Welcome to Procurement Paradise, where your bottom line is our top priority. Tropic is a cost-savings platform that automatically finds ways for businesses to reduce and control costs. By combining procurement workflows, supplier management, and benchmark data together in one place, Tropic makes savings opportunities easy to find and act on.

Tropic

HackerNoon Reporter: Please tell us briefly about your background. After failing as a novelist (more than once), I got into software sales. I was one of first 20 employees at a martech startup call BounceX (now Wunderkind) and grew into VP. I met my co-founder Justin (VP of Ops), so the two of us learned how to scale a company from 20 people to 350 people and ~$80M in revenue. After that I was a Global Business Manager at Microsoft with a huge territory selling into 200K-person banks. At the end of my tenure there, I realized that I’d sold to companies of every size and stage and EVERYONE hated buying and managing software at EVERY LEVEL - too confusing, too opaque, salespeople suck, etc. Seemed like a natural opportunity fix a problem, since literally everyone was mad about it. What's your startup called? And in a sentence or two, what does it do? We’re called Tropic; we make companies fall in love with procurement. On our purchasing platform CFOs can approve, manage, renew and all of their SaaS tools in one place. They can also activate , where Tropic resources renegotiate contracts for clients to reduce SaaS expenses, all without a single call or email to a vendor. Assisted Purchasing What is the origin story? After I left BounceX for Microsoft, I saw a client lay off 2,000 people because they couldn’t afford to pay their SaaS vendors. I quit that day, recognizing that software management is a challenge for every company of every size. I sort of felt like I’d been one of the “bad guys” all along without realizing it and resolved to focus my career on solving problems for the buyer. What do you love about your team, and why are you the ones to solve this problem? I’ve seen every type of software sale at every level in the market. My co-founder Justin ran RFPs while he was consulting at Bain and built our procurement process from scratch at BounceX. Together, we’ve looked at this problem from every angle on both sides. We’ve also somehow managed to hire some of the most brilliant people in the industry to solve this problem, including sourcing people from Twitter, Tesla, Carta, Boeing, and Oracle, and they all bring a unique and brilliant perspective to the business. If you weren’t building your startup, what would you be doing? Building a different startup :) At the moment, how do you measure success? What are your core metrics? Well, as a recovering sales rep, revenue is still #1. We also look at total spend managed for clients ( ), total vendors in our database ( ), churn (only one logo so far…), average savings per customer ( ) $200M+ 1,200+ 23% What’s most exciting about your traction to date? We got to in our first 12 months in the market, and of that was just last quarter. $2.6M $1.1M What technologies are you currently most excited about, and most worried about? And why? Most excited: CRISPR because… no genetic diseases ever again? Most worried: AI - while real AI doesn’t exist yet, credible researchers predict a singularity in our lifetimes. Skynet, anyone? What drew you to get published on HackerNoon? What do you like most about our platform? We love HackerNoon’s no-nonsense approach and engaged audience of people just like us. What advice would you give to the 21-year-old version of yourself? Never expect failure, but ALWAYS embrace it. Failure is on your critical path. You need it. No one got anywhere by taking the safe option anyway. What is something surprising you've learned this year that your contemporaries would benefit from knowing? The book “The Exorcist” is actually better than the movie (which I also loved) and is totally worth reading. Vote for Tropic as the . startup of the year from New York, NY

"Software Management Is A Challenge For Every Company Of Every Size," David Campbell, Tropic CEO

Author:
(1) CALLA TSCHANZ. Table of Links Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References 2. Background on tropical perspective We briefly introduce here the language of tropical and logarithmic geometry in the context of this problem. For more details on the contents of this section, see the article [Log], lecture notes [Ran22a], as well as the first section of [MR20]. 2.1 Tropicalisation and expansion Subdivisions of the tropicalisation define expansions of X. In the following, we will want to study possible birational modifications of the scheme X around the divisor D. In the tropical language, these are expressed as subdivisions. A subdivision of the tropicalisation defines a birational modification of X in the following way. The subdivision 2.2 Maulik-Ranganathan construction We will briefly recall some key points of [MR20]. The aim of their work is to study the moduli space of ideal sheaves of fixed numerical type which meet the boundary divisor transversely. Some key motivations for the study of such an object come from enumerative geometry. For example, a common method used to address problems of curve counting in a given smooth variety is to degenerate this variety to a singular union of simpler irreducible components. The property of transversality is then crucial to ensure that all interesting behaviour of the ideal sheaves on the degenerate object occurs with support in the interior of the simpler irreducible components, which allows us to study it with more ease. One of the main difficulties with this approach is that often, as in this setting, the space of transverse ideal sheaves with respect to D is non-compact. Constructing the appropriate compactification will yield a space which is flat and proper over C. In [MR20], Maulik and Ranganathan formulate the Donaldson-Thomas theory of the pair (X, D), starting by constructing compactifications of the space of ideal sheaves in X transverse to D. We discuss [MR20] specifically with respect to the case which interests us here, namely that of a degeneration X ! C as described above, where we seek to study the moduli space of ideal sheaves with fixed constant Hilbert polynomial m, for some m ∈ N with respect to the boundary divisor D = X0. The key idea is to construct the tropicalisation of X, denoted ΣX , and a corresponding tropicalisation map, which is used to understand how to obtain the desired transversality properties in our compactifications. Existence and uniqueness of transverse limits. Maulik and Ranganathan introduce notions of dimensional transversality and strong transversality, which, in the specific case of Hilbert schemes of points, happen to be equivalent to Li-Wu stability (see Section 5.3 for a definition of this stability condition). In general for higher dimensional subschemes this will not be the case, however. This operation results in non-uniqueness, as we are making a choice of polyhedral subdivision and there is in general no canonical choice. The addition of these tube vertices in the tropicalisation means that there are more potential components in each expansion, which interferes with the previously set up uniqueness results. Indeed, recall that trop(Z ◦ ) gave us exactly the right number of vertices in the dual complex in order for each family of subschemes Z ◦ ⊂ X◦ to have a unique limit representative. Therefore, to reflect this, Donaldson-Thomas stability asks for subschemes to be DT stable if and only if they are tube schemes precisely along the tube components. We say that a 1-dimensional subscheme is a tube if it is the schematic preimage of a zerodimensional subscheme in D. In the case of Hilbert schemes of points, this condition will translate simply to a 0-dimensional subscheme Z being DT stable if and only if no tube component contains a point of the support of Z and every other irreducible component expanded out by our blow-ups contains at least one point of the support of Z. Maulik and Ranganathan define a subscheme to be stable if it is strongly transverse and DT stable. For fixed numerical invariants the substack of stable subschemes in the space of expansions forms a Deligne-Mumford, proper, separated stack of finite type over C. Comparison with the results of this paper. The construction we present in this paper has the surprising property that we do not need to label any components as tubes in order for the stack of stable objects we define to be proper. This is an artifact of the specific choices of blow-ups to be included in our expanded degenerations. The work of Maulik and Ranganathan shows us that this is not expected in general. As mentioned in Section 1.3, we will discuss in an upcoming paper how to construct proper stacks of stable objects in cases where different choices of expansions are made and it becomes necessary for us as well to introduce a Donaldson-Thomas stability condition. This paper is available on arxiv under CC 4.0 license. Author: (1) CALLA TSCHANZ. Author: Author: (1) CALLA TSCHANZ. Table of Links Abstract and Intro Background on tropical perspective The expanded construction GIT stability Stack perspective The canonical moduli stack References Abstract and Intro Abstract and Intro Background on tropical perspective Background on tropical perspective The expanded construction The expanded construction GIT stability GIT stability Stack perspective Stack perspective The canonical moduli stack The canonical moduli stack References References 2. Background on tropical perspective We briefly introduce here the language of tropical and logarithmic geometry in the context of this problem. For more details on the contents of this section, see the article [Log], lecture notes [Ran22a], as well as the first section of [MR20]. 2.1 Tropicalisation and expansion Subdivisions of the tropicalisation define expansions of X. In the following, we will want to study possible birational modifications of the scheme X around the divisor D. In the tropical language, these are expressed as subdivisions. A subdivision of the tropicalisation defines a birational modification of X in the following way. The subdivision 2.2 Maulik-Ranganathan construction We will briefly recall some key points of [MR20]. The aim of their work is to study the moduli space of ideal sheaves of fixed numerical type which meet the boundary divisor transversely. Some key motivations for the study of such an object come from enumerative geometry. For example, a common method used to address problems of curve counting in a given smooth variety is to degenerate this variety to a singular union of simpler irreducible components. The property of transversality is then crucial to ensure that all interesting behaviour of the ideal sheaves on the degenerate object occurs with support in the interior of the simpler irreducible components, which allows us to study it with more ease. One of the main difficulties with this approach is that often, as in this setting, the space of transverse ideal sheaves with respect to D is non-compact. Constructing the appropriate compactification will yield a space which is flat and proper over C. In [MR20], Maulik and Ranganathan formulate the Donaldson-Thomas theory of the pair (X, D), starting by constructing compactifications of the space of ideal sheaves in X transverse to D. We discuss [MR20] specifically with respect to the case which interests us here, namely that of a degeneration X ! C as described above, where we seek to study the moduli space of ideal sheaves with fixed constant Hilbert polynomial m, for some m ∈ N with respect to the boundary divisor D = X0. The key idea is to construct the tropicalisation of X, denoted ΣX , and a corresponding tropicalisation map, which is used to understand how to obtain the desired transversality properties in our compactifications. Existence and uniqueness of transverse limits. Maulik and Ranganathan introduce notions of dimensional transversality and strong transversality, which, in the specific case of Hilbert schemes of points, happen to be equivalent to Li-Wu stability (see Section 5.3 for a definition of this stability condition). In general for higher dimensional subschemes this will not be the case, however. This operation results in non-uniqueness, as we are making a choice of polyhedral subdivision and there is in general no canonical choice. The addition of these tube vertices in the tropicalisation means that there are more potential components in each expansion, which interferes with the previously set up uniqueness results. Indeed, recall that trop(Z ◦ ) gave us exactly the right number of vertices in the dual complex in order for each family of subschemes Z ◦ ⊂ X◦ to have a unique limit representative. Therefore, to reflect this, Donaldson-Thomas stability asks for subschemes to be DT stable if and only if they are tube schemes precisely along the tube components. We say that a 1-dimensional subscheme is a tube if it is the schematic preimage of a zerodimensional subscheme in D. In the case of Hilbert schemes of points, this condition will translate simply to a 0-dimensional subscheme Z being DT stable if and only if no tube component contains a point of the support of Z and every other irreducible component expanded out by our blow-ups contains at least one point of the support of Z. Maulik and Ranganathan define a subscheme to be stable if it is strongly transverse and DT stable. For fixed numerical invariants the substack of stable subschemes in the space of expansions forms a Deligne-Mumford, proper, separated stack of finite type over C. Comparison with the results of this paper. The construction we present in this paper has the surprising property that we do not need to label any components as tubes in order for the stack of stable objects we define to be proper. This is an artifact of the specific choices of blow-ups to be included in our expanded degenerations. The work of Maulik and Ranganathan shows us that this is not expected in general. As mentioned in Section 1.3, we will discuss in an upcoming paper how to construct proper stacks of stable objects in cases where different choices of expansions are made and it becomes necessary for us as well to introduce a Donaldson-Thomas stability condition. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Expansions for Hilbert Schemes: Background on Tropical Perspective

