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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915 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 ...
11
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2answers
131 views

For a (not necessarily affine) scheme $X = \bigcup_{i=1}^n X_{f_i}$, does $(f_1, \ldots, f_n) = (1)$ in $\mathcal{O}_X(X)$?

This is of course true in the affine case, so it seems like it should be true in general, because $\mathcal{O}_X(X)$ should be "smaller" for a non-affine scheme than for a similar affine scheme (e.g. ...
11
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1answer
363 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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2answers
195 views

What is a “subscheme”?

Every source I've looked at defines open subschemes and closed subschemes, but the definitions always look ad-hoc and not closely related to one another. Are there other kinds of subschemes? If not, ...
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1answer
1k views

Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
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3answers
975 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 ...
11
votes
1answer
460 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 ...
11
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2answers
263 views

Can there be a point on a Riemann surface such that every rational function is ramified at this point?

Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset. Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$? I'm ...
11
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1answer
810 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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451 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
11
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202 views

algebraic versus analytic line bundles

If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective? I assume the latter ...
11
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1answer
395 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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2answers
797 views

Is there a more elementary proof of this special case of Riemann-Roch?

I'm looking for an elementary proof of the fact that $\ell(nP) = \dim L(nP) = n$, where $L(nP)$ is the linear (Riemann-Roch) space of certain rational functions associated to the divisor $nP$, where ...
11
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1answer
184 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 ...
11
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1answer
357 views

principal G-bundles in zariski vs etale topology

Let $G$ be an (affine) algebraic group over say $\mathbb{C}$. A principal $G$-bundle is a scheme $P$ with a $G$ action and a $G$-invariant morphism of schemes $\pi:P \to X$ that is etale locally on ...
11
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1answer
137 views

What's the intution behind defining the cotangent sheaf as $\Delta^\ast(\mathscr{I}/\mathscr{I}^2)$?

This definition seems to be given all over the place (e.g. Hartshorne II.8, Vakil 21.2.20, Wikipedia, McKernan's lecture notes from MIT), and never with any explanation as to why the map $\Delta : X ...
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1answer
250 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 $\mathbb{C}(x)$ ...
11
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1answer
361 views

Problem about Complete Intersection in $\textbf P^n$ (from Hartshorne).

I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe ...
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1answer
950 views

Folium of Descartes

A colleague came to me with an interesting observation: Consider the folium of Descartes, $$x^3+y^3=3axy$$ which upon implicit differentiation of the latter yields $$\frac{\mathrm dy}{\mathrm ...
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2answers
272 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 ...
11
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1answer
293 views

Is the composition of blowing-up a blowing-up?

Is the composition of blowing-up of algebraic varieties itself a blowing-up ? I think this is true but I am surprised not to have found any reference, though it seems to be an interesting property. ...
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248 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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3answers
337 views

Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb ...
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258 views

Is every algebraic curve birational to a planar curve

Let $X$ be an algebraic curve over an algebraically closed field $k$. Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$? I think I can prove this ...
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375 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
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3answers
966 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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993 views

A plane algebraic curve with all four kinds of double points

During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a node, a cusp (for my purposes I do not care which of the two kinds of ...
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3answers
1k 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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2answers
757 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
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499 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
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2answers
532 views

Why are affine varieties except points not compact in the standard topology on $C^n$ ?

I am starting to learn algebraic geometry and in the notes I am reading there is the following remark: " Over the complex numbers and with the strong topology we see that $A^n$ and affine varieties ...
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3answers
750 views

Picture of spec?

I recall seeing a hand-drawn picture of spec of a ring (maybe of $\mathbb Z$?) that had been passed around in the early days of the Zariski topology. Does anyone know where I can find a copy?
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1answer
681 views

Simple example of an ample line bundle that is not very ample

I am looking for a very concrete and simple example of a line bundle $L$ (on a curve or a surface) which is ample, but not very ample. I would also like that $L^{\otimes k}$ is very ample for a small ...
10
votes
1answer
207 views

Is $\operatorname{Gal}(\mathbb{Q}_p^{un})\cong \hat{\mathbb{Z}}$?

Is the absolute Galois group of $\mathbb{Q}_p^{un}$ the profinite completion of $\mathbb{Z}$? I was never quite sure... In similar cases, it is true. Namely, $\mathbb{C}((t))$ does have absolute ...
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3answers
337 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
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659 views

About the definition of Cech Cohomology

Let $X$ be a topological space with and open cover $\{U_i\}$ and let $\mathcal F$ be a sheaf of abelian groups on $X$. A $n$-cochain is a section $f_{i_0,\ldots,i_n}\in U_{i_0,\ldots,i_n}:= ...
10
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1answer
375 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 ...
10
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1answer
182 views

Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
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2answers
291 views

Hartshorne exercise III.6.2 (b) - $\mathfrak{Qco}(X)$ need not have enough projectives

Let $X=\mathbb P^1_k$, with $k$ an infinite field. Show that there does not exist a projective object $\mathcal P$ either in $\mathfrak{Qco}(X)$ or $\mathfrak{Coh}(X)$ together with a surjection ...
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1answer
881 views

Geometric meaning of primary decomposition

In the book "Commutative Algebra with a view toward Algebraic Geometry of David Eisenbud, he wrote about the Geometric interpretation of primary decomposition. I summary as follows : Let ...
10
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1answer
281 views

Geometrical interpretation of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$, $X_i$ algebraic sets in $\mathbb{A}^n$

Edit: I should point out that I'm working over an algebraically closed field $k$. Let $X_1,X_2\subset\mathbb{A}^n$ be affine algebraic sets. Show that $I(X_1\cap X_2)=\sqrt{I(X_1)+I(X_2)}$. Show ...
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1answer
105 views

In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic ...
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584 views

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
846 views

The Picard Group of the Affine line with double origin

Let $X$ be the affine line with double origin over a field $k$. It is the scheme obtained gluing two copies of the affine line $\mathbb{A}^1_k$ along the open sets $U_1 = U_2 =\mathbb{A}^1_k - (x)$, ...
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212 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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2answers
102 views

How to write down a non-degenerate cubic surface in P^4

I want to do something very concrete: write down a smooth scheme of given degree and dimension in projective n-space. A natural way to go about this is to try and write down a complete intersection, ...
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1answer
471 views

Gaining insight into the Inverse Image Sheaf

Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is ...
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290 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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1answer
470 views

Weil and Cartier divisors on a curve

I'm trying to understand the relationship between Weil divisors and Cartier divisors, and I would like to see why these are the same in the simple case where $X$ is a nonsingular projective curve over ...
10
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1answer
861 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? ...