), gender
(Male, and Female), and political affiliation (Conservative, Moderate, and
Liberal).222We removed social groups with fewer than 50 annotators from this
analysis for lack of sufficient data points. These include other racial groups
such as ‘Middle Eastern’ with 2 annotators, ‘Native Hawaiian or Pacific
Islander’ with 4 annotators, and non-binary gender identity with one
annotator).

We perform three one-way ANOVA tests to test whether annotators belonging to
different demographic groups have significantly different agreement scores
with the aggregated labels, on average. The results show significant
differences among racial groups ($F(2,2387)$=3.77, $p$=0.02); in particular,
White annotators show an average agreement of 0.42 ($SD$=0.26), significantly
higher ($p$=0.03 according to a post-hoc Tukey test) than Black annotators
with average of 0.37 ($SD$=0.27). The difference between average agreement
scores across different political groups are not statistically significant,
although moderate annotators on average have higher agreement (0.42) compared
to conservative and liberal annotators (0.40 and 0.38, respectively).
Similarly, annotation agreements of male and female annotators are not
significantly different.

##  4 Utility of Annotator-level Labels

Another argument in favor of retaining annotator-level labels is their utility
in modeling disagreement during training and evaluation. Prabhakaran et al.
(2012) and Plank et al. (2014) incorporated annotator disagreement in the loss
functions used during training to improve predictive performance. Cohn and
Specia (2013) and Fornaciari et al. (2021) use a multi-task approach to
incorporate annotator disagreements to improve machine translation and part-
of-speech tagging performance, respectively. Chou and Lee (2019) and Guan et
al. (2018) developed learning architectures that model individual annotators
as a way to improved performance. Wich et al. (2020) show the utility of
detecting clusters of annotators in hate-speech detection based on how often
they agree with each other. Finally, Davani et al. (2021) introduce a multi-
annotator architecture that models each annotators’ perspectives separately
using a multi-task approach. They demonstrate that this architecture helps to
model uncertainty in predictions, without any significant loss of accuracy or
efficiency. This array of recent work further demonstrates the utility of
retaining annotator-level information in the datasets for downstream modeling
steps.

##  5 Discussion and Conclusion

Building models to predict or measure subjective phenomena based on human
annotations should involve explicit consideration for the unique perspectives
each annotator brings forth in their annotations. Annotators are not
interchangeable– that is, they draw from their socially-embedded experiences
and knowledge when making annotation judgments. As a result, retaining their
perspectives separately in the datasets will enable dataset users to account
for these differences according to their needs. We demonstrated that
annotation aggregation may unfairly disregard perspectives of certain
annotators, and sometimes certain socio-demographic groups. Based on our
analysis, we propose three recommendations aimed to avoid these issues:

#### Annotator-level labels:

We urge dataset developers to release the annotator-level labels, preferably
in an anonymous fashion, and leave open the choice of whether and how to
utilize or aggregate these labels for the dataset users.

#### Socio-demographic information:

Sociodemographic identity of the annotators is crucial to ascertain that the
datasets (and the models trained on them) equitably represent perspectives of
various social groups. We urge dataset developers to include socio-demographic
information of annotators, when viable to do so responsibly.

#### Documentation about recruitment, selection, and assignment of annotators:

Finally, we urge dataset developers to document how the annotators were
recruited, the criteria used to select them and assign data to them, and any
efforts to ensure representational diversity, through transparency artefacts
such datasheets Gebru et al. (2018) or data statements Bender and Friedman
(2018).

## Acknowledgements

We thank Ben Hutchinson for valuable feedback on the manuscript. We also thank
the anonymous reviewers for their feedback.

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# Requirements for Open Political Information: Transparency Beyond Open Data

Andong Luis Li Zhao 1, Andrew Paley 1, Rachel Adler 1 2, Harper Pack 1,
Sergio Servantez 1, Alexander Einarsson 1, Cameron Barrie 1, Marko Sterbentz
1,
Kristian Hammond 1

###### Abstract

A politically informed citizenry is imperative for a well-developed democracy.
While the US government has pursued policies for open data, these efforts have
been insufficient in achieving an open government because only people with
technical and domain knowledge can access information in the data. In this
work, we conduct user interviews to identify wants and needs among
stakeholders. We further use this information to sketch out the foundational
requirements for a functional political information technical system.

## Introduction

A modern democratic country’s political system is an intricate tapestry of
multiple, often conflicting, tangible and intangible institutions, including
an organized state, an independent bureaucracy, an adherence to rule of law
(Fukuyama 2014), and, ideally, an informed citizenry.

To support such a citizenry, the UN Sustainable Development Goals make it
clear: transparent government is a requirement of a well-developed and
sustainable society. Specifically in the US, the principles of transparency
have proliferated in various forms, from the Freedom of Information Act (FOIA)
in 1966 (Yu and Robinson 2011) to the more recent advent of open data
initiatives (Data.gov ; Federal Election Commission ; McDermott 2010). There
have also been significant third party efforts to make various forms of raw
data available (GovTrack.us ; OpenSecrets ; Vote Smart ). Across these
initiatives, a common theme is a focus on issues of data access.

Access to data is far from a solved issue – sometimes the available data is
constrained and requires specific requests (Kwoka 2015), sometimes it’s locked
behind paywalls (PACER ), and often available repositories are decentralized
or in formats that are difficult to combine or use. Further, there are issues
of scope: despite the above, federal data is significantly more available than
data about state and local governments.

Nevertheless, even if all of these problems are solved, “open data” alone is
an insufficient policy for achieving “open government” – the actual
requirement is open information. This distinction is important; to a citizen
without technical skills (or the resources to acquire them), a zip file of a
hundred thousand documents or database credentials is a relatively opaque form
of transparency. And, even for those with domain knowledge and technical
skills – such as data-driven journalists – the sheer volume and pace of raw
data generated can become insurmountable for one individual (GovTrack.us ). In
fact, it is people who can afford to hire teams of technical and domain
experts who profit most from this opaque data (Janssen, Charalabidis, and
Zuiderwijk 2012; Kwoka 2015), indicating that open data alone might be
reinforcing inequalities in the political system.

Hence, truly open government requires that we rethink the means of access,
moving from providing data to providing information. This is all the more
important in our currently fraught information landscape. With the advent of
social media misinformation and threats to basic institutions that we
historically have taken for granted, societies around the world have realized
the importance of upholding and strengthening our political institutions – and
part of doing so is making sure they’re accessible, understandable and
transparent to all of their stakeholders.

In this paper, we explore the issue of identifying the sociotechnical
requirements for political information systems, taking into account new types
of user experiences afforded by AI. As a first step, we focus our research on
information about Illinois state politics. We conducted interviews with domain
stakeholders, political scientists and journalists, to understand their needs.
We report the current problems in the space of technical systems for political
transparency based on real user feedback. Finally, we summarize our findings
in a set of sociotechnical requirements that political information systems
should fulfill to facilitate political transparency.

## Related Works

The government itself keeps records available to all (Data.gov ; Federal
Election Commission ), although sometimes behind paywalls (PACER ). This is an
important first step. However, these have significant issues (Talukder et al.
2019), and require skills or resources to make use of the data (Janssen,
Charalabidis, and Zuiderwijk 2012; Kitchin 2013).

Outside of government, various civil institutions have initiatives aimed at
access to campaign finance and legislative data at state and federal levels
(FollowTheMoney ; OpenStates ; ProPublica Data ), supplemented by efforts to
standardize data schemas (OpenCivicData ). However, technical skills and
political knowledge are required to make use of these APIs. Further
initiatives that allow users to track politicians and legislation (GovTrack.us
; OpenSecrets ; Vote Smart ) are mostly limited to federal officials and rely
heavily on human-curated material, limiting scalability. Also, these
initiatives appear to lack a user-centered design approach that leads to more
usable interfaces (Norman and Draper 1986).

In the AI and government space, there has been research exploring user needs
(Smith 2019), with some of these works even proposing related solutions
(Faruqe, Watkins, and Medsker 2020). However, most of these works have not
conducted user interviews and also tend to be broadly focused on public sector
applications, yielding potentially overbroad requirements not informed by
their stakeholders’ needs.

## User Interviews

To ensure that we identify and address our users’ needs, we took a user-
centered design approach. Our main goal for the interviews was to understand
what questions users had regarding Illinois politics and the main impediments
to getting answers . We also asked them to describe their background and
skills to better understand their range of technical capabilities. These
interviews lasted roughly 30 to 60 minutes.

### Stakeholders

Our target users are people who have interest and expertise in political
information but lack either the technical skills necessary for more robust
data wrangling and analysis or the time or resources necessary to do so). We
interviewed 15 users: 4 political science professors, 4 political science
undergraduate students, 2 political science PhD students, 1 journalism
professor, and 4 reporters. All but one user were based in Illinois (one
political science professor was based in New York). We obtained this sample of
users via our professional networks. Below are the highlights from the
interviews categorized by the three main stakeholders.

#### Journalists

Journalists were one of our primary focuses because they exemplified our
target user. All four journalists were focused on Illinois state or local
politics, had interests in data-driven journalism, and often did their own
entry-level analyses with platforms like Microsoft Excel. However, they all
still had many questions that they were unable to answer due to lack of skills
or resources to collect, collate, and analyze complex datasets.

