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I got admitted in a PhD program in Europe last year. But due to serious mental health issues , I was deemed unfit by the mathematics department to continue the program. I am from a 3rd world nation and I got these problems there due to some issues I would not like to mention.

So, I have decided to take a break and work on my health and to take a job to sustain myself before I apply again. The reason why I want to return to mathematics ( in future years only not in current or upcoming year) because I love studying mathematics, solving exercises.

I have a fine background in abstract algebra and algebraic geometry and I was wondering if you can let me know some books/ websites/ blogs which lists some open problems in Algebraic Geometry?

Having to know some problems in algebraic geometry and current literature will help me to keep in touch with the field and a little motivated.

J. W. Tanner
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Arnold
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    I would go to mathematics section in https://arxiv.org.. click "algebraic geometry" option and see names and abstracts.. Doing this for 2/3 months should give an idea of what is happening in the area and may be a starting point for an interesting problem – Praphulla Koushik Jul 05 '23 at 12:13
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    Algebraic Geometry has been one of the most active area in mathematics for the last 70 years and, in my opinion, it has reached such a high level of technicality that it seems very very difficult to attack research-style problem without the guidance and help of an enthusiastic professional researcher. – Libli Jul 05 '23 at 14:39
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    If you are interested in application of algebraic geometry, I would however advice you to read any of Bernd Sturmfels paper. They are wonderfully well-written, require a modest amount of technicalities in AG and usually contain a lot of interesting open problems at the boundary between applied maths and AG. – Libli Jul 05 '23 at 14:42
  • @Libli Thanks for your reply. I am sorry but I am not interested in applications. – Arnold Jul 06 '23 at 09:20
  • @Libli Is there some book in Algebraic Geometry like following in number theory( by Richard K Guy) : https://www.amazon.com/Unsolved-Problems-Number-Problem-Mathematics/dp/1441919287 – Arnold Jul 06 '23 at 11:10
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    Related: What are some open problems in algebraic geometry? – Timothy Chow Jul 06 '23 at 12:27
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    @Arnold : not that I am aware of (though the question linked by Timothy Chow seems to be close to what you are looking for). The problem is that even reading a open problem in AG is not easy as it requires to understand many technicalities. Understanding why the problem is open seems even more difficult. – Libli Jul 07 '23 at 08:47
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    @Arnold : maybe the book "recent developments in Algebraic Geometry" edited by cn Cambridge may be of interest. Though the correct title of the book should be " recent developments in birational geometry". The four editirs ate birational geometers (and three of them are co-authors) which explain why the choice of subject in the book areca litle biaised. For instance, derived categories and mirror symmetry are strikingly under-represented in that book. – Libli Jul 07 '23 at 08:56
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    @Arnold : as far as Mirror Symmetry is concerned, the book "Mirror Symmetry" in the Clay Mathematics institute collection looks interesting. However, due to the research interests of the editors, it almost solely focuses on the "counting invariants" aspects of the story. The homological, tropical and birational sides of Mirror Symmetry are completely ignored in that book. – Libli Jul 07 '23 at 09:02
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    @Arnold: as far as the derived categories and classical AG are concerned, the book by Huybrechts (Fourier-Mikai transforms) seems to be a must-have that has never been surpassed since. Though it has become outdated on the Hodge theoretic aspects of the story (and completely ignores the surprising number-theoretic implications of derived equivalences between schemes over the integers). – Libli Jul 07 '23 at 09:11
  • @Libli I am grateful for your comments! They are very helpful for me. Thank you very much. – Arnold Jul 07 '23 at 12:04

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This book has a "motivating statement" that resonates with the OP:

Open Problems in Arithmetic Algebraic Geometry (2019)

This book originated in the idea that open problems act as crystallization points in mathematical research. Mathematical books usually deal with fully developed theories. But here we present work at an earlier stage—when challenging questions can give new directions to mathematical research.
In mathematics, significant progress is often made by looking at the underlying structures of open problems and discovering new directions that are developed to find solutions. In that process, the search for finding the "true" nature of the problem at hand is the impetus for our thoughts. It is only much later, in retrospect, that we see the "flow of mathematics"—from problem to theory and new insight. This is the gist of the present volume.

Some of the problems from the book are collected at Open problems in Algebraic Geometry

Carlo Beenakker
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  • The last reference should be called "Open problems in arithmetic geometry from the late 90's" I think this reference is misleading (especially when given to a beginner) as it hardly highlights the current state of affairs in algebraic geometry. – Libli Jul 19 '23 at 06:45