হিলবার্ট স্কিমগুলির সম্প্রসারণ: গ্রীষ্মমন্ডলীয় দৃষ্টিভঙ্গির পটভূমি

Erweiterungen für Hilbert-Schemata: Hintergrundinformationen zur tropischen Perspektive

Ampliaciones de los esquemas Hilbert: antecedentes de la perspectiva tropical

Extensions des projets Hilbert : contexte sur la perspective tropicale

हिल्बर्ट योजनाओं के लिए विस्तार: उष्णकटिबंधीय परिप्रेक्ष्य पर पृष्ठभूमि

ヒルベルトスキームの拡張：熱帯の視点の背景

힐베르트 방식의 확장: 열대 관점의 배경

Expansões para Esquemas de Hilbert: Antecedentes da Perspectiva Tropical

Расширения схем Гильберта: предыстория тропической перспективы

Hilbert Şemaları için Genişletmeler: Tropikal Perspektifin Arka Planı

Mở rộng các sơ đồ Hilbert: Bối cảnh về quan điểm nhiệt đới

希尔伯特方案的扩展：热带视角的背景

Tropicalization is a process by which a company adds local flavor to its product, giving it a more regional feel. Tropicalization is often used in digital products to appeal to local customers and make them more easily adaptable to the market. Unfortunately, many companies get it wrong, which leads them to spend money on products that don't meet their needs or expectations as consumers. In this article, I'll talk about three main reasons why most tropicalization fails in LatAm and how companies can avoid these mistakes and become more successful in developing tailor-made digital products for Latin America! But what happens when tropicalization fails? There are three main reasons that most tropicalization fails in LatAm. Companies need to understand their market: It's not enough to sell your product in a new country; you need to understand the needs of that specific market. For example, if you're launching a new app in Brazil that doesn't support Portuguese, people won't buy it because they can't use it--even though many people speak English there! This is an example of how companies make assumptions about what local populations want without talking with them first (or at all). Companies need to understand those needs: Even if you research your target audience, it may not be enough because cultural differences can make even basic things like communication difficult between companies and consumers. This means there needs to be more than one person involved in decision-making processes explicitly related to tropicalization; instead, multiple stakeholders should collaborate across departments, so everyone understands each other well enough to work together effectively towards a shared agenda. First, most companies need help understanding the region's audience drivers and basic needs. It's crucial for any company looking to succeed in Latin America (or any market) to understand what makes its customers tick and the market dynamics. An excellent place to start is with local market research--conducting interviews with potential users or customers, surveys and polls, etc.--but understanding local culture also plays an essential role in getting this right. For example: In Brazilians' minds, there's no such thing as bad weather; if you're complaining about how hot it is outside, you're missing out on all of life's opportunities! Argentinians love talking about themselves, but they'll never tell anyone directly what they want or need; instead, they'll hint at those things through subtle hints ("Oh my gosh, I'm so busy these days" translates roughly into "I need more help around here!"). In another example, a Peruvian business newspaper article says, "Colombians look for the greater good, instead of a personal agenda" when comparing the two countries' politics. This could mean that when you market to the Peruvian market, the campaign needs to focus more on the YOU and the Colombian market on the WE. Second, companies often misuse products from other markets in LatAm and do not adapt them to be relevant for the country. The product should be localized and adapted to the local market. You must take into account the local culture, needs of your customers, and language (languages in LatAm are very diverse). Do you know how to say popcorn in Spanish? So, if you are going to have an American English-speaking media buyer translate a word to Spanish and use the same in each ad, you will be making a mistake. Finally, cultural differences can make it hard for companies to develop a product for the market that is suitable for consumers there. For example, in Brazil and Colombia, people value their privacy more than in other countries; they're less likely to share personal information on social media or download apps with permissions like location tracking. This makes it more challenging for companies like Facebook or Google, who rely heavily on user data as part of their business model--and many tech startups have failed because of this issue alone. In a personal experiment: We tryed a social media challenge in Colombia for Fandango Latam that definitely would have worked in Peru, but failed in Colombia, because people didn't want to overshare in social media. The success of any digital product depends on its ability to meet local needs and adapt well to local customers' preferences. To do this, it's essential that you understand the culture of your target market--and more importantly, get under its skin by understanding what makes them tick. This can be difficult for companies who are used to working in their home markets or have limited knowledge of other cultures outside their own. If you're thinking about launching a new app in Latin America but haven't researched how people behave here yet, there's no time like now! You need all the information possible if you want that product to succeed here (or at least significantly increase its chances). My recommendation here is to surround yourself with people that understand the market. You need a Latinx leading a Latam initiative to win this region. Conclusion The success of any digital product depends on its ability to meet local needs and adapt well to local customers' preferences. Tropicalization is a great way to do this, but it requires careful planning and execution so that the result will be something people want. The three main reasons that most tropicalization fails in LatAm are because companies fail to understand the region's audience drivers and basic needs; misuse products from other markets instead of adapting them for the area; or cannot develop a product suitable for consumers there because of cultural differences between countries or regions within them