#### Professors

Professors (5 total) were on the higher end of users who had technical and
domain expertise. Three of them were interested in exploring questions about
political trends in legislation, finances, and accountability to voters. Three
professors discussed data challenges – including staleness and lack of
interoperability – but only one professor had significant data analysis
experience.

The remaining two professors we interviewed had more methodological rather
than domain-specific questions, including how to survey information from
constituents, methods for modelling political behavior, and use of natural
language processing to capture political information.

#### Political Science Students

Graduate students had some experience with research, technical tools, and the
domain. They had systemic questions, and focused on political processes’
trends rather than election-focused or daily political questions. However,
they still had significant issues with political transparency tools,
especially due to their cumbersome user interfaces and the complexity of
conducting database-wide and cross-dataset analyses.

Undergraduate students were the stakeholders who relied least on primary data
for their political information, favoring lower-barrier sources, such as
Twitter or other social media feeds. Almost all had minimal experience with
data analysis, with only one student indicating they had performed statistical
analysis to research a question.

### User Interview Themes

#### Questions they want to answer

Of the users who were interested in Illinois State politics, the three most
common topics of interest were the role money plays in politics, legislative
process, and voting behavior. Questions include:

  * •

What groups of donors are most influential? How does this vary across time and
political party?

  * •

How does the donor profile of a legislator affect their voting pattern? Can we
look at this relationship for different interest group and topics (e.g.
environmental regulation, healthcare, immigration)?

  * •

What bills are being repeatedly introduced to the legislature even though they
consistently fail to pass? What bill changes were implemented such that these
previously failing bills are able to pass?

  * •

How do voting patterns change according to different demographics (e.g. race,
rural vs. urban, economic class)?

There were also users who had interests in other domains such as international
relations, landlord records, police misconduct, and criminal arrest records.

#### Working with primary data sources

From our set of users, 6 of them had worked with or were familiar with
political data sources such as the Illinois State Board of Elections (ISBE) or
the Illinois General Assembly (ILGA). The other users dealt with more
experimental datasets (e.g. research surveys) or obtained information from
news sources (e.g. FiveThirtyEight or Twitter).

The most common problems that these sources had were that the data was
difficult to obtain and that it was difficult to understand the data and the
platform. Obtaining the data from original sources is difficult for many
reasons. For instance, one of the reporters mentioned wanting to work a story
about pensions for public sector workers in Chicago, yet the data are
scattered across many departments’ websites; similarly, we heard the Chicago
police data was unlinked from the criminal court data of Cook County. The data
can also be quite unreliable and out of date, such as that in the Chicago Data
portal.

Perhaps the biggest challenge to overcome was that user experience was a
significant barrier to entry. Though some of the websites, like the Center for
Tax and Budget Accountability, had good capabilities and interfaces, many
others had significant issues. In fact, one of the users said that the ILGA
website was “a nightmare” and that learning how to use the ISBE website took
several years.

#### Working with auxiliary platforms

The same set of 6 users who were familiar with primary political data sources
were also familiar with auxiliary platforms that compile and process data from
main sources; platforms like these include OpenStates, Reform for Illinois,
and Legistar.

Overall users reported more satisfaction with these platforms. However, some
users were unsure if they could trust the platforms’ data. One user worked
with the website FindMyLandlord, which helps users find Chicago building
landlords, but they felt uncomfortable using it because it was unclear how the
data was obtained and processed. Additionally, the user found errors in the
data and were unable to suggest changes, further making them distrust the data
and the platform. These issues sometimes were significant enough that users
would rather deal with the problems of the primary sources to ensure
reliability and trust in their data.

#### Time, skills, and resource constraints

Users reported that even after getting the needed data, they had difficulty
using the data due to time, skills, and resource constraints. For many users,
data analysis was not the primary function of their job nor was it part of
their educational background. Hence, even entirely open, updated, and
documented data would be difficult for them to take full advantage of, let
alone stores of unstructured documents.

Understanding the data schema and properties was a significant issue for all
users. Once the data was understood, users reported that they were able to run
simple queries and visualizations. The most advanced user was able to use Data
Wrapper to display information; all other users did their analysis with
platforms like Microsoft Excel.

Some users reported they would like to have an experienced data scientist help
them and some went even further and wanted a tool that would just answer the
questions they had, such as “which donors gave money to legislator X and how
did X vote on legislation related to the donors?”

## System Requirements

From these user interviews, we identified main principles that political
information systems should follow. These can be broken down into the four main
areas of data requirements, usability, transparency, and domain
personalization. Though our user interviews were mostly focused on Illinois
state politics, these requirements broadly apply to political information
systems, and we believe these requirements can be applied to other domains as
well. However, consideration of other domains is outside of the scope of this
research and will not be explored in this paper.

#### Data Requirements

  1. 1.

Documentation: Because political data are both timely and include various
terms of art, having thorough documentation, such as data schemas and a
glossary, helps avoid ambiguities and lowers the barrier to entry to new
users.

  2. 2.

Access to and interoperability with other data: Since political data are
scattered, it is critical that platforms can utilize decentralized data for
more holistic analyses.This includes data on legislation, executive and
judicial actions, electoral finances, electoral results, demographics, and
media content.

  3. 3.

Access to up-to-date data: Because political systems constantly generate new
data, current approaches that focus on a data snapshot make it difficult to
get information beyond that dataset’s scope.

#### Domain Personalization

  1. 4.

Accommodate different ontologies: There are many equally valid categorizations
for the same set of entities (e.g. there are multiple valid categorizations of
bills by topic) that ought to be available. Some users might also want to
define their own categorizations (e.g. considering all donors from the same
address as a single entity).

  2. 5.

Integration of user preferences and knowledge: This includes adapting what the
platform shows users based on their preferences, as well as users inputting
their own knowledge into the system to run new analyses.

#### Usability and Analytics

  1. 6.

Adherence to UX best practices: Though a rather basic requirement, we found
that few platforms follow these best practices that reduce cognitive overhead
and make the system more usable.

  2. 7.

Allow follow up questions and flexible analyses paths: Especially in the
interconnected world of political information, data analyses are rarely a
single-question endeavour; rather, they are explorations that follow non-
linear paths.

  3. 8.

Analysis of unstructured data: Because much of the political data landscape
involves unstructured data (e.g. social media, transcripts, bill text),
systems should have a process in place to extract meaningful analysis out of
these, be it automatically or by facilitating human intervention on these
processes (e.g. building ontologies, training machine learning models).

  4. 9.

Attend to different levels of user technical background: Some users might be
satisfied with simply obtaining data whereas others will want to ask more
complex questions about the data that does not require them to manage the data
itself. For each user group, the main use cases are:

    1. (a)

Technically-savvy domain experts: Allowing users to download the data to use
it on other platforms.

    2. (b)

Domain experts without technical skills: Aiding users with conducting
analyses.

    3. (c)

Users without domain or technical expertise: Identifying user questions and
mapping these to analyses, as well as providing technical and domain context.

  5. 10.

Export analysis: To maximize the utility of the information, users should be
able to share the information or use it in other platforms.

#### Transparency and Trust

  1. 11.

Transparent analysis and processing: Documentation on analysis procedures, AI
techniques, data scope, and data processing should be readily available.
Building trustworthy and explainable AI is especially important when handling
sensitive data such as electoral finances and records.

  2. 12.

Identification and communication of limitations: This includes reporting
limits on technical insight (e.g. uncertainty in analyses, strength of
correlation) as well as domain significance (e.g. when looking at finances it
is important to look at contributions and expenditures). This is especially
pertinent to the system’s AI, since these are known to have significant issues
with fairness and generalization outside their training data,

  3. 13.

Allow verification of results by domain or technical experts: Though we want
to allow flexibility and lower barriers to entry, politics is a domain rife
with misleading statements and outright fake information. Hence, there need to
be balancing mechanisms to identify and highlight trustworthy content.

### Use Cases following the System Requirements

Consider a user interested in exploring trends in financial contributions to
state congressional candidates by environmental interest groups in the state
of Illinois. Given that the user knows a bit about the domain, they are
interested in total contributions by environmental groups donated to each
candidate committee, and different breakdowns (e.g. by time, by district, by
political party). The user asks the platform these analytical questions, and
the platform provides answers in graphs or text. The platform also identifies
potential follow up analytical questions that might interest the user.

Consider a different user who has little knowledge about the domain, but wants
to know more about their representatives in the state Congress and their
record regarding immigration. The platform deduces the user’s representatives
based on their address. Given the users’ limited domain knowledge, the
platform also gives an overview of different roles (e.g. difference between
senators and representatives) and definitions (e.g. what is a PAC). The
platform then provides an overview of the representatives’ finances, voting
records, and statements that relate to immigration.

## Conclusions and Future Work

In this work we have identified the main sociotechnical requirements political
information systems based on user interviews with domain stakeholders. These
principles can be used to develop platforms for enabling users with interest
in the domain to answer questions about politics. Development of said platform
will be explored in future work.

A significant group of users that we did not interview were users who had no
interest in politics. We excluded this group to be able to better focus on
those who we do not need to persuade to seek political information, and due to
inherent problems in trying to interview users who are not interested in
politics about politics. Though outside the scope of this work, identifying
requirements for these users is an imperative line of work that we also leave
for future work.

