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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Motivation for stable curves

I was looking at Deligne-Mumford's paper on the irreducibility of the space of curves of a given genus, and it seems that they generalize the notion of a smooth curve to a "stable curve." I'm a little ...
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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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691 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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148 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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170 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
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398 views

Stacks are just sheaves up to Isomorphism

I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" ...
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753 views

vector bundles on affine schemes

Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that ...
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379 views

What is the homotopy type of the affine space in the Zariski topology..?

I'm asking this question out of curiosity, as I was unable to come to a conclusion. Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
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411 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 ...
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615 views

Inverse Limit of Sheaves

It is well-known that if you have an inverse system of abelian groups $(A_n)$ (this works in several other nice categories) in which all the maps are surjective (or at least satisfy the Mittag-Leffler ...
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171