# Tagged Questions

231 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
648 views

### The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor ...
867 views

### Using schemes to prove things about rings

I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment. The question is as follows: What are examples of strict applications ...
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### Pathological Examples of Dimension

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
1k views

### What information is contained in the Kazhdan-Lusztig polynomials?

The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations. For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
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### What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

This is, basically, me trying to generalize "Why should I care for sheaves and schemes?" into a reasonable question. Whether successfully, time will tell, but let me hope that if not the question, ...
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I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
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### What formal properties should resolution of singularities have?

If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
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### What is a good introductory text for moduli theory?

Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...
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### Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
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### More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
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### Facts from algebraic geometry that are useful to non-algebraic geometers

A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...
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### Algebraic geometry used “externally” (in problems without obvious algebraic structure).

This is a request for a list of examples of problems (or other mathematical situations) that are not initially of algebro-geometric nature, but can be solved or understood by using algebraic geometry. ...
15k views

### What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons ...
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### References for complex analytic geometry?

I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc. A ...
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### Interesting results in algebraic geometry accessible to 3rd year undergraduates

On another thread I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies. Willie Wong proposed me ...
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### A list of machineries for computing cohomology

This is not a question, but I just hope to hear more from everyone here on it. A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I ...
722 views

### Collecting various theories on toy examples: Projective space

I am looking for text books/notes/papers/documents playing with toy examples: projective space, in particular, $P^{1}$. Because I think this is really a cute example. Although algebraic geometry on ...