## References

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$n=d-k\in\left\\{\left\lfloor\frac{d}{2}\right\rfloor+1,\left\lfloor\frac{d}{2}\right\rfloor+2\right\\},\quad\text{
thus }\quad
k\in\left\\{\left\lceil\frac{d}{2}\right\rceil-1,\left\lceil\frac{d}{2}\right\rceil-2\right\\}.$
(50)

By the definition of blow up, when $n>1$, the embedding dimension of $A$ is
$r-1$ (one less than the embedding dimension of $C$) and by the surjectivity
of $\pi$ embedding dimension of $T$ is at most $r-1$. This yields the
following upper bounds on the Hilbert functions of $A$ and $T$:

$\displaystyle\dim_{\mathbb{F}}A_{i}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\binom{r-2+i}{r-2},\dim_{\mathbb{F}}A_{d-i}\right\\}$
(51) $\displaystyle\dim_{\mathbb{F}}T_{i}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\binom{r-2+i}{r-2},\dim_{\mathbb{F}}T_{k-i}\right\\}.$
(52)

By Theorem 4.6 we have

$H(C)=H(A)+H(T)[1]+\cdots+H(T)[n-1].$ (53)

Evaluating the above identity in degrees
$i\leq\left\lfloor\frac{k}{2}\right\rfloor+1$ combined with the above
inequalities yields

$\binom{r+i-1}{r-1}=H(A)[i]+\sum_{j=1}^{i}H(T)[i-j]\leq\sum_{j=0}^{i}\binom{r+j-2}{r-2}=\binom{r+i-1}{r-1},$

where the last equality is a well-known combinatorial identity. This implies
that equality must hold both in (51) and in (52) for
$i\leq\left\lfloor\frac{k}{2}\right\rfloor+1$. In degree
$i=\left\lfloor\frac{k}{2}\right\rfloor+2$ since $r\geq 3$ the inequality (52)
yields

$H(T)[i-1]=H(T)[k-i+1]=H(T)\left[\left\lceil\frac{k}{2}\right\rceil-1\right]\leq\binom{r-3+\left\lceil\frac{k}{2}\right\rceil}{r-2}<\binom{r+i-2}{r-2}$

Evaluating equation (53) in degree $i=\left\lfloor\frac{k}{2}\right\rfloor+2$
and combining the result with the inequality (51) and the inequality displayed
above gives

$H(C)[i]=H(A)[i]+\sum_{j=1}^{i}H(T)[i-j]<\sum_{j=0}^{i}\binom{r+j-2}{r-2}=\binom{r+i-1}{r-1}.$

This contradicts our assumption that $C$ is compressed, provided that

$\left\lfloor\frac{k}{2}\right\rfloor+2\leq\left\lfloor\frac{d}{2}\right\rfloor,\text{
for
}k\in\left\\{\left\lceil\frac{d}{2}\right\rceil-1,\left\lceil\frac{d}{2}\right\rceil-2\right\\}.$

The above inequality is satisfied if and only if $d=4$ or $d\geq 6$. ∎

The following example shows that there exists compressed algebras of socle
degree $5$ which are cohomological blow-ups.

######  Example 7.3.

Let

$\pi\colon A=\frac{{\mathbb{F}}[x,y]}{(x^{4},y^{3})}\rightarrow
T=\frac{{\mathbb{F}}[x,y]}{(x^{2}-xy,y^{2})}$

be the map defined by $\pi(x)=x$ and $\pi(y)=y$ and distinguished socle
generators $\sigma_{A}=x^{3}y$ and $\sigma_{T}=x^{2}$. Then $\tau=xy$ and
$K=(x^{2}-xy,y^{2})$, and hence taking parameters $t_{1}=t_{2}=0$ and
$\lambda=1$ we get the cohomological blow-up

$\tilde{A}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{4},y^{3},\xi(x^{2}-xy),\xi
y^{2},\xi^{2}-xy)}$

which has Hilbert function $H(\tilde{A})=(1,3,6,6,3,1)$, and hence $\tilde{A}$
is a compressed BUG of socle degree $5$.

##  8 The Lefschetz Property

The strong Lefschetz property for graded AG algebras is an algebraic version
of a property of cohomology rings of smooth complex projective varieties
stemming from the Hard Lefschetz theorem in algebraic geometry.

######  Definition 8.1.

A graded Artinian ${\mathbb{F}}$-algebra $A=\bigoplus_{i=0}^{d}A_{i}$ is said
to satisfy the strong Lefschetz property (SLP) if there is a linear form
$\ell\in A_{1}$ for which the multiplication maps $\times\ell^{j}\colon
A_{i}\rightarrow A_{i+j}$ have full rank
$\operatorname{rank}(\times\ell^{j})=\min\\{\dim_{\mathbb{F}}A_{i},\dim_{\mathbb{F}}A_{i+j}\\}$
for each degree $i$ and each exponent $j$. A linear form $\ell$ with this
property is called a strong Lefschetz element for $A$.

If the multiplication maps $\times\ell\colon A_{i}\rightarrow A_{i+1}$ have
full rank for each degree $i$ then $A$ is said to satisfy the weak Lefschetz
property (WLP).

More generally, given any graded Artinian algebra $A$, and any linear form
$\ell\in A_{1}$ we can define its _Jordan type_ $P_{\ell}$ to be the partition
corresponding to the block decomposition of the Jordan canonical form for the
nilpotent linear operator $\times\ell\colon A\rightarrow A$. It is well known
that for a standard-graded AG algebra with unimodal Hilbert function then
$\ell$ is strong Lefschetz if and only if the Jordan type $P_{\ell}$ is equal
to $H^{\vee}$, the conjugate of the Hilbert function regarded as a partition
(switch rows and columns in the Ferrers graph) [26, Proposition 2.10] and
$\ell$ is weak Lefschetz if $P_{\ell}$ has number of parts equal to the
Sperner number of $H$ [19, Proposition 3.5].

In this section we study the strong Lefschetz Property for cohomological blow-
up algebras. To attain this goal, we observe the behavior of these rings in
families. Our strategy is to consider every cohomological blow-up algebra as
the general fiber in a certain flat family. Interestingly, all fibers in these
familes will be AG algebras with the exception of the special fiber. Our proof
of the SLP for the cohomological blow up then employs the following well known
fact: in a flat family if SLP holds on the special fiber, then it must also
hold on a sufficiently general fiber by semicontinuity of Jordan type, e.g.
[26, Corollary 2.44].

We continue with the notation established in previous sections, namely
$\pi:A\to T$ is a surjective map of oriented AG algebras with Thom class
$\tau$ and kernel $K$, $f_{A}(\xi)\in A[\xi]$ is a homogeneous monic
polynomial yielding the cohomological blow-up algebra

$\tilde{A}=\frac{A[\xi]}{(\xi\cdot K,f_{A}(\xi))}=\frac{A[\xi]}{I},$

as in Construction 3.1.

Consider a weighted monomial order $<$ on $A[\xi]$ obtained by assigning
weight $1$ to the variable $\xi$ and weight $0$ to each element of $A$. Then
the weight of a monomial $\mu$ is $w(\mu)=\max\\{n:\,\ \xi^{n}\mid\mu\\}$ and
the weight of a polynomial $g=\sum c_{i}\mu_{i}\in A[\xi]$ with $c_{i}\in
F^{\times}$ and $\mu_{i}$ monomials is $w(g)=\max\\{w(\mu_{i})\\}$. Two
monomials $\mu,\mu^{\prime}$ satisfy $\mu<\mu^{\prime}$ if and only if
$w(\mu)<w(\mu^{\prime})$. The initial form of a polynomial $g=\sum
c_{i}\mu_{i}\in A[\xi]$ is ${\rm
in}_{<}(g)=\sum_{w(\mu_{j})=w(g)}c_{j}\mu_{j}$ and the initial ideal of an
ideal $J\subseteq A[\xi]$ is

${\rm in}_{<}(J)=({\rm in}_{<}(g)\mid g\in J).$

While it is not usually the case that the generators of the initial ideal of
an ideal $J$ are the initial forms of the generators of $J$, this is
nevertheless the case for $I$ since, as we show below, the generators of $I$
form a Gröbner basis with respect to $<$. We obtain the following description
for the initial ideal of $I$.

######  Lemma 8.2.

The set $\\{\xi\cdot K,f_{A}(\xi)\\}$ is a Gröbner basis for $I$ with respect
to $<$, i.e. the initial ideal of $I$ is ${\rm in}_{<}(I)=(\xi\cdot
K,\xi^{n})$.