Why Tropicalization Often Fails in LatAm and What Needs to be Done for Success in Digital Products

Por qué la tropicalización suele fallar en LatAm y qué se debe hacer para tener éxito en los productos digitales

Pourquoi la tropicalisation échoue souvent en Amérique latine et ce qui doit être fait pour réussir dans les produits numériques

ट्रॉपिकलाइज़ेशन लैटम में अक्सर विफल क्यों होता है और डिजिटल उत्पादों में सफलता के लिए क्या करने की आवश्यकता है

中南米で熱帯化が失敗することが多い理由と、デジタル製品で成功するために必要なこと

Por que a tropicalização geralmente falha na América Latina e o que precisa ser feito para ter sucesso em produtos digitais

Tại sao nhiệt đới hóa thường thất bại ở LatAm và những việc cần làm để thành công trong các sản phẩm kỹ thuật số

为什么热带化在拉丁美洲经常失败以及数字产品成功需要做些什么

The HackerNoon editorial team has launched this interview series with to celebrate their achievements and share their struggles. We need more women in technology, and by sharing stories, we can encourage many girls to follow their dreams. ! women in tech Share your story today Tell us about yourself! Hi. I'm Ellen Cibula; my career has taken a non-linear path to where I am today. I started as a chemical analyst at a laboratory for a chemical demilitarization site (Johnston Atoll Chemical Agent Demilitarization System (JACADS)). JACADS was on a remote island in the central Pacific. I was in charge of maintaining all of the non-agent air monitoring equipment at the plant and around the island. I stayed on this tiny island for 13 years, leaving only when the project was completed and I was laid off. After a few years back at my new home stateside, I decided I needed to start a new chapter. I have always been interested in computers and programming, so I got a job as a computer programmer, with little experience. I quickly worked my way up, from beginner coder to senior software engineer, lead software engineer, and solutions architect. I enjoyed programming - who wouldn't want to solve puzzles all day and get paid for it? But I detested the non-effective meetings, constantly changing requirements, and long hours. I had planned to quit my job but was laid off just a few days before I could take that action for myself. After a few years of recovering from burnout, I decided to take matters into my own hands and start my own business. I am now a full-time blogger writing about , payments, and security, but who knows what the future will hold - by running my own business, I am free to pivot to wherever I need to. FinTech Why did you choose this field in the first place? My parents both worked in science fields, my father worked at (and retired from) NASA and my mother was a nurse, and education was very important to them both. My father would take us to his laboratories at both work and the university where he taught classes, and I remember helping set up his equipment and assisting with the experiments. He would help me with my science projects, helping me grow my interest in science from an early age. My mother started college while I was in elementary school, and I used to spend my spare time perusing her nursing textbooks. Many of these were probably not appropriate for my age, but I was intrigued all the same. I would also spend time with her at the hospital after she became a nurse, talking with the other employees. Although I enjoyed all areas of the hospital, the laboratory was one of my favorite areas. There was no doubt in my mind I was going to college and would study science. I started in Chemistry, mostly because my parents knew one of the professors and he became my advisor, but I wasn't really into Chemistry (the irony is not lost on me that my first job was in a chemical laboratory). I switched my major to Biology, dreaming of becoming a marine biologist. I also took various math classes along the way, just because I enjoyed them. So I ended up with a B.S. degree in Biology and Math, with a minor in Chemistry. While I was on the island, I was able to complete my M.S. degree in Computer Systems Management, which was a mesh between Computer Science and Management. What tech are you most excited/passionate about right now and why? I am excited about electric vehicles - I like the impact that these can have on the environment. But I also like the technical aspect of having a quiet car that does not use gasoline. I also am excited about smart homes and home automation. I guess I'm lazy and it just thrills me to tell the AI assistant to turn on or off the lights (or whatever), without having to get up to turn on the light switch. I also am intrigued by the DeFi trend - anytime there is an opportunity to shake up a core industry, I think it tends to move society forward. It is too easy for society to become entrenched in how things have always been; change is hard, but it is the only constant in our lives. What tech are you most worried about right now and why? My biggest concerns revolve around cybersecurity and privacy. As a former developer in the financial sector, I know how critical it is to build security in systems from the beginning; it is extremely difficult to enhance security after the fact. I also know that some developers are not as well versed in security best practices as they should be (as evidenced by SQL injection still in the top 3 of the OWASP Top 10 web application security risks. Still. Easy to mitigate, but it is still there. At least it dropped from #1 to #3. Sigh.). This worries me because so much of our personal and financial information is out there, and if the developers of the systems that contain our data don't understand security, then all of that information is at risk. What are your hobbies and interests outside of tech? I love swimming; I try to swim two miles, three days a week. I also enjoy camping (well, glamping, in an RV with power and water), hiking, and anything that gets me away from the computer and into the outdoors. I also love to read, mostly sci-fi and fantasy, but occasionally a random math or science book. My wife and I have 2 rescue pets - a yellow lab/border collie mix (Sprocket) and a newly-rescued former tom cat (Gryffin) who is figuring out his place in the family and learning to live his best life. They keep us entertained. Let's talk about breaking the glass ceiling. What were the biggest challenges you faced as a woman in tech, and how did you deal with them? For most of my career, I was always in the minority; on Johnston Island, the ratio of men to women was about 10:1 when I got there, and about 4:1 when I left. The ratio of women in the laboratory was higher, but the plant was very similar to a construction site, with the same cat calls and innuendo whenever a woman would walk through the plant. When I became a programmer, the ratio of men to women was even higher; I was often the only woman on the team, especially when I went to meetings with C-level executives. Also, as a lesbian in the tech world, again, I was usually the only member representing in the IT department; at one job, there was one gay guy in product who was out - and he was the only other one I knew about. Maybe there were more in the closet as much as I was, but I wasn't aware of them. I was not out during my programming career. I live in a religious, conservative state and I was worried that I would lose my job. Sometimes conversations with others became torture, trying to talk about my wife without using pronouns; it was "my spouse this" or a generic "they". I think only two coworkers clued in with my weird phrasing, but they could have cared less when I came out to them privately. It was difficult because I never did anything with others outside of work. After all, it would have been hard to have a personal relationship with someone without informing them of my personal life. I dealt with these challenges by working harder than almost anyone else in the department. I was usually the first to arrive at the office