2
votes
0answers
45 views

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 ...
3
votes
2answers
155 views

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 ...
3
votes
1answer
69 views

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 $S_n$ ...
3
votes
1answer
99 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
2
votes
1answer
63 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
2
votes
1answer
157 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
1
vote
1answer
60 views

Beggining in Algebraic Geometry

My question is about sources for start the study of algebraic geometry. I know that it requieres so much algebra, but, is there any book which can be readed without many tolos of modules, Galois, ...
6
votes
1answer
72 views

Affine varieties, the Brauer group and gerbes

For an affine variety $V$, the Brauer group $Br(V)$ and the cohomological Brauer group $Br'(V)$ (i.e. the torsion subgroup of $H^2_{et}(V,\mathbb{G}_m)$) coincide, by results of various authors ...
3
votes
1answer
72 views

Learning roadmap for classical algebraic geometry (italian school)

Can someone suggest a learning roadmap for classical algebraic geometry as develop by the great italian school? Severi has a few books but in italian. I would like to know what are the best English ...
0
votes
0answers
19 views

irreducibility of a regular scheme over a local field

Are there any criterions allowing us to detect whether a regular scheme over a local field is irreducible ?
3
votes
0answers
84 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
2
votes
1answer
58 views

Explicit Kähler forms and Kähler cone of one-point blowup of $\mathbb{CP}^2$

I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone ...
1
vote
1answer
32 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
3
votes
0answers
56 views

$h^{p,q}$ of projective space

How can we calculate the Hodge number $h^{p,q}= \dim H^p(\mathbb{P^n},\Omega^q_{\mathbb{P}^n})$ of projective space? Is there a reference for that?
2
votes
0answers
78 views

Books on algebraic surfaces

I am interested in learning about algebraic surfaces (e.g. their classification in characteristic 0), and I was wondering whether any knowledgeable people would be so kind as to give their thoughts ...
2
votes
1answer
111 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
3
votes
1answer
80 views

Resolution of Singularity

Consider $y^8=x^{11}$. So, $y^8-x^{11}=0$. Define $f(x,y)=y^8-x^{11}$. Then, $\nabla f = (f_x,f_y)=(-11x^{10},8y^7)$. For $(f_x,f_y)=(0,0)$, we must have $(x,y)=(0,0)$; the singular point. ...
6
votes
4answers
197 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
1
vote
0answers
44 views

A question on the notation in Hartshorne's Algebraic Geometry

My confusion is in III.Corollary $9.4$. By $\mathcal{F}_y$ does he mean the pull-back of $\mathcal{F}$ by the closed immersion of $X_y$ into $X$? By $\mathcal{F} \otimes k(y)$ does he mean the ...
0
votes
1answer
39 views

surjective morphism of varieties

Let $A,B$ be two varieties (integral finite type separated scheme) over a field $k$ and let $f : A \to B$ be a proper morphism. Is it true that $f$ is surjective if and only if $dim(\overline{f(A)}) ...
2
votes
2answers
102 views

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 ...
5
votes
0answers
177 views

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 ...
7
votes
2answers
109 views

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 ...
3
votes
0answers
38 views

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 ...
1
vote
1answer
39 views

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 ...
2
votes
1answer
86 views

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 ...
0
votes
0answers
102 views

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.
3
votes
0answers
104 views

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 ...
2
votes
0answers
44 views

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 ...
1
vote
1answer
96 views

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??
14
votes
1answer
181 views

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 ...
5
votes
1answer
83 views

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? ...
1
vote
0answers
72 views

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 ...
2
votes
1answer
55 views

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 ...
5
votes
0answers
54 views

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?
4
votes
1answer
54 views

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 ...
4
votes
1answer
89 views

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.
5
votes
1answer
117 views

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?
2
votes
2answers
109 views

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?
2
votes
0answers
197 views

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 ...
14
votes
7answers
458 views

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 ...
8
votes
1answer
220 views

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 ...
0
votes
1answer
131 views

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 ...
27
votes
1answer
475 views

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.
2
votes
0answers
205 views

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 ...
9
votes
2answers
213 views

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 ...
5
votes
2answers
168 views

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 ...
16
votes
1answer
408 views

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 ...
0
votes
1answer
70 views

Deformation Theory reference

I was recommended a book by Oort sometime ago to read with reference to deformation theory, in particular in positive characteristic with hyperelliptic curves. I couldn't find anything that seemed to ...
0
votes
0answers
56 views

point are Zariski-dense in projective space

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 ...