###### Proof.

To show that the set $\\{\xi\cdot K,f_{A}(\xi)\\}$ is a Gröbner basis for $I$
one utilizes Buchsberger’s criterion; see [12, Theorem 15.8]. Since any
element in $\xi\cdot K$ is equal to its initial form, the S-polynomial of any
two such elements is $0$. It remains to compute the S-polynomial of $u\xi$
(with $u\in K$) and $f_{A}(\xi)$ which is

$S(f_{A}(\xi),u\xi)=u(\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n-1}\xi+\lambda\tau)-u\xi\cdot\xi^{n-1}=ua_{1}\xi^{n-1}+\cdots+ua_{n-1}\xi$

since $K\cdot\lambda\tau=0$ by Lemma 2.2. Since
$S(f_{A}(\xi),u\xi)\in\left(\xi\cdot K\right)$, this polynomial reduces to $0$
modulo the the set $\\{\xi\cdot K,f_{A}(\xi)\\}$, concluding the proof that
this is a Gröbner basis. It follows that ${\rm in}_{<}(I)=({\rm
in}_{<}(\xi\cdot K),{\rm in}_{<}(f_{A}(\xi)))=(\xi\cdot K,\xi^{n}).$ ∎

The relationship between $\tilde{A}=A[\xi]/I$ and its initial algebra ${\rm
in}_{<}(\tilde{A}):=A[\xi]/{\rm in}_{<}(I)$ is established by means of the
following well-known construction; see [12, p. 343]. Given a parameter $t$ one
considers an ideal $\mathcal{I}$ of $A[\xi,t]$ given by

$\mathcal{I}=\left(\xi\cdot K,\xi^{n}+a_{1}t\xi^{n-1}+\cdots+a_{n-1}\xi
t^{n-1}+\lambda\tau t^{n}\right).$

Note that setting $t=1$ in $\tilde{I}$ recovers $I$ while setting $t=0$ gives
${\rm in}_{<}(I)$. We recall some key properties of this construction cf. [12,
Theorem 15.17].

######  Theorem 8.3.

The following ring $\mathcal{A}$ is free and hence flat as a
${\mathbb{F}}[z]$-algebra

$\mathcal{A}=\frac{A[\xi,z]}{\mathcal{I}}=\frac{A[\xi,z]}{(\xi\cdot
K,\xi^{n}+a_{1}\xi^{n-1}z+\cdots+a_{n-1}\xi z^{n-1}+a_{n}z^{n})}.$ (54)

Thus $\mathcal{A}$ can be viewed as a flat family of algebras with fibers
$\mathcal{A}_{c}=\mathcal{A}\otimes_{{\mathbb{F}}[z]}F[z]/(z-c)$ given by

$\mathcal{A}_{c}\cong\begin{cases}A[\xi]/I=A[\xi]/(\xi\cdot
K,f_{A}(\xi))=\tilde{A}&\text{if}\ c\in{\mathbb{F}}^{\times}\\\ A[\xi]/{\rm
in}_{<}(I)=A[\xi]/(\xi\cdot K,\xi^{n})={\rm in}_{<}(\tilde{A})&\text{if}\
c=0\\\ \end{cases}$

In the literature, flat families in which the general fibers are isomorphic
are sometimes called _jump deformations_. We remark that Theorem 8.3 remains
true if we replace $\tilde{A}$ with any $\hat{A}$ from Construction 3.1, but
recall that $\hat{A}$ is Gorenstein if and only if $\hat{A}=\tilde{A}$ by
Theorem 3.6.

Since $\tilde{A}$ is Gorenstein, Theorem 8.3 implies that the general fibers
of the family $\mathcal{A}$ are Gorenstein. However, Theorem 3.6 implies that
the special fiber $\mathcal{A}_{0}$ is not Gorenstein, but is boundary-
Gorenstein (Remark 3.8). Moreover, Corollary 3.7 implies that all the fibers
have the same Hilbert function given by

$H(\mathcal{A}_{c})=H(A)+H(T)[1]+\cdots+H(T)[n-1]\qquad\text{ for all
}c\in{\mathbb{F}}.$

We now consider the strong Lefschetz property for the special fiber. The ring
${\rm in}_{<}(\tilde{A})$ can thus be recognized as a fibered product

${\rm in}_{<}(\tilde{A})=A\times_{T}B\qquad\text{ with}\qquad
B=\frac{T[\xi]}{(\xi^{n})}$

with respect to the projections $\pi_{A}=\pi:A\to T$ and $\pi_{B}:B\to
T=B/(\xi)$, where $\pi_{B}$ is the canonical projection. Our proof of Lemma
8.4 below closely resembles the proof of [30, Theorem 5.12], where the SLP is
established for fibered products of certain AG algebras of the same socle
degree. This is not the case here as $\rm{soc.deg.}(A)=\rm{soc.deg.}(B)+1$.

######  Lemma 8.4.

Let $\pi:A\to T$ be a surjective homomorphism of graded AG
${\mathbb{F}}$-algebras of socle degrees $d>k$ respectively, such that both
$A$ and $T$ have SLP. Assume that ${\mathbb{F}}$ has characteristic zero, or
characteristic $p>d>k$. Then the initial algebra of $\tilde{A}$, $A[\xi]/{\rm
in}_{<}(I)$ has SLP as well.

###### Proof.

Recall from Lemma 8.2 that

${\rm in}_{<}(\tilde{A})=\frac{A[\xi]}{{\rm
in}_{<}(I)}=\frac{A[\xi]}{(\xi\cdot K,\xi^{n})}.$

This algebra admits a decomposition

${\rm in}_{<}(\tilde{A})=A\oplus J,\text{ where }J=T\xi\oplus\cdots\oplus
T\xi^{n-1}=\xi\cdot\frac{T[\xi]}{(\xi^{n-2})}.$

The important point here is that in the algebra ${\rm in}_{<}(\tilde{A})$, the
vector space $J$ is actually an ideal, and thus multiplication by a linear
form $\ell=\ell_{A}+\xi$ with $\ell_{A}\in A_{1}$ is represented with respect
to the above decomposition by a block matrix

$\begin{pmatrix}\ell_{A}&0\\\ *&\ell|_{J}\end{pmatrix}$ (55)

where $\ell_{A}$ stands for the map given by multiplication by $\ell_{A}$ on
$A$ and $\ell|_{J}$ denotes the restriction of multiplication by $\ell$ to
$J$. Since $A$ and $T$ have SLP, one can pick $\ell_{A}$ so that both
$\ell_{A}$ and its image $\ell_{T}=\overline{\ell_{A}}$ in $T$ are strong
Lefschetz elements. Then by [20, Theorem 6.1] if characteristic ${\mathbb{F}}$
is zero, or [28, Theorem 2.6] in general $\ell=\ell_{T}+\xi$ is also a strong
Lefschetz element on the free extension $T[\xi]/(\xi^{n-2})$ of $T$, and hence
on $J$. Since the Hilbert functions of $A$ and $J$ are both symmetric around
$d/2$ (due to both being AG) and unimodal (due to both having SLP) we conclude
that $\ell_{A}^{j}:A_{i}\to A_{i+j}$ and $\ell|_{J}^{j}:J_{i}\to J_{i+j}$ are
simultaneously injective or simultaneously surjective. It follows from (55)
that $\ell$ is a strong Lefschetz element on ${\rm in}_{<}(\tilde{A})$. ∎

We are now ready to prove that cohomological blow-up algebras have SLP.

######  Theorem 8.5.

Let ${\mathbb{F}}$ be an infinite field and let $\pi:A\to T$ be a surjective
homomorphism of graded AG ${\mathbb{F}}$-algebras of socle degrees $d>k$
respectively such that both $A$ and $T$ have SLP. Assume that characteristic
${\mathbb{F}}$ is zero or characteristic $F$ is $p>d$. Then every
cohomological blow-up algebra of $A$ along $T$ satisfies SLP.

###### Proof.

We have seen that $\mathcal{A}$ is a flat family. By Lemma 8.4, the special
fiber $\mathcal{A}_{0}$ has SLP. Therefore by semicontinuity of Jordan type
[26, Corollary 2.44], and since ${\mathbb{F}}$ is infinite, we deduce that
there is some $c\neq 0$ for which the general fiber $\mathcal{A}_{c}$ has SLP.
By Theorem 8.3 it follows that the cohomological blow
$\tilde{A}=\mathcal{A}_{1}$ has SLP as well. ∎

The following example shows that our assumptions on the characteristic of the
field in Theorem 8.5 are necessary.