and the last to leave. There was nothing more frustrating than working on a critical, time-sensitive project, and turn around to ask a coworker a question, and realizing I was the only one who was still at the office. Fortunately, I worked for managers that appreciated my work ethic and regularly promoted me. Any questionable misogynistic story/situation you faced/handled, and you want to share with the HackerNoon Fam? There were several instances of misogyny and sexual harassment when I was on the island. One, in particular, stands out; I was calibrating some equipment that was located in an isolated corner in the plant; there was a group of men relaxing in an adjacent area; they were telling sex stories, complete with explicit braggadocio about their encounters. I was able to hear all of it, but they did not realize I was there (even though I had to walk through them to get to my work area). Eventually, they realized I was there and changed their conversation. Before I left, I talked to their supervisor (who was with them) and explained that I did not appreciate having to hear that while I was working. He expressed remorse and said it wouldn't happen again. Again, I had to walk through them to leave, which was a whole new level of awkwardness. Later that day, I relayed the event to a friend, who insisted I tell my manager. I did not want to, because the island population was very small and I didn't want everyone to think I was a "tattle tale". Against my better judgment, I told my manager, insisting that he not do anything about it unless it happened again. Well, he went to the lab manager, who went to the plant manager, and by the end of the day, everyone at the plant knew I "tattled." I now had a reputation as someone not to be trusted if I overheard something. For a while, it was very uncomfortable working there. I ended up wearing earplugs whenever I walked around the plant, just so I didn't have to hear the cat calls, wolf whistles, and other inane comments as I did my job. Another incident occurred when I was working as a developer. I remember when I started a new job, one of my new male colleagues, also a new hire, kept telling me stories about another woman programmer that he worked with - she couldn't do her job, she would just put her head on the desk and sleep, she never finished her assignments - and he hoped I wouldn't be like her. He and I butted heads most of our time there together; he was condescending to me, insinuating that there was no way I could be a good programmer because I was a woman. He never said that out loud, but that was the undercurrent based on his comments to me and other coworkers. It didn't help when I was promoted over him. The sad part was that we shared some of the same interests and could have interesting conversations. He just didn't seem to be comfortable that I was doing the same job he was, but I was doing it better. What was the biggest setback/failure that you faced, and how did you manage it? I would say my biggest failure was not being able to be my true self at work; being closeted was difficult, to say the least. I was often promoted, but only in tech roles, never as a manager. Was it because I could never make personal connections with others at work? I don't know. What's your biggest achievement that you're really proud of? Professionally, my biggest achievement was to help rid the world of chemical weapons. Working at JACADS was one of the best things that could have happened to me because among other reasons, I met my wife there. I am also really proud that during my software career, I was promoted quickly and above others who had been developers for years. I credit that to my ability to learn quickly and apply what I learned to the project at hand. I've always said I was able to see the big picture while being detail-oriented. I also was able to understand the business requirements behind the project and steer the project toward those requirements. Many of my coworkers were wowed by the latest development paradigm and neglected to achieve the project goals. Personally, my biggest achievement was that my wife and I were able to both retire early. Much of that was because we spent so many years on an island where everything was provided for us (food, housing, activities) and we had nowhere to spend our money. We are both naturally frugal, though, and when we came back stateside, we were able to live well within our salaries; we also hired some top-notch financial planners that have helped us realize our dreams of retiring. Being retired has opened many doors for us, from starting our own business to having the time to pursue whatever activities we want. And I finally have the time to see what exciting things I can do with that Raspberry Pi that is sitting behind my monitors. In your opinion, why do we see this huge gender gap in the tech industry, and how can we reduce it? Everything starts with education, but our education system (in the US) is flawed; it is funded based on property taxes, but poorer areas have fewer property taxes, therefore, less funding. This limits the opportunities of the children in these schools, and because ~50% are girls, these girls are limited even more. We also have a cultural problem where girls are not steered into STEM/STEAM classes. I include the "arts" in STEAM, because, as a lover of all things math, I understand the importance of music in developing math skills. Sadly, many girls are not even exposed to STEM/STEAM topics. I get frustrated that we are still selling "girl" toys (dolls, makeup, purses) and "boy" toys (robots, RC cars/trucks, rockets). We should be selling just . I make a conscious effort to buy educational (but fun) genderless toys for the children in my extended family. toys However, there are also problems once women get into the workforce: lack of comprehensive health care, no cohesive national maternity leave policy, inflexible work hours, and misogynistic attitudes that women can't be as good as, or successful as, men in leadership roles. Many women I know have had to quit their jobs or reduce their work hours while they raise their children. This puts them behind in their careers when they return to full work after their children are older. Without having the ability to work flexible hours or hybrid work (remote + office), they have few options. During the pandemic, many worked remotely from home, but now many organizations are pulling workers back into the office, not even allowing a hybrid work environment. This will again, put many women in a position where they have to choose between a career and their family. We (as a people) can and should do better in finding ways to provide work environments that work for all who wish to pursue careers. Who is your tech idol? Why? I have always admired Sally Ride, the first American woman in space and the first LGBTQ+ NASA astronaut. Even with her success in the Space Shuttle program, she felt she needed to hide a part of who she was. I can relate. I am also grateful for Reshma Saujani, founder and former CEO of Girls Who Code. Providing opportunities to girls to learn how to code is so critical in breaking the gender gap in technology. Every business sector is powered by applications, and those applications need software developers. Why not fill those positions with women who code? and changing the image of what a computer programmer looks like Do you have any advice for aspiring girls who want to join the field? Just go for it. There will be pushback but do it anyhow. Read everything you can get your hand on (even if it is a college nursing book and you are only in 5th grade). Be curious. Try something new. You will fail, and it's okay. Every time you fail you have an opportunity to learn something. Don't listen to those making fun of you because you want to be a programmer, scientist, mathematician, whatever. Just do it (apologies to Nike).

