# Tagged Questions

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### Quotient of smooth variety is smooth if fixed point set is a divisor?

I've heard (a variant of) the following result being mentioned , but haven't been able to find a reference. I would like to know if the following is true, and if so, I'd very much appreciate a good ...
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### What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
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### The family of schemes $\mathrm{spec} \ A[x]/(x^n)$

Consider the family $S_n:=\mathrm{spec} \ A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for $S_n$ ...
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### Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big?

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ? Is there a formula relating the dimension of the Zariski tangent space and the order of ...
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### Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
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### Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
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### Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
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### Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
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### Mumford-Oda - Algebraic Geometry II . There will be a complete book?

Online there is the draft of a book written by Mumford and Oda that should be the continuation of "Algebraix Geometry I complex projetive varieties" (Mumford,1976). Do you know if and when this book ...
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### Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
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### Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
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### Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
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### Book Recommend Differential Geometry of Algebraic Manifolds

I just want to study Differential Geometry of Algebraic Manifolds. but I can`t find a book about that. Is there any good book for studying Differential Geometry of Algebraic Manifolds??
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### Global Optimization and Real Algebraic Geometry

Wikipedia suggests that: "Methods based on real algebraic geometry" are some of the "most successful general strategies" for solving global optimization problems. Could someone suggest an reference ...
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### Which pullbacks preserve the sheaf of differentials?

Let $f : X \to Y$ be a morphism of $S$-schemes. Assume that the canonical morphism $\alpha : f^* \Omega^1_{Y/S} \to \Omega^1_{X/S}$ is an isomorphism. What can we say about $f$? Is $f$ formally étale? ...
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### Coherent sheaves of finite length

Where can I find a treatment of coherent sheaves of finite length over, say, Noetherian schemes? Just things as basic as their definition and elementary facts about them. I am familiar with modules of ...
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### Orbits that 'coalesce'

Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$. Suppose $S$ is another ...
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### If $U\to X$ is a closed immersion, $U$ is dense in $X$ and $X$ is reduced, why is the closed immersion an isomorphism?

This came up in the Reduced-to-separated theorem. If $U\to X$ is a closed immersion, $U$ is dense in $X$ and $X$ is reduced, why is the closed immersion an isomorphism?
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### What is the Bi-affine plane

I want to know the definition of the Bi-affine plane. In an article it says that semi-symmetric plane is same as bi-affine plane. But I want to the exact definition and axioms. Also There are two ...
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### Does anyone have a copy of SGA 4?

It's not on Laszlo's site, although the latest message says that it is indeed going to be published, that was way back in 2010.
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### Book with color pictures of algebraic surfaces

I have a pretty specific question: I'm looking for a book with color pictures of algebraic surfaces. Could anyone point me in the right direction?
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### Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
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### Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
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### Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
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### Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
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### Are there online English proofs of preservation of properties of morphisms by fpqc descent?

Accroding to Wikipedia's article, the following propositions are proved in EGA. Are there online proofs written in English? Suppose that $f\colon X \rightarrow Y$ is a morphism of $S$-schemes. Let ...
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### Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.
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### Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
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### Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification ...
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### Does Hartshorne *really* not define things like the composition or restriction of morphisms of schemes?

So far as I can tell, Hartshorne's Algebraic Geometry doesn't define the composition of morphisms of schemes, or the restriction of a morphism to an open subset. Of course it's easy enough to define ...
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### Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
Statement i read a few times : if $k$ is a field, then the rational points are Zariski-dense in the projective space $\mathbb{P}^n$. Does anybody could provide a proof of this fact or a reference? (i ...