######  Example 8.6.

Let $d$ and $k$ be integers satisfying ${2k<d}$, and, following the usual
notation, write ${n=d-k}$. Let ${\mathbb{F}}$ be a field of characteristic
$p$, and suppose there is an integer ${m\geq 1}$ such that ${n<p^{m}\leq
d-2}$. Consider the AG algebras

$A=\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(X^{d}+Y^{d})}=\frac{{\mathbb{F}}[x,y]}{\left(xy,x^{d}-y^{d}\right)}\qquad\text{and}\qquad
T=\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(X^{k})}=\frac{{\mathbb{F}}[x,y]}{\left(y,x^{k+1}\right)}\cong\frac{{\mathbb{F}}[x]}{\left(x^{k+1}\right)}.$

The Thom class of the natural surjection ${\pi:A\to T}$ is ${\tau=x^{n}}$, and
choosing ${f_{R}(\xi)=\xi^{n}-x^{n}}$ and ${h_{R}(\xi)=1}$ as in Construction
4.2, we get a cohomological blow-up algebra

$\tilde{A}=\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(X^{d}+Y^{d}+X^{k}\Xi^{n})}=\frac{{\mathbb{F}}[x,y,\xi]}{\left(xy,x^{d}-y^{d},y\xi,x^{k+1}\xi,\xi^{n}-x^{n}\right)}.$

We can easily check that ${\ell_{A}=x+y}$ and ${\ell_{T}=x}$ are Lefschetz
elements in $A$ and $T$, respectively. We also know that the Hilbert function
of $\tilde{A}$ satisfies
${H\bigl{(}\tilde{A}\bigr{)}_{1}=H\bigl{(}\tilde{A}\bigr{)}_{d-1}=3}$. This
means that if $\tilde{A}$ satisfies the SLP, multiplication by ${\ell^{d-2}}$,
for a general ${\ell\in\tilde{A}_{1}}$ must have rank $3$. However, writing
${\ell=ax+by+c\xi}$, we have
${\ell^{p^{m}}=a^{p^{m}}x^{p^{m}}+b^{p^{m}}y^{p^{m}}+c^{p^{m}}\xi^{p^{m}}=a^{p^{m}}x^{p^{m}}+b^{p^{m}}y^{p^{m}}}$,
becuse ${\xi^{p^{m}}}$ is zero in $\tilde{A}$. Therefore ${\ell^{p^{m}}\xi=0}$
(note that ${p^{m}>k}$), meaning that multiplication by $\ell^{p^{m}}$ has
rank less than $3$, hence so does multiplication by ${\ell^{d-2}}$. Therefore
$\tilde{A}$ does not satisfy the SLP.

Theorem 8.5 once again singles out cohomological blow-up algebras among
connected sums. Connected sums $A\\#_{\mathbb{F}}B$ over a field
${\mathbb{F}}$ of two strong or weak Lefschetz algebras $A,B$ are strong or
weak Lefschetz, respectively; however taking connected sum $A\\#_{T}B$ over an
arbitrary AG algebra $T$ may not in general preserve SLP or WLP [30, Section
5.2].

The following example shows that the converse of Theorem 8.5 is not true: if
the cohomological blowup $\hat{A}$ has SLP it does not follow that $A$ has
SLP. In other words, while the process of blowing up preserves SLP, the
process of blowing down does not preserve SLP, nor even WLP.

######  Example 8.7.

The following example, originally due to U. Perazzo, but re-examined more
recently by R. Gondim and F. Russo [15], is an AG algebra with unimodal
Hilbert function which does not have SLP or WLP:

$A=\frac{{\mathbb{F}}[x,y,z,u,v]}{\operatorname{Ann}(XU^{2}+YUV+ZV^{2})}=\frac{{\mathbb{F}}[x,y,z,u,v]}{\left(x^{2},xy,y^{2},xz,yz,z^{2},u^{3},u^{2}v,uv^{2},v^{3},xv,zu,xu-
yv,zv-yu\right)}.$

Taking the quotient $T$ of $A$ given by the Thom class $\tau=u^{2}$ yields

$T=\frac{{\mathbb{F}}[x,y,z,u,v]}{\operatorname{Ann}(X)}=\frac{{\mathbb{F}}[x,y,z,u,v]}{\left(x^{2},y,z,u,v\right)}\cong\frac{{\mathbb{F}}[x]}{(x^{2})}.$

Fix a parameter $\lambda\in{\mathbb{F}}$ and define polynomials $f_{T}(\xi)\in
T[\xi]$ and $f_{A}(\xi)\in A[\xi]$ by

$f_{T}(\xi)=\xi^{2}-\lambda x\xi\quad\text{ and }f_{A}(\xi)=\xi^{2}-\lambda
x\xi+u^{2}.$

Denoting the ideal of relations of $A$ by $I$ we obtain the cohomological
blowup

$\tilde{A}=\frac{{\mathbb{F}}[x,y,z,u,v,\xi]}{I+\xi(y,z,u,v)+(f_{A}(\xi))},$

which has Hilbert function $H(\tilde{A})=H(A)+H(T)[1]=(1,6,6,1)$. Fix
${\mathbb{F}}$-bases

$\tilde{A}_{1}=\operatorname{span}_{{\mathbb{F}}}\left\\{x,y,z,u,v,\xi\right\\},\
\ \text{and}\ \
\tilde{A}_{2}=\operatorname{span}_{{\mathbb{F}}}\left\\{u^{2},uv,v^{2},yv,yu,-x\xi\right\\}$

and let $\ell\in\hat{A}_{1}$ be a general linear form

$\ell=ax+by+cz+du+ev+f\xi.$

Then the matrix for the Lefschetz map
$\times\ell\colon\hat{A}_{1}\rightarrow\hat{A}_{2}$ and its determinant are
given by

$M=\left(\begin{array}[]{cccccc}0&0&0&d&0&-f\\\ 0&0&0&e&d&0\\\ 0&0&0&0&e&0\\\
d&e&0&a&b&0\\\ 0&d&e&b&c&0\\\ -f&0&0&0&0&-(a+\lambda f)\\\
\end{array}\right)\Rightarrow\det(M)=f^{2}e^{4}.$

Thus $\ell$ is a strong Lefschetz element for $\tilde{A}$ if and only if
$e\cdot f\neq 0$. In particular $\tilde{A}$ satisfies SLP and also WLP.

Surprisingly, the analogous result to Theorem 8.5 does not hold for the weak
Lefschetz property. The obstruction to establishing such a result via an
analogue of Lemma 8.4 is that the tensor product of two weak Lefschetz
algebras need not be weak Lefschetz. Examples of non-AG quadratic algebras
which demonstrate this are given in [40, §4.1]. Here we point out an AG
example. The AG algebra $T$ in Example 8.8 below is an example of R. Gondim,
quoted as [26, Example 3.4]. Using the Clebsch-Gordan Theorem [19, Theorem
3.29], one can see that if $\mathrm{char}\,{\mathbb{F}}=0$ or
$\mathrm{char}\,{\mathbb{F}}\geq 7$, then the tensor product $T\otimes B$,
where $B={\mathbb{F}}[\xi]/(\xi^{2})$, has Jordan type (for generic linear
form) the tensor product of $P_{T}=(5,3,3,3,2,2)$ and $P_{B}=(2)$, which is
$(6,4^{4},3^{2},2^{3},1^{2})$ with 12 parts; on the other hand $H(T\otimes
B)=(1,6,11,11,6,1)$, hence $H(T\otimes B)^{\vee}=(6,4^{5},2^{5})$ has so
$T\otimes B$ is not weak Lefschetz.

Building upon this, we give an example illustrating that blowing up does not
preserve WLP.

######  Example 8.8.

Consider the following algebra

$A=\frac{{\mathbb{F}}[x,y,z,u,v]}{\operatorname{Ann}(XU^{6}+YU^{4}V^{2}+ZU^{5}V)}=\frac{{\mathbb{F}}[x,y,z,u,v]}{\left(yz,xz,xy,vy-
uz,vx,ux-vz,u^{5}y,u^{5}v^{2},u^{6}v,u^{7},v^{3},x^{2},y^{2},z^{2}\right)}$

with $H(A)=(1,5,6,6,6,6,5,1)$ and its quotient corresponding to the Thom class
$\tau=u^{3}$

$T=\frac{{\mathbb{F}}[x,y,z,u,v]}{\operatorname{Ann}(XU^{3}+YUV^{2}+ZU^{2}V)}=\frac{{\mathbb{F}}[x,y,z,u,v]}{\left(z^{2},yz,xz,y^{2},xy,vy-
uz,x^{2},vx,ux-vz,u^{2}y,v^{3},u^{2}v^{2},u^{3}v,u^{4}\right)}$

with $H(T)=(1,5,6,5,1)$. Both $A$ and $T$ satisfy WLP, but not SLP: for a
generic $\ell$ the Jordan types are $P_{A,\ell}=(8,6,6,6,5,5)$, and
$P_{T,\ell}=(5,3,3,3,2,2)$. Denoting the ideal of relations of $A$ by $I$ and
the ideal of relations of $T$ by $K$, consider the cohomological blow-up
algebra

$\tilde{A}=\frac{{\mathbb{F}}[x,y,z,u,v,\xi]}{I+\xi\cdot K+(\xi^{3}-u^{3})},$

which has Hilbert function
$H(\tilde{A})=H(A)+H(T)[1]+H(T)[2]=(1,6,12,17,17,12,6,1)$ and dual Macaulay
generator $\tilde{F}=\Xi^{3}G+\Xi^{6}X+F$, here $\lambda=-1$ in the
Construction 4.2. Fix the following bases for the 17-dimensional vector spaces

$\displaystyle\tilde{A}_{3}$ $\displaystyle=$
$\displaystyle\operatorname{span}_{{\mathbb{F}}}\left\\{x\xi^{2},yu^{2},yuv,yu\xi,yv^{2},yv\xi,y\xi^{2},zv\xi,z\xi^{2},u^{2}v,u^{2}\xi,uv^{2},uv\xi,u\xi^{2},v^{2}\xi,v\xi^{2},\xi^{3}\right\\}$
$\displaystyle\tilde{A}_{4}$ $\displaystyle=$
$\displaystyle\operatorname{span}_{{\mathbb{F}}}\left\\{x\xi
3,yuv\xi,yu\xi^{2},yv2\xi,yv\xi^{2},y\xi^{3},zv\xi^{2},z\xi^{3},u^{2}v^{2},u^{2}v\xi,u^{2}\xi^{2},uv^{2}\xi,uv\xi^{2},u\xi^{3},v2\xi^{2},v\xi^{3},\xi^{4}\right\\}$

