The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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Varieties as schemes

Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne. Firstly, my main question. I understood that Grothendiecks introduction of schemes ...
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1answer
699 views

How to compute localizations of quotients of polynomial rings

At the moment I'm trying to understand the concept of localizations of rings / modules. I have done some exercises (using the book of Atiyah / MacDonald) and I will do some more, but a more practical ...
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444 views

Does Hom commute with stalks for locally free sheaves?

This is somewhat related to the question Why doesn't Hom commute with taking stalks?. My question is this: If $F$ and $G$ are locally free sheaves of $\mathcal{O}_X$ -modules on an arbitrary ...
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1answer
290 views

What is the line bundle $\mathcal{O}_{X}(k)$ intuitively?

I am always confused about how to understand the line bundle $\mathcal{O}_{X}(k)$ on a projective scheme $X=\mathrm{Proj}(\oplus_{n=0}^\infty A_{n})$. Of course this is by definition the ...
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850 views

Toy sheaf cohomology computation

I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology One thing that really helped in learning the Serre SS ...
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201 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
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386 views

On the definition of the structure sheaf attached to $Spec A$

Let $A$ be a ring (commutative with $1$),if $X=Spec A$ we want to attach a sheaf of rings to $X$. If $f\in A$, $D(f)=X\setminus V(f)$ is an element of the base and we define $$\mathcal ...
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356 views

Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
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759 views

(Ir)reducibility criteria for homogeneous polynomials

Suppose I have a homogeneous polynomial in at least 3 variables over some algebraically closed field (of characteristic 0, if need be). Question: How may I test — by hand — whether it is irreducible? ...
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1k views

My first course in algebraic geometry: two simple questions

I'm attending my first course in algebraic geometry, and my professor has chosen an approach which is a middle-way between the basic algebraic geometry done in $\mathbb A^n_k$ and the approach with ...
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257 views

Why study schemes?

Why study schemes instead of only affine/projective varieties, given by zeros of polynomials in the affine/projective space? I mean, what is gained by introducing the concept of schemes? Thank you!
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222 views

Theorems in algebraic geometry which have been proved only by using cohomology

There are many theorems in algebraic geometry which were proved using cohomology. I would like to know examples of such theorems which have been proved only by using cohomology. In other words, those ...
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238 views

Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
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1answer
232 views

Intersection of powers of maximal ideals

Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid ...
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1answer
915 views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
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143 views

Is the product of non-separated schemes non-separated?

My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
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3answers
880 views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
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1answer
556 views

How to think of the pullback operation of line bundles?

Recall that give a map $f : (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ of ringed spaces and a sheaf $\mathcal{F}$ on $Y$ we can form the pullback $f^\ast \mathcal{F} := ...
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425 views

Atiyah-MacDonald help with exercise 5.10

This is an exercise from Atiyah-MacDonald, if someone can give an idea on how to prove that $a)\Rightarrow b)$: Let $f:A\rightarrow B$ a ring homomorphism. a) ...
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1answer
167 views

Weil conjectures - motivation?

Can anyone explain (heuristically, intuitively is fine) what the importance of the Weil conjectures is? I realize they have motivated much of recent algebraic geometry. I don't really understand why ...
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1answer
318 views

Concrete example of calculation of $\ell$-adic cohomology

Let $p$ and $\ell$ be distinct prime numbers. Consider in the affine plane $\mathbb{A}^2_{\mathbb{F}_p}$ with coordinates $(x,y)$ the union $L$ of the axes $x = 0$ and $y = 0$. How does one compute ...
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1answer
407 views

Stacks in arithmetic geometry [closed]

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
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228 views

Sard's theorem for algebraic varieties

(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular values ...
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1answer
213 views

Finite extensions of rational functions

I know that finite extensions of $\mathbb{C}(x)$ correspond to finite branched covers of $\mathbb{P}^1$, and this leads to an abstract characterization of the absolute Galois group of $\mathbb{C}(x)$ ...
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1answer
580 views

Cohomology of a tensor product of sheaves

Say I have two locally free sheaves $F,G$ on projective variety $X$. I know the cohomology groups $H^i(X,F)$ and $H^i(X,G)$. Is this enough to give me information about $H^i(X,F\otimes G)$? In ...
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169 views

What are the “correct” modules over locally ringed spaces?

$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
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1answer
132 views

Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus ...
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137 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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219 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
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352 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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285 views

Visualizing a Calabi Yau

I would like to understand how I can visualize the quintic threefold $$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$ For a similar problem, Hanson proposes the following: ...
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285 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
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2answers
279 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
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188 views

Curves and Sums-of-Powers Representations

Jacobi first noticed the connection between the functions that bear his name and counting the representations of sums-of-squares, \begin{eqnarray} \theta_{3}^{n}(q) = \left( \sum_{k \in \mathbb{Z}} ...
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Are “$n$ by $n$ matrices with rank $k$” an affine algebraic variety?

Identify the set of all complex $n$ by $n$ matrices with $\mathbb{C}^{n^2}$. We say a subset $S \subset \mathbb{C}^{n^2}$ is an affine algebraic variety if $S$ is the common zero set of a collection ...
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862 views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
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831 views

Why doesn't Hom commute with taking stalks?

I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, ...
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4answers
142 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
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308 views

Why did Hartshorne bother on schemes in his textbook when he worked only over an algebraically closed field?

In his textbook Algebraic Geometry, he wrote in p. 58: Now that we have seen a litle bit of what algebraic geometry is about, we should discuss the degree of generality in which to develop the ...
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318 views

Complex analysis book for Algebraic Geometers

I know that there exist many questions on this site on complex analysis books but my question is more specific than that. I am looking for recommendations for a concise complex analysis book but with ...
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5answers
544 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
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225 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 ...
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3answers
799 views

Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
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735 views

Question on sheafification of a presheaf

In chapter 2 of GTM 52 by Robin Hartshone there are definition of presheaf and the associated sheaf of a given presheaf. I found that the definition of the sheafification is rather less natural and ...
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502 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
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701 views

rational points of an algebraic variety

In http://en.wikipedia.org/wiki/Rational_point we read : a $K$-rational point is a point on an algebraic variety where each coordinate of the point >belongs to the field $K$. This means that, if ...
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Dominant rational maps

The University I go to doesn't have any courses in (classical) Algebraic Geometry so I am trying to learn myself. I am fairly comfortable with the content I have covered so far aside from a so called ...
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791 views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
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180 views

Can an algebraic variety be described as a category, in the same way as a group?

Can an algebraic variety be described as a category, in the same way as a group? A group can be considered a category with one object, with elements of the group the morphisms on the object.
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Zariski topology in the complex plane: an example

I want to find the closure under the zariski topology, of this set $ \left\{ {\left( {x,y} \right) \in {\Bbb C}^2 ;\left| x \right| + \left| y \right| = 1} \right\} $ I have no idea what I can do