From a Tropical Island to FinTech: Interview with Ellen Cibula

Author:
(1) CALLA TSCHANZ. Table of Links Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References 5. Stack perspective 5.1 Stacks and stability conditions Before we describe the expanded degenerations as stacks, we comment on the role of stacks in this problem and on the stability conditions defined in Section 5.3. Base change. Our aim is to construct degenerations of Hilbert schemes of points as good moduli spaces. In the proofs of Propositions 6.1.2 and 6.1.4, we will use the valuative criterion to prove universal closure and separatedness. We will see that this argument holds only up to base change, which is why it is necessary for us to work with stacks instead of schemes. In the following sections, when studying quotient stacks, we will therefore want to consider the sublocus of the GIT stable stable locus containing only length m zero-dimensional subschemes which are smoothly supported in a given fibre of X[n]. Building a compactification in which limits are represented by smoothly supported subschemes will also be useful for future applications as it allows us to break down the problem of a Hilbert scheme of m points on a singular surface into products of Hilbert schemes of fewer than m points on smooth components. Definition 5.1.1. We say that a fibre in some expanded degeneration X[n] has base codimension k if exactly k basis directions vanish at this fibre. This is independent of the value n. Making the expanded degenerations large enough. Finally, if we construct a unique GIT quotient in which not all limit subschemes are smoothly supported then the limits given by orbit closures containing only subschemes with singular support will not lie in a fibre of the expected base codimension. This gives an intuition that the degeneration we have chosen is too small. That being said, it can be useful to think about this GIT quotient if what we are trying to do is simply to resolve singularities in a way that preserves some good properties of the space, e.g. in the context of constructing minimal models for type III degenerations of Hilbert schemes of points on K3 surfaces. 5.2 Expanded construction for stacks 5.3 Stability conditions. We remark that since the smooth locus of the fibres of X[n] is G-invariant, restricting the functor to this locus preserves the G-invariance. The restrictions of the semistable and stable loci to the loci of smoothly supported subschemes therefore yield G-invariant open subschemes. We have the following inclusions: Li-Wu stability. We recall here the notion of stability used in [LW15], in order to compare it with the GIT stability and construct an appropriate stability condition for this case. Modified GIT stability. As stated above, we only want to allow length m zerodimensional subschemes to be stable if their support lies in the smooth locus of a fibre. However, restricting the GIT stability condition to this locus makes the space of stable subschemes no longer uiversally closed. Indeed, there is no single GIT condition which can represent all desired length m zero-dimensional subschemes as smoothly supported subschemes. We must therefore define a modified GIT stability condition which patches together several GIT stability conditions in order to obtain the desired stable locus. This paper is available on arxiv under CC 4.0 license. Author: (1) CALLA TSCHANZ. Author: Author: (1) CALLA TSCHANZ. Table of Links Abstract and Intro Background on tropical perspective The expanded construction GIT stability Stack perspective The canonical moduli stack References Abstract and Intro Abstract and Intro Background on tropical perspective Background on tropical perspective The expanded construction The expanded construction GIT stability GIT stability Stack perspective Stack perspective The canonical moduli stack The canonical moduli stack References References 5. Stack perspective 5.1 Stacks and stability conditions Before we describe the expanded degenerations as stacks, we comment on the role of stacks in this problem and on the stability conditions defined in Section 5.3. Base change . Our aim is to construct degenerations of Hilbert schemes of points as good moduli spaces. In the proofs of Propositions 6.1.2 and 6.1.4, we will use the valuative criterion to prove universal closure and separatedness. We will see that this argument holds only up to base change, which is why it is necessary for us to work with stacks instead of schemes. Base change In the following sections, when studying quotient stacks, we will therefore want to consider the sublocus of the GIT stable stable locus containing only length m zero-dimensional subschemes which are smoothly supported in a given fibre of X[n]. Building a compactification in which limits are represented by smoothly supported subschemes will also be useful for future applications as it allows us to break down the problem of a Hilbert scheme of m points on a singular surface into products of Hilbert schemes of fewer than m points on smooth components. Definition 5.1.1. We say that a fibre in some expanded degeneration X[n] has base codimension k if exactly k basis directions vanish at this fibre. This is independent of the value n. Definition 5.1.1. Making the expanded degenerations large enough. Finally, if we construct a unique GIT quotient in which not all limit subschemes are smoothly supported then the limits given by orbit closures containing only subschemes with singular support will not lie in a fibre of the expected base codimension. This gives an intuition that the degeneration we have chosen is too small. That being said, it can be useful to think about this GIT quotient if what we are trying to do is simply to resolve singularities in a way that preserves some good properties of the space, e.g. in the context of constructing minimal models for type III degenerations of Hilbert schemes of points on K3 surfaces. Making the expanded degenerations large enough. 5.2 Expanded construction for stacks 5.3 Stability conditions. We remark that since the smooth locus of the fibres of X[n] is G-invariant, restricting the functor to this locus preserves the G-invariance. The restrictions of the semistable and stable loci to the loci of smoothly supported subschemes therefore yield G-invariant open subschemes. We have the following inclusions: Li-Wu stability. We recall here the notion of stability used in [LW15], in order to compare it with the GIT stability and construct an appropriate stability condition for this case. Li-Wu stability. Modified GIT stability. As stated above, we only want to allow length m zerodimensional subschemes to be stable if their support lies in the smooth locus of a fibre. However, restricting the GIT stability condition to this locus makes the space of stable subschemes no longer uiversally closed. Indeed, there is no single GIT condition which can represent all desired length m zero-dimensional subschemes as smoothly supported subschemes. We must therefore define a modified GIT stability condition which patches together several GIT stability conditions in order to obtain the desired stable locus. Modified GIT stability. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Expansions for Hilbert Schemes: Stack Perspective