Taking a general linear form $\ell\in\tilde{A}_{1}$ as
$\ell=ax+by+cz+du+ev+f\xi$ we compute the matrix for the Lefschetz map
$\times\ell\colon\tilde{A}_{3}\rightarrow\hat{A}_{4}$ and its determinant,
which shows $\tilde{A}$ does not satisfy WLP

$\small
M={\left({\begin{array}[]{ccccccccccccccccc}f&0&e&0&d&0&0&0&0&c&0&b&0&0&0&0&a\\\
0&0&f&e&0&d&0&0&0&0&c&0&b&0&0&0&0\\\ 0&0&0&f&0&0&d&0&0&0&0&0&0&b&0&0&0\\\
0&0&0&0&f&e&0&d&0&0&a&0&c&0&b&0&0\\\ 0&0&0&0&0&f&e&0&d&0&0&0&0&c&0&b&0\\\
0&d&0&0&0&0&f&0&0&0&0&0&0&0&0&0&b\\\ d&0&0&0&0&0&0&f&e&0&0&0&0&a&0&c&0\\\
0&e&d&0&0&0&0&0&f&b&0&0&0&0&0&0&c\\\ 0&0&0&0&0&0&0&0&0&e&0&d&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&f&e&0&d&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&f&0&0&d&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&f&e&0&d&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&f&e&0&d&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&f&0&0&d\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&f&e&0\\\
0&0&0&0&0&0&0&0&0&d&0&0&0&0&0&f&e\\\ 0&0&0&0&0&0&0&0&0&0&d&0&0&0&0&0&f\\\
\end{array}}\right)}\Rightarrow\det(M)=0.$

A Macaulay2 [18] calculation gives the generic Jordan type
$P_{\tilde{A}}=(8,6^{5},4^{6},2^{4},1^{2})$ with $18$ parts, whereas the
conjugate partition of the Hilbert function is
$H(\tilde{A})^{\vee}=(8,6^{5},4^{6},2^{5})$ with $17$ parts, which implies
that $\tilde{A}$ has neither SLP nor WLP.

In Example 8.8 the Thom class of the map $A\to T$ has degree 3. This is the
minimal possible value for such an example based on the following result.

######  Theorem 8.9.

Let ${\mathbb{F}}$ be an infinite field and let $\pi:A\to T$ be a surjective
homomorphism of graded AG ${\mathbb{F}}$-algebras such that the difference
between the socle degrees of $A$ and $T$ is at most 2 and $A$ and $T$ both
satisfy WLP. Then every cohomological blow-up algebra of $A$ along $\pi$
satisfies WLP.

###### Proof.

The hypothsis on the socle degreed of $A$ and $T$ translates into $n\leq 2$.
The proofs of Lemma 8.4 in the spacial cases $J=0$ (for $n=1$) or $J=T$ (for
$n=2$) and Theorem 8.5 go through upon replacing SLP by WLP throughout. ∎

##  9 Geometric View and Examples

As in the Introduction, our motivation for studying the cohomological blow-up
algebras stems from the blow up construction in algebraic geometry. That the
cohomology ring of the blow up of a compact complex manifold along a closed
complex submanifold satisfies the conditions of Theorem 3.14 can be pieced
together from results in the book of P. Griffiths and J. Harris [17, Chapter
4, Section 6]; see also the paper of D. McDuff [37, Proposition 2.4] for the
statements in the symplectic category. The cohomology of the blow up was also
studied by S. Gitler who has obtained a presentation, as in Equation (13), of
the cohomology of the blow up of a complex manifold along a complex
submanifold in the case of a surjective restriction map as in Equation (13)
[14, Theorem 3.11]. An analogous presentation for the Chow ring of the blow up
of an algebraic variety along a regularly embedded subvariety with a
surjective restriction map also appears in the paper of S. Keel [32, Appendix,
Theorem 1].

Blowing up and blowing down are fundamental building blocks in birational
geometry. For example, in the theory of algebraic surfaces, a classical result
states that every birational map between algebraic surfaces (smooth complex
projective variety of dimension two) admits a strong factorization, meaning it
factors as a sequence of blow ups followed by a sequence of blow downs [6,
Corollary II.12]. More recently D. Abramovich et al. have proved the weak
factorization conjecture, a higher dimensional analogue which states that
every birational map between complete non-singular algebraic varieties over an
algebraically closed field of characteristic zero factors as a product of blow
ups and blow downs (in no specific order) [1, Theorem 0.1.1]. In the theory of
(smooth, projective) toric varieties, one can be quite explicit with these
factorizations by working with the associated (simplicial, polytopal) fan,
where blowing up along a toric subvariety corresponds to subdividing a cone of
the fan [11, Proposition 3.3.15]. As we have seen in Section 8 concerning the
strong Lefschetz Property, and will show in a sequel [34] concerning the
Hodge-Riemann bilinear relations (HRR), factorizations in terms of blow ups
and blow downs can be useful in establishing SLP and HRR.

P. McMullen [38] in his proof of SLP and HRR for the polytope algebra, an AG
algebra which he shows is isomorphic to a certain Artinian reduction of the
Stanley-Reisner ring of the corresponding simplicial polytopal fan, has given
an explicit formula for a (weak) factorization of the birational map
${\mathbb{P}}^{n}\supset({\mathbb{C}}^{*})^{n}\rightarrow X$ where $X$ is a
smooth projective toric variety of dimension $n$; see also V. A. Timorin [44]
for an exposition from the point of view of Macaulay duality. K. Karu [31]
used similar arguments in his proof of SLP and HRR for non-simplicial
polytopal fans; see also [5, 9]. More recently F. Ardila, G. Denham, and J.
Huh [2] have exploited blow up factorizations to prove SLP and HRR for the
Chow ring of the Bergman fan associated to a matroid. Especially important in
their work is the special case of blow ups which correspond to edge
subdivisions in these Bergman fans, and in fact they factor any blow up as a
sequence of successive edge subdivison blow ups and blow downs. Geometrically,
these edge subdivisions correspond to cohomological blow ups with $n=2$.
Theorem 8.9 indicates that this $n=2$ scenario is particularly favorable in
terms of ascent of the weak Lefschetz property to the cohomological blow-up
algebra but it does not necessarily guarantee the descent of WLP or SLP to the
blow-down, see Example 8.7.

Algebraically, one might say that two oriented graded AG algebras $A$ and
$A^{\prime}$ are _birationally equivalent_ if there is a sequence of oriented
graded AG algebras $(A_{0},A_{1},\ldots,A_{m})$ of some fixed socle degree
$d$, where $A_{0}=A$ and $A_{m}=A^{\prime}$ and for each $i$, $A_{i}$ is
either a cohomological blow up or a cohomological blow down of $A_{i-1}$. We
give an example below of several birationally equivalent AG algebras using
fans corresponding to smooth projective toric surfaces; in fact we derive a
(strong) factorization of the birational map
${\mathbb{P}}^{2}\dashrightarrow{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. First
we recall a few fundamental facts on toric varieties and their associated
fans; for further details we refer the reader to the book [11]. In these
geometric examples, as in the Introduction, we take cohomology with
coefficients over the rationals ${\mathbb{F}}={\mathbb{Q}}$.

A presentation for the cohomology ring of a complete simplicial toric variety
can be obtained from the associated fan in the following explicit manner. Let
$\Sigma$ be a complete simplicial fan and let $X$ be the corresponding toric
variety. Let $\rho_{1},\ldots,\rho_{r}$ be the rays of $\Sigma$, each
$\rho_{i}$ having minimal generator $u_{i}$. Introduce a variable $x_{i}$ for
each $\rho_{i}$. In the ring $\mathbb{Q}[x_{1},\ldots,x_{r}]$, let $I$ be the
square-free monomial ideal

$I=(x_{i_{1}}\cdots x_{i_{s}}\mid i_{1}<\cdots<i_{s}\text{ and
}\rho_{i_{1}},\ldots,\rho_{i_{s}}\text{ are not part of the same cone of
}\Sigma).$

We call $I$ the Stanley-Reisner ideal of $\Sigma$. Let $J$ be the ideal
generated by the linear forms

$J=\left(\sum_{i=1}^{r}\langle m,u_{i}\rangle x_{i}\mid
m\in\sum_{i=1}^{r}u_{i}\mathbb{Z}\right).$

By [11, Theorem 12.4.1] the singular cohomology ring of $X$ can be presented
as

$H^{{2\bullet}}(X,\mathbb{{\mathbb{Q}}})\cong\frac{\mathbb{{\mathbb{Q}}}[x_{1},\ldots,x_{r}]}{I+J}.$
(56)

The following example shows that the oriented graded AG algebras
$A_{1}={\mathbb{Q}}[x]/(x^{3})$, $A_{2}={\mathbb{Q}}[y,z]/(y^{2},z^{2}-yz)$,
$A_{3}={\mathbb{Q}}[r,s,t]/(rt,st,r^{2},s^{2},t^{2}+rs)=A_{4}$, and
$A_{5}={\mathbb{Q}}[u,v]/(u^{2},v^{2})$ are all birationally equivalent to one
another. This corresponds to the well known fact from algebraic geometry that
if $\mathbb{P}^{2}$ is blown up at two points and the proper transform of the
line joining the two points is blown down, the resulting surface is isomorphic
to $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We work through the details of this
example from the perspective of cohomological blow ups.

