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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.

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    $\begingroup$ I would go to mathematics section in 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 $\endgroup$
    – Praphulla Koushik
    Commented Jul 5, 2023 at 12:13
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    $\begingroup$ 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. $\endgroup$
    – Libli
    Commented Jul 5, 2023 at 14:39
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    $\begingroup$ 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. $\endgroup$
    – Libli
    Commented Jul 5, 2023 at 14:42
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    $\begingroup$ Related: What are some open problems in algebraic geometry? $\endgroup$
    – Timothy Chow
    Commented Jul 6, 2023 at 12:27
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    $\begingroup$ @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. $\endgroup$
    – Libli
    Commented Jul 7, 2023 at 8:47

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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

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  • $\begingroup$ 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. $\endgroup$
    – Libli
    Commented Jul 19, 2023 at 6:45

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