হিলবার্ট স্কিমগুলির সম্প্রসারণ: স্ট্যাক পরিপ্রেক্ষিত

Erweiterungen für Hilbert-Schemata: Stapelperspektive

Ampliaciones para esquemas de Hilbert: perspectiva de pila

Extensions pour les schémas Hilbert : perspective de pile

हिल्बर्ट योजनाओं के लिए विस्तार: स्टैक परिप्रेक्ष्य

ヒルベルトスキームの拡張: スタックの観点

Hilbert 계획의 확장: 스택 관점

Expansões para esquemas de Hilbert: perspectiva de pilha

Расширения схем Гильберта: перспектива стека

Hilbert Şemaları için Genişletmeler: Yığın Perspektifi

Mở rộng cho các sơ đồ Hilbert: Phối cảnh ngăn xếp

希尔伯特方案的扩展：堆栈视角

Author:
(1) CALLA TSCHANZ. Table of Links Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References 4. GIT stability In this section, we set up some results analogous to those of [GHH19] to describe various GIT stability conditions on the scheme X[n] with respect to the possible choices of G-linearised line bundles described in the previous section. In particular, we show that these stability conditions do not depend on the scheme structure of the length m zero-dimensional subschemes, but instead can be reduced to combinatorial criteria on configurations of n points. 4.1 Hilbert-Mumford criterion In this section, we shall recall the definition of Hilbert-Mumford invariants and give a numerical criterion for stability and semi-stability in terms of these invariants. Let H be a reductive group acting on a scheme S, which is proper over an algebraically closed field k. Let L be a H-linearised ample line bundle. Then a 1-parameter subgroup of H (denoted 1-PS for convenience) is defined to be a homomorphism 4.2 Action of 1-parameter subgroup 4.3 Bounded and combinatorial weights In this section, we explain the relation between what [GHH19] call the bounded and combinatorial weights of the Hilbert-Mumford invariants. Keeping the notation as consistent as possible with [GHH19], let be the universal family, with first and second projections p and q. The line bundle is relatively ample when l ≫ 0 and is G-linearised, exactly as in Section 2.2.1 of [GHH19]. Relationship between bounded and combinatorial weights. The following lemmas describe how the Hilbert-Mumford invariant can be decomposed into a sum of invariants. Note that, whereas the combinatorial weight depends on the choice of linearised line bundle, the bounded weight does not. Similarly to [GHH19], we can show that the bounded weight, as its name suggests, can be given an upper bound. The following result is based on Lemma 2.3 of [GHH19], with some slight modifications to suit our setting. Let us discuss now how the bounded weight affects the overall stability condition. The following lemma is immediate from [GHH19], but we recall their proof here for convenience. Proof. As we have shown that the bounded weight can be expressed as it is just a matter of choosing a big enough value of l to make the combinatorial weight overpower the bounded weight. This allows us effectively to treat the bounded weight as negligible and ignore it in our computations. Remark 4.3.5. Note, here, that such a Z will not necessarily be smoothly supported, nor will every point of the support of Z necessarily be contained in a ∆-component. Repeating this process over all k ∈ {1, . . . , n} will give us a description of L and we may form the G-linearised line bundle M from this line bundle in the way described at the start of this section. For more details on why this yields a positive combinatorial weight, see the proof of the following lemma. Note that this is not the only GIT stability condition for which Z is stable. Proof. It is clear that the combinatorial weight may be written as a sum 4.4 Semistable locus and GIT quotient Proof. This follows from Lemmas 4.3.3 and 4.3.7. Indeed, by Lemma 4.3.3, if the combinatorial weight can be written in the form Proof. Let us choose an arbitrary G-linearised line bundle M, not necessarily constructed as above, with respect to which Z has Hilbert-Mumford invariant Proof. This follows directly from Lemmas 4.4.1 and 4.4.2. We can now describe the GIT quotients resulting from these constructions. Let Then we recall from Lemma 3.1.13, the isomorphism For all choices of linearised line bundle described in the above, the GIT quotient on the base therefore behaves as follows Proof. This result follows directly from the relative Hilbert-Mumford criterion of [GHH15]. This paper is available on arxiv under CC 4.0 license. Author: (1) CALLA TSCHANZ. Author: Author: (1) CALLA TSCHANZ. Table of Links Abstract and Intro Background on tropical perspective The expanded construction GIT stability Stack perspective The canonical moduli stack References Abstract and Intro Abstract and Intro Background on tropical perspective Background on tropical perspective The expanded construction The expanded construction GIT stability GIT stability Stack perspective Stack perspective The canonical moduli stack The canonical moduli stack References References 4. GIT stability In this section, we set up some results analogous to those of [GHH19] to describe various GIT stability conditions on the scheme X[n] with respect to the possible choices of G-linearised line bundles described in the previous section. In particular, we show that these stability conditions do not depend on the scheme structure of the length m zero-dimensional subschemes, but instead can be reduced to combinatorial criteria on configurations of n points. 4.1 Hilbert-Mumford criterion In this section, we shall recall the definition of Hilbert-Mumford invariants and give a numerical criterion for stability and semi-stability in terms of these invariants. Let H be a reductive group acting on a scheme S, which is proper over an algebraically closed field k. Let L be a H-linearised ample line bundle. Then a 1-parameter subgroup of H (denoted 1-PS for convenience) is defined to be a homomorphism 4.2 Action of 1-parameter subgroup 4.3 Bounded and combinatorial weights In this section, we explain the relation between what [GHH19] call the bounded and combinatorial weights of the Hilbert-Mumford invariants. Keeping the notation as consistent as possible with [GHH19], let be the universal family, with first and second projections p and q. The line bundle is relatively ample when l ≫ 0 and is G-linearised, exactly as in Section 2.2.1 of [GHH19]. Relationship between bounded and combinatorial weights. The following lemmas describe how the Hilbert-Mumford invariant can be decomposed into a sum of invariants. Relationship between bounded and combinatorial weights. Note that, whereas the combinatorial weight depends on the choice of linearised line bundle, the bounded weight does not. Similarly to [GHH19], we can show that the bounded weight, as its name suggests, can be given an upper bound. The following result is based on Lemma 2.3 of [GHH19], with some slight modifications to suit our setting. Let us discuss now how the bounded weight affects the overall stability condition. The following lemma is immediate from [GHH19], but we recall their proof here for convenience. Proof . As we have shown that the bounded weight can be expressed as Proof it is just a matter of choosing a big enough value of l to make the combinatorial weight overpower the bounded weight. This allows us effectively to treat the bounded weight as negligible and ignore it in our computations. l Remark 4.3.5. Note, here, that such a Z will not necessarily be smoothly supported, nor will every point of the support of Z necessarily be contained in a ∆-component. Remark Repeating this process over all k ∈ {1, . . . , n} will give us a description of L and we may form the G-linearised line bundle M from this line bundle in the way described at the start of this section. For more details on why this yields a positive combinatorial weight, see the proof of the following lemma. Note that this is not the only GIT stability condition for which Z is stable. Proof . It is clear that the combinatorial weight may be written as a sum Proof 4.4 Semistable locus and GIT quotient Proof . This follows from Lemmas 4.3.3 and 4.3.7. Indeed, by Lemma 4.3.3, if the combinatorial weight can be written in the form Proof Proof . Let us choose an arbitrary G-linearised line bundle M, not necessarily constructed as above, with respect to which Z has Hilbert-Mumford invariant Proof Proof . This follows directly from Lemmas 4.4.1 and 4.4.2. Proof We can now describe the GIT quotients resulting from these constructions. Let Then we recall from Lemma 3.1.13, the isomorphism For all choices of linearised line bundle described in the above, the GIT quotient on the base therefore behaves as follows Proof . This result follows directly from the relative Hilbert-Mumford criterion of [GHH15]. Proof This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Expansions for Hilbert Schemes: GIT Stability