######  Example 9.1.

The algebraic varieties featured in this example correspond to the following
fans

$u_{1}$$u_{2}$$u_{3}$$X_{1}=\mathbb{P}^{2}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$X_{2}=\tilde{X}_{1}$$u_{1}$$u_{2}$$u_{3}$$u_{5}$$u_{4}$$X_{3}=\tilde{X}_{2}=X_{4}=\tilde{X}_{5}$$u_{1}$$u_{2}$$u_{5}$$u_{4}$$X_{5}=\mathbb{P}^{1}\times\mathbb{P}^{1}$

As a toric variety, $X_{1}=\mathbb{P}^{2}$ is defined by the fan $\Sigma$ with
ray generators $u_{1}=e_{1},u_{1}=e_{2},u_{3}=-e_{1}-e_{2}$, where
$e_{1}=(1,0)$ and $e_{2}=(0,1)$ are the standard basis vectors; see [11,
Example 12.4.2]. According to the formula above its cohomology algebra is

$A_{1}=H^{2\bullet}(X_{1})\cong\frac{\mathbb{Q}[x_{1},x_{2},x_{3}]}{(x_{1}x_{2}x_{3},x_{1}-x_{3},x_{2}-x_{3})}\cong\frac{\mathbb{Q}[x]}{(x^{3})}.$

The blow up $X_{2}=\tilde{X}_{1}$ of $X_{1}=\mathbb{P}^{2}$ at a point is
obtained by subdividing a cone of $\Sigma$ by adding the ray generated by
$u_{4}=-e_{2}$. Then

$A_{2}=H^{2\bullet}(X_{2})\cong\frac{\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]}{(x_{1}x_{3},x_{2}x_{4},x_{1}-x_{3},x_{2}-x_{3}-x_{4})}=\frac{{\mathbb{Q}}[x_{1},x_{2}]}{(x^{2}_{1},x_{2}^{2}-x_{2}x_{1})}.$
(57)

In terms of cohomological blow ups, take $T_{1}=\mathbb{Q}$ and let $\pi\colon
A_{1}\rightarrow T_{1}$ be the natural projection, with Thom class
$\tau_{1}=x^{2}$ and kernel $K_{1}=(x)$. The normal bundle of a point in
${\mathbb{P}}^{2}$ has total Chern class $c=1$, and hence
$f_{T_{1}}(\xi)=\xi^{2}$ and $f_{A_{1}}(\xi)=\xi^{2}+x^{2}$, and hence the
cohomological blowup of $A$ along $\pi_{1}$ is

$\tilde{A}_{1}=\frac{\mathbb{Q}[x,\xi]}{(x^{3},x\xi,\xi^{2}+x^{2})}=\frac{\mathbb{Q}[x,\xi]}{(x\xi,\xi^{2}+x^{2})}\cong\frac{{\mathbb{Q}}[x_{1},x_{2}]}{(x_{1}^{2},x_{2}^{2}-x_{1}x_{2})}=A_{2}$
(58)

where the last isomorphism sends $x\mapsto x_{2}$ and $\xi\mapsto
x_{1}-x_{2}$.

Next define $X_{3}=\tilde{X}_{2}$ as the blow up of $X_{2}$ obtained by adding
the ray generated by $u_{5}=-e_{1}$. Then its cohomology algebra is given by

$\displaystyle A_{3}=H^{2\bullet}(X_{3})=$
$\displaystyle\frac{\mathbb{Q}[x_{1},x_{2},x_{3},x_{4},x_{5}]}{(x_{1}x_{3},x_{1}x_{5},x_{2}x_{3},x_{2}x_{4},x_{4}x_{5},x_{1}-x_{3}-x_{5},x_{2}-x_{3}-x_{4})}$
$\displaystyle=$
$\displaystyle\frac{{\mathbb{Q}}[x_{1},x_{2},x_{3}]}{(x_{1}x_{3},x_{1}^{2},x_{2}x_{3},x_{2}^{2},x_{3}^{2}+x_{1}x_{2})}.$

Again let $T_{2}={\mathbb{Q}}$ and $\pi_{2}\colon A_{2}\rightarrow T_{2}$ be
the canonical projection, with Thom class $\tau_{2}=x_{1}x_{2}$ and kernel
$K=(x_{1},x_{2})$. Then we have $f_{A_{2}}(\xi)=\xi^{2}+x_{1}x_{2}$ and the
cohomological blow up of $A_{2}$ along $\pi_{2}$ is

$\tilde{A}_{2}=\frac{{\mathbb{Q}}[x_{1},x_{2},\xi]}{(x_{1}^{2},x_{2}^{2}-x_{2}x_{1},\xi
x_{1},\xi
x_{2},\xi^{2}+x_{1}x_{2})}\cong\frac{{\mathbb{Q}}[x_{1},x_{2},x_{3}]}{(x_{1}x_{3},x_{1}^{2},x_{2}x_{3},x_{2}^{2},x_{3}^{2}+x_{1}x_{2})}=A_{3}$

where the last isomorphism sends $x_{1}\mapsto x_{1}$, $x_{2}\mapsto
x_{2}-x_{3}$ and $\xi\mapsto x_{3}$.

Finally let $X_{5}={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, defined by the fan
$\Sigma^{\prime}$ with ray generators $u_{1}=e_{1}$, $u_{2}=e_{2}$,
$u_{4}=-e_{2}$, and $u_{5}=-e_{1}$, and let $X_{4}=\tilde{X}_{5}$ be the blow
up at a point obtained by adding the ray to $\Sigma^{\prime}$
$u_{3}=-e_{1}-e_{2}$. Since the fans for $X_{4}$ and $X_{3}$ are identical, it
follows that the toric varieties coincide as well. Hence their cohomology
algebras are

$A_{5}=H^{2\bullet}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})=\frac{\mathbb{Q}[x_{1},x_{2},x_{4},x_{5}]}{(x_{1}x_{5},x_{2}x_{4},x_{1}-x_{5},x_{2}-x_{4})}=\frac{{\mathbb{Q}}[x_{1},x_{2}]}{(x_{1}^{2},x_{2}^{2})}$

and

$A_{4}=A_{3}=H^{2\bullet}(X_{4})=H^{2\bullet}(X_{3})=\frac{{\mathbb{Q}}[x_{1},x_{2},x_{3}]}{(x_{1}x_{3},x_{1}^{2},x_{2}x_{3},x_{2}^{2},x_{3}^{2}+x_{1}x_{2})}\cong\frac{{\mathbb{Q}}[x_{1},x_{2},\xi]}{(x_{1}^{2},x_{2}^{2},x_{1}\xi,x_{2}\xi,\xi^{2}+x_{1}x_{2})}=\tilde{A}_{5}$

where the last isomorphism is the obvious $x_{1}\mapsto x_{1}$, $x_{2}\mapsto
x_{2}$, and $x_{3}\mapsto\xi$.

It follows that the oriented graded AG algebras $A_{1}$, $A_{2}$,
$A_{3}=A_{4}$, and $A_{5}$ are all birationally equivalent, corresponding to
the (strong) factorization of the birational map

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
58.25803pt\raise-13.10681pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.3886pt\raise-15.83783pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
17.6264pt\raise-26.58766pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
98.16739pt\raise-15.83783pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-17.6264pt\raise-32.57788pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{1}={\mathbb{P}}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
144.07677pt\raise-32.57788pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
46.3233pt\raise-32.57788pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
98.16739pt\raise-32.57788pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
144.07677pt\raise-32.57788pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}=X_{5}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Remark 9.2.

The cohomology algebra of the blow up of a smooth toric variety $X$ along a
smooth torus invariant subvariety $Y\subset X$ will always agree with the
Constructions 3.1, 4.2, or 6.2 of this paper since in that case the
restriction map $\pi^{*}\colon H^{2\bullet}(X)\rightarrow H^{2\bullet}(Y)$ is
always surjective. Indeed in that case, the associated fan of $Y$ corresponds
to a subfan of $X$, and the surjectivity follows from the combinatorial
presentation of cohomology algebras as in (56). The following examples shows
what can happen in cases where that restriction map is not surjective.

######  Example 9.3.

Let $\pi\colon
Y={\mathbb{P}}^{1}\times{\mathbb{P}}^{2}\hookrightarrow{\mathbb{P}}^{5}=X$ be
the Segre embedding. Then we have a short exact sequence of vector bundles on
$Y$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{*}\mathcal{T}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{Y/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\mathcal{T}_{Y}$ is the tangent bundle of $Y$, $\pi^{*}\mathcal{T}_{X}$
is the restriction of the tangent bundle of $X$ to $\pi(Y)$ and
$\mathcal{N}_{Y/X}$ is the normal bundle to $\pi(Y)\subset X$. If we identity
the cohomology algebras as the oriented graded AG algebras

$A=H^{2\bullet}(X)\cong\frac{{\mathbb{Q}}[x]}{(x^{6})},\ \
T=H^{2\bullet}({\mathbb{P}}^{1}\times{\mathbb{P}}^{2})\cong\frac{{\mathbb{Q}}[y,z]}{(y^{2},z^{3})}$

where $x$, $y$, and $z$ are the classes of a hyperplane in
$H^{2}({\mathbb{P}}^{5})$, and the factors $H^{2}({\mathbb{P}}^{1})$, and
$H^{2}({\mathbb{P}}^{2})$ of
$H^{2\bullet}({\mathbb{P}}^{1}\times{\mathbb{P}}^{2})\cong
H^{2\bullet}({\mathbb{P}}^{1})\otimes_{{\mathbb{Q}}}H^{2\bullet}({\mathbb{P}}^{2})$,
then the induced map $\pi^{*}\colon A\rightarrow T$ satisfies
$\pi^{*}(x)=y+z$. Note that $\pi^{*}$ is not surjective here. From the Euler
sequence we compute the total Chern classes
$c(\pi^{*}\mathcal{T}_{X})=(1+\pi^{*}(x))^{6}$ and
$c(\mathcal{T}_{Y})=(1+y)^{2}\cdot(1+z)^{3}$. It follows from the Whitney
product formula that the total Chern class for the normal bundle is

$\displaystyle
c(\mathcal{N}_{Y/X})=\frac{c(\pi^{*}\mathcal{T}_{X})}{c(\mathcal{T}_{Y})}=$
$\displaystyle\frac{\left(1+y+z\right)^{6}}{(1+y)^{2}\cdot(1+z)^{3}}$
$\displaystyle=\left(1+y+z\right)^{6}\cdot\left(1-y\right)^{2}\cdot\left(1-z+z^{2}\right)^{3}$
$\displaystyle=(6yz+3z^{2})+(4y+3z)+1$

and hence the Chern classes are

$\begin{cases}c_{1}(\mathcal{N}_{Y/X})=&4y+3z\\\
c_{2}(\mathcal{N}_{Y/X})=&6yz+3z^{2}.\\\ \end{cases}$

Hence if we blow up $X$ along $Y$ then, according to Equation (10), the
cohomology algebra of the exceptional divisor $\tilde{Y}$ is given by

$\tilde{T}=\frac{{\mathbb{Q}}[y,z,\xi]}{(y^{2},z^{3},\xi^{2}-(4y+3z)\xi+(6yz+3z^{2}))}\cong
H^{2\bullet}(\tilde{Y}).$

Moreover using the conditions of Theorem 3.14, we can derive a presentation of
the cohomology algebra of the blow up manifold $\tilde{X}$:

$\tilde{A}=\frac{{\mathbb{Q}}[x,\xi]}{\left(\xi^{3}-6x\xi^{2}+12x^{2}\xi-8x^{3},3\xi^{4}-9x\xi^{3}+6x^{2}\xi^{2}+4x^{3}\xi\right)}\cong
H^{2\bullet}(\tilde{X}).$ (59)

where the restriction map $\tilde{\pi}\colon\tilde{A}\rightarrow\tilde{T}$
defined by $\tilde{\pi}(x)=y+z$ and $\tilde{\pi}(\xi)=\xi$ has Thom class
$\tilde{\tau}=-\xi$, the blow up map $\beta\colon A\rightarrow\tilde{A}$
defined by $\beta(x)=x$ is injective with
$\beta(a_{soc})=\tilde{a}_{soc}=x^{5}$, and the Hilbert function satisfies
$H(\tilde{A})=H(A)+H(T)[1]=(1,2,3,3,2,1)$. Furthermore, a Macaulay2 [18]
calculation computes the Macaulay dual generator of $\tilde{A}$ as

$\tilde{F}=X^{5}-3X^{3}\Xi^{2}-10X^{2}\Xi^{3}-24X\Xi^{4}-48\Xi^{5}.$

Note that $\tilde{A}$ in (59) does not fit the model described by our
Construction 3.1; in particular the defining ideal of $\tilde{A}$ does not
contain any monic polynomial of degree $n=2$.

######  Remark 9.4.

The algebra $\tilde{A}$ computed in Equation (59) might be termed a
cohomological blow up along the _non-surjective_ map $\pi$. In that case $A$
and $T$ are both standard graded, and $\tilde{A}$ is too, but this need not
hold in general. For example, if we blow up $X={\mathbb{P}}^{8}$ along the
Segre embedding of $Y={\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$, then again we
have a non-surjective restriction map

$\pi\colon
A=H^{2\bullet}(X)={\mathbb{Q}}[x]/(x^{9})\rightarrow{\mathbb{Q}}[y,z]/(y^{3},z^{3})=H^{2\bullet}(Y)=T$

but the cohomology of the blow up of $X$ along $Y$, $\tilde{A}\cong
H^{2\bullet}(\tilde{X})$ has Hilbert function

$H(\tilde{A})=H(A)+H(T)[1]+H(T)[2]+H(T)[3]=(1,2,4,7,8,7,4,2,1),$

which implies that $\tilde{A}$ cannot be standard graded.

Motivated by these examples, we pose some problems for further research.

######  Problem 9.5.

Generalize Example 9.1 and find other algebras which are birationally
equivalent to $A_{0}={\mathbb{F}}[x]/(x^{d+1})$. Can one classify them?

######  Problem 9.6.

Generalize Example 9.3 and find a construction, similar in spirit to
Construction 3.1, for a cohomological blow up of an AG algebra $A$ along _any_
(i.e. possibly non-surjective) restriction map $\pi\colon A\rightarrow T$.
Does it have similar properties as the cohomological blow up along a
surjective map, i.e. flat family, strong Lefschetz, connected sum, minimal
generators?

##  Appendix A A Guide To Our Examples

Given below is a list of the examples in this paper together with a brief
description of the idea that example is attempting to illustrate.

  1. 1.

Example 3.3 shows that without further qualifications, Construction 3.1 can
produce non-Gorenstein, Gorenstein, or boundary-Gorenstein algebras, i.e. non-
Gorenstein algebras in the closure of the Gorenstein locus of the Hilbert
scheme of that Hilbert function.

  2. 2.

Example 3.12 shows that over non-algebraically closed fields, distinct
parameter values $\lambda$ can produce non-isomorphic cohomological blow up
algebras.

  3. 3.

Example 4.5 shows that
$\tilde{T}=R[\xi]/\operatorname{Ann}\left(h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)\right)$ is not necessarily a free extension over
$T=R/\operatorname{Ann}(G)$ for any choice of $h_{R}(\xi)\in R[\xi]$.

  4. 4.

Example 4.7 provides an algebra $\hat{A}_{MD}$ from Construction 4.2 which is
not a cohomological blow up.

  5. 5.

Example 5.3 shows a cohomological blow up as a connected sum.

  6. 6.

Example 5.7 shows a cohomological blow down of Hilbert function
$H(A)=(1,3,6,3,1)$ as a connected sum.

  7. 7.

Example 5.8 shows that the cohomological blow up $\hat{A}$ may be standard
graded even if $A$ is not.

  8. 8.

Example 6.1 gives a cohomological blow up that is a complete intersection.

  9. 9.

Example 6.7 gives ideals $I$ and $(I:\tau)$ with homology groups $H$ and
$H^{\prime}$ equal to zero, but where the cohomogical blow up ideal $\hat{I}$
is not generated by a regular sequence.

  10. 10.

Example 6.9 gives examples of exact pairs of zero divisors.

  11. 11.

Example 6.14 gives a CI with exact zero divisors which is not a BUG.

  12. 12.

Example 7.3 shows a compressed AG algebra of socle degree $5$ and embedding
dimension $3$ can be a cohomological blow up of a standard graded AG algebra.

  13. 13.

Example 8.6 shows if $A$ and $T$ have SLP over ${\mathbb{F}}_{p}$, then
$\hat{A}$ may fail SLP.

  14. 14.

Example 8.7 gives algebras in which $\hat{A}$ and $\hat{T}$ both have SLP, but
the cohomological blow down $A$ does not have SLP.

  15. 15.

Example 8.8 shows that if $A$ and $T$ both have WLP (but fail SLP), then
$\hat{A}$ may fail WLP.

  16. 16.

Example 9.1 gives a geometric example of a strong factorization of a
birational map between toric varieties which yields several birationally
equivalent AG algebras, i.e. a sequence of AG algebras in which each one is
either a cohomological blow up or blow down of the previous.

  17. 17.

Example 9.3 computes a presentation of the blow up of ${\mathbb{P}}^{5}$ along
the Segre embedding
${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}\hookrightarrow{\mathbb{P}}^{5}$ in
which case the restriction map on cohomology is not surjective.

######  Acknowledgment.

The authors are grateful to the series of annual Lefschetz Properties In
Algebra and Combinatorics conferences and workshops, some of which we each
participated in, beginning with Göttingen (2015), where the fifth author
proposed the Bold Conjecture 6.15, and followed by meetings at Mittag Leffler
(2017), Levico (2018), CIRM Luminy (2019), and Oberwolfach (2020).

The second author was partially supported by CIMA – Centro de Investigação em
Matemática e Aplicações, Universidade de Évora, project UIDB/04674/2020
(Fundação para a Ciência e Tecnologia). The fourth author is supported by NSF
grant DMS–2101225.

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