হিলবার্ট স্কিমগুলির সম্প্রসারণ: জিআইটি স্থিতিশীলতা

Erweiterungen für Hilbert-Schemata: GIT-Stabilität

Ampliaciones para los esquemas Hilbert: estabilidad de GIT

Extensions pour les schémas Hilbert : stabilité GIT

हिल्बर्ट योजनाओं के लिए विस्तार: जीआईटी स्थिरता

ヒルベルトスキームの拡張: GIT の安定性

Hilbert 체계 확장: GIT 안정성

Expansões para esquemas de Hilbert: estabilidade do GIT

Расширения схем Гильберта: стабильность GIT

Hilbert Şemaları için Genişletmeler: GIT Kararlılığı

Mở rộng cho các sơ đồ Hilbert: Tính ổn định của GIT

希尔伯特方案的扩展：GIT 稳定性

Author:
(1) CALLA TSCHANZ. Table of Links Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References Abstract The aim of this paper is to extend the expanded degeneration construction of Li and Wu to obtain good degenerations of Hilbert schemes of points on semistable families of surfaces, as well as to discuss alternative stability conditions and parallels to the GIT construction of Gulbrandsen, Halle and Hulek and logarithmic Hilbert scheme constructions of Maulik and Ranganathan. We construct a good degeneration of Hilbert schemes of points as a proper Deligne-Mumford stack and show that it provides a geometrically meaningful example of a construction arising from the work of Maulik and Ranganathan. 1. Introduction The study of moduli spaces is a central topic in algebraic geometry; among moduli spaces, Hilbert schemes form an important class of examples. They have been widely studied in geometric representation theory, enumerative and combinatorial geometry and as the two main examples of hyperk¨ahler manifolds, namely Hilbert schemes of points on K3 surfaces and generalised Kummer varieties. A prominent direction in this area is to understand the local moduli space of such objects and, in particular, the ways in which a degeneration of smooth Hilbert schemes may be given a modular compactification. For example, we may consider the geometry of relative Hilbert schemes on a degeneration whose central fibre has normal crossing singularities. We may then ask how the singularities of such a Hilbert scheme may be resolved while preserving certain of its properties or how it may be expressed as a good moduli space. This then becomes a compactification problem with respect to the boundary given by the singular locus. Historically, an important method used in moduli and compactification problems has been Geometric Invariant Theory (GIT). More recently, the work of Maulik and Ranganthan [MR20] has explored how methods of tropical and logarithmic geometry can be used to address such questions for Hilbert schemes. This builds upon previous work of Li [Li13] and Li and Wu [LW15] on expanded degenerations for Quot schemes and work of Ranganathan [Ran22b] on logarithmic Gromov-Witten theory with expansions. Briefly stated, the aim of this paper is to provide explicit examples of such compactifications and explore the connections between these methods. 1.1 Basic setup As is mentioned in Section 1.3, this type of construction can be applied to construct type III degenerations of Hilbert schemes of points on K3 surfaces. This will be described in future work. 1.2 Previous work in this area Following on from [LW15], Gulbrandsen, Halle and Hulek [GHH19] present a GIT version of the above construction in the case of Hilbert schemes of points. They construct an explicit expanded degeneration, i.e. a modified family over a larger base, whose fibres correspond to blow-ups of components of X0 in the family. They present a linearised line bundle on this space for the natural torus action and they are able to show that in this case the Hilbert-Mumford criterion simplifies down to a purely combinatorial criterion. Using this, they impose a GIT stability condition which recovers the transverse zero-dimensional subschemes of Li and Wu and prove that the corresponding stack quotient is isomorphic to that of Li and Wu. A motivation for this work was to construct type II degenerations of Hilbert schemes of points on K3 surfaces. Indeed, type II good degenerations of K3 surfaces present these types of singularities in the special fibre, which is a chain of surfaces intersecting along smooth curves. There is more recent work of Maulik and Ranganathan [MR20], building upon earlier ideas of Ranganathan [Ran22b] and results of Tevelev [Tev07], in which they use techniques of logarithmic and tropical geometry to construct appropriate expansions of X ! C. This allows them to define moduli stacks of transverse subschemes starting from the case where X0 is any simple normal crossing variety. They show that the stacks thus constructed are proper and Deligne-Mumford. For more details on this, see Section 2.2. 1.3 Main results Let X ! C be a semistable degeneration of surfaces. In the following sections, we propose explicit constructions of expanded degenerations and stacks of stable length m zero-dimensional subschemes on these expanded families, which we show to have good properties. Allowing for different choices of expansions. In this paper, we discuss only a specific choice of model for the Hilbert scheme of points which we call the canonical moduli stack. In upcoming work, we will investigate how these methods can be extended to describe other choices of models. We will consider an approach which parallels work of Kennedy-Hunt on logarithmic Quot schemes [Ken23], as well as recover certain geometrically meaningful choices of moduli stacks arising from the methods of Maulik and Ranganathan [MR20]. In particular, we will discuss how tube components and DonaldsonThomas stability enter the picture in these more general cases (see Section 2.2 for definitions). 1.4 Organisation We start, in Section 2, by giving some background on logarithmic and tropical geometry, and an overview of the work of Maulik and Ranganathan from [MR20] which we will want to refer to in later sections. Then, in Section 3, we set out an expanded construction on schemes and, in 4, we discuss how various GIT stability conditions can be defined on this construction. In Section 5, we describe a corresponding stack of expansions and family over it, building on the expanded degenerations we constructed as schemes. In Section 6, we extend our stability conditions to this setting. We then show that the stacks of stable objects defined have the desired Deligne-Mumford and properness properties. Acknowledgements. I would like to thank Gregory Sankaran for all his support throughout this project. Thank you also to my PhD examiners, Alastair Craw and Dhruv Ranganathan, for their many helpful comments. This work was undertaken while funded by the University of Bath Research Studentship Award. I am also grateful to Patrick Kennedy-Hunt and Thibault Poiret for many interesting conversations. This paper is available on arxiv under CC 4.0 DEED license. Author: (1) CALLA TSCHANZ. Author: Author: (1) CALLA TSCHANZ. Table of Links Abstract and Intro Background on tropical perspective The expanded construction GIT stability Stack perspective The canonical moduli stack References Abstract and Intro Abstract and Intro Background on tropical perspective Background on tropical perspective The expanded construction The expanded construction GIT stability GIT stability Stack perspective Stack perspective The canonical moduli stack The canonical moduli stack References References Abstract The aim of this paper is to extend the expanded degeneration construction of Li and Wu to obtain good degenerations of Hilbert schemes of points on semistable families of surfaces, as well as to discuss alternative stability conditions and parallels to the GIT construction of Gulbrandsen, Halle and Hulek and logarithmic Hilbert scheme constructions of Maulik and Ranganathan. We construct a good degeneration of Hilbert schemes of points as a proper Deligne-Mumford stack and show that it provides a geometrically meaningful example of a construction arising from the work of Maulik and Ranganathan. 1. Introduction The study of moduli spaces is a central topic in algebraic geometry; among moduli spaces, Hilbert schemes form an important class of examples. They have been widely studied in geometric representation theory, enumerative and combinatorial geometry and as the two main examples of hyperk¨ahler manifolds, namely Hilbert schemes of points on K3 surfaces and generalised Kummer varieties. A prominent direction in this area is to understand the local moduli space of such objects and, in particular, the ways in which a degeneration of smooth Hilbert schemes may be given a modular compactification. For example, we may consider the geometry of relative Hilbert schemes on a degeneration whose central fibre has normal crossing singularities. We may then ask how the singularities of such a Hilbert scheme may be resolved while preserving certain of its properties or how it may be expressed as a good moduli space. This then becomes a compactification problem with respect to the boundary given by the singular locus. Historically, an important method used in moduli and compactification problems has been Geometric Invariant Theory (GIT). More recently, the work of Maulik and Ranganthan [MR20] has explored how methods of tropical and logarithmic geometry can be used to address such questions for Hilbert schemes. This builds upon previous work of Li [Li13] and Li and Wu [LW15] on expanded degenerations for Quot schemes and work of Ranganathan [Ran22b] on logarithmic Gromov-Witten theory with expansions. Briefly stated, the aim of this paper is to provide explicit examples of such compactifications and explore the connections between these methods. 1.1 Basic setup As is mentioned in Section 1.3, this type of construction can be applied to construct type III degenerations of Hilbert schemes of points on K3 surfaces. This will be described in future work. 1.2 Previous work in this area Following on from [LW15], Gulbrandsen, Halle and Hulek [GHH19] present a GIT version of the above construction in the case of Hilbert schemes of points. They construct an explicit expanded degeneration, i.e. a modified family over a larger base, whose fibres correspond to blow-ups of components of X0 in the family. They present a linearised line bundle on this space for the natural torus action and they are able to show that in this case the Hilbert-Mumford criterion simplifies down to a purely combinatorial criterion. Using this, they impose a GIT stability condition which recovers the transverse zero-dimensional subschemes of Li and Wu and prove that the corresponding stack quotient is isomorphic to that of Li and Wu. A motivation for this work was to construct type II degenerations of Hilbert schemes of points on K3 surfaces. Indeed, type II good degenerations of K3 surfaces present these types of singularities in the special fibre, which is a chain of surfaces intersecting along smooth curves. There is more recent work of Maulik and Ranganathan [MR20], building upon earlier ideas of Ranganathan [Ran22b] and results of Tevelev [Tev07], in which they use techniques of logarithmic and tropical geometry to construct appropriate expansions of X ! C. This allows them to define moduli stacks of transverse subschemes starting from the case where X0 is any simple normal crossing variety. They show that the stacks thus constructed are proper and Deligne-Mumford. For more details on this, see Section 2.2. 1.3 Main results Let X ! C be a semistable degeneration of surfaces. In the following sections, we propose explicit constructions of expanded degenerations and stacks of stable length m zero-dimensional subschemes on these expanded families, which we show to have good properties. Allowing for different choices of expansions. In this paper, we discuss only a specific choice of model for the Hilbert scheme of points which we call the canonical moduli stack. In upcoming work, we will investigate how these methods can be extended to describe other choices of models. We will consider an approach which parallels work of Kennedy-Hunt on logarithmic Quot schemes [Ken23], as well as recover certain geometrically meaningful choices of moduli stacks arising from the methods of Maulik and Ranganathan [MR20]. In particular, we will discuss how tube components and DonaldsonThomas stability enter the picture in these more general cases (see Section 2.2 for definitions). Allowing for different choices of expansions. 1.4 Organisation We start, in Section 2, by giving some background on logarithmic and tropical geometry, and an overview of the work of Maulik and Ranganathan from [MR20] which we will want to refer to in later sections. Then, in Section 3, we set out an expanded construction on schemes and, in 4, we discuss how various GIT stability conditions can be defined on this construction. In Section 5, we describe a corresponding stack of expansions and family over it, building on the expanded degenerations we constructed as schemes. In Section 6, we extend our stability conditions to this setting. We then show that the stacks of stable objects defined have the desired Deligne-Mumford and properness properties. Acknowledgements . I would like to thank Gregory Sankaran for all his support throughout this project. Thank you also to my PhD examiners, Alastair Craw and Dhruv Ranganathan, for their many helpful comments. This work was undertaken while funded by the University of Bath Research Studentship Award. I am also grateful to Patrick Kennedy-Hunt and Thibault Poiret for many interesting conversations. Acknowledgements This paper is available on arxiv under CC 4.0 DEED license. This paper is available on arxiv under CC 4.0 DEED license. available on arxiv

Expansions for Hilbert Schemes: Abstract and Intro

হিলবার্ট স্কিমগুলির সম্প্রসারণ: বিমূর্ত এবং ভূমিকা

Erweiterungen für Hilbert-Schemata: Zusammenfassung und Einführung

Ampliaciones para esquemas de Hilbert: resumen e introducción

Extensions pour les schémas Hilbert : résumé et introduction

हिल्बर्ट योजनाओं के लिए विस्तार: सार और परिचय

ヒルベルトスキームの展開: 概要と導入

Hilbert 체계의 확장: 개요 및 소개

Expansões para esquemas de Hilbert: resumo e introdução

Разложения схем Гильберта: Аннотация и введение

Hilbert Şemaları için Genişletmeler: Özet ve Giriş

Mở rộng cho các sơ đồ Hilbert: Tóm tắt và giới thiệu

希尔伯特方案的扩展：摘要和简介

Author:
(1) CALLA TSCHANZ. Table of Links Abstract and Intro
Background on tropical perspective
The expanded construction
GIT stability
Stack perspective
The canonical moduli stack
References 3. The expanded construction Output of expanded construction. The expanded degeneration X[n] ! C[n] which we construct in this section has the following properties: 3.1 The blow-ups In this expanded degeneration construction, we will be blowing up schemes along Weil divisors. A consequence of the way these blow-ups are defined is that the blow-up morphisms contract only components of codimension at least 2. the morphisms corresponding to each individual blow-up. We therefore have the equality We now fix the following terminology. Proposition 3.1.5. The following blow-up diagram commutes Proof. This is immediate from the local description of the blow-ups above. We now extend the definition of ∆1-components to the schemes X[n] and fix some additional terminology. Before we continue we fix some terminology which will help us describe the expanded components. Definition 3.1.11. We refer to an irreducible component of a ∆-component as a bubble. The notions of two bubbles being equal and a bubble being expanded out in a certain fibre are as in Definitions 3.1.4 and 3.1.9. Now, we note that there is a natural inclusion which, in turn, induces a natural inclusion on the basis directions, and acts by on the ∆-components. Proof. This follows immediately from [GHH19]. we described in the previous section are equivariant under the group action. Lemma 3.1.13. We have the isomorphism Proof. This is immediate from the above description of the group action. Remark 3.1.14. We abuse notation slightly by referring to the group acting on X[n] by G, instead of G[n]. It should always be clear from the context what group G is meant. 3.2 Embedding into product of projective bundles Lemma 3.2.1. There is an embedding From this, we deduce that there are embeddings Hence we have embeddings Linearisations. The following lemma gives a method to construct all the linearised line bundles we will need to vary the GIT stability condition. This paper is available on arxiv under CC 4.0 license. Author: (1) CALLA TSCHANZ. Author: Author: (1) CALLA TSCHANZ. Table of Links Abstract and Intro Background on tropical perspective The expanded construction GIT stability Stack perspective The canonical moduli stack References Abstract and Intro Abstract and Intro Background on tropical perspective Background on tropical perspective The expanded construction The expanded construction GIT stability GIT stability Stack perspective Stack perspective The canonical moduli stack The canonical moduli stack References References 3. The expanded construction Output of expanded construction. The expanded degeneration X[n] ! C[n] which we construct in this section has the following properties: 3.1 The blow-ups In this expanded degeneration construction, we will be blowing up schemes along Weil divisors. A consequence of the way these blow-ups are defined is that the blow-up morphisms contract only components of codimension at least 2. the morphisms corresponding to each individual blow-up. We therefore have the equality We now fix the following terminology. Proposition 3.1.5. The following blow-up diagram commutes Proposition 3.1.5. The following blow-up diagram commutes Proof . This is immediate from the local description of the blow-ups above. Proof We now extend the definition of ∆1-components to the schemes X[n] and fix some additional terminology. Before we continue we fix some terminology which will help us describe the expanded components. Definition 3.1.11. We refer to an irreducible component of a ∆-component as a bubble. The notions of two bubbles being equal and a bubble being expanded out in a certain fibre are as in Definitions 3.1.4 and 3.1.9. Definition 3.1.11. Now, we note that there is a natural inclusion which, in turn, induces a natural inclusion on the basis directions, and acts by on the basis directions, and acts by on the ∆-components. on the ∆-components. Proof . This follows immediately from [GHH19]. Proof we described in the previous section are equivariant under the group action. Lemma 3.1.13. We have the isomorphism Lemma 3.1.13. We have the isomorphism Proof . This is immediate from the above description of the group action. Proof Remark 3.1.14. We abuse notation slightly by referring to the group acting on X[n] by G, instead of G[n]. It should always be clear from the context what group G is meant. Remark 3.2 Embedding into product of projective bundles Lemma 3.2.1. There is an embedding Lemma 3.2.1. There is an embedding From this, we deduce that there are embeddings Hence we have embeddings Linearisations . The following lemma gives a method to construct all the linearised line bundles we will need to vary the GIT stability condition. Linearisations This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Expansions for Hilbert Schemes: the Expanded Construction

হিলবার্ট স্কিমগুলির সম্প্রসারণ: প্রসারিত নির্মাণ

Erweiterungen für Hilbert-Schemata: die erweiterte Konstruktion

Ampliaciones para los proyectos Hilbert: la construcción ampliada

Extensions des projets Hilbert : la construction élargie

हिल्बर्ट योजनाओं का विस्तार: विस्तारित निर्माण

ヒルベルトスキームの拡張：拡張構築

힐베르트 계획의 확장: 확장된 구성

Expansões para Esquemas Hilbert: a Construção Expandida

Разложения схем Гильберта: расширенная конструкция

Hilbert Şemaları için Genişletmeler: Genişletilmiş Yapı

Mở rộng cho Đề án Hilbert: Xây dựng mở rộng

希尔伯特格式的扩展：扩展构造

Tropic

tropicapp.io

201-500 employees

Since 2019

COMPANY RANKING

TROPIC

EVERGREEN INDEX #11983

Tropic WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!

Tropic

tropicapp.io

201-500 employees

Since 2019

COMPANY RANKING

TROPIC

EVERGREEN INDEX #11983

RELATED COMPANIESRANK

STORIES ABOUT TROPIC

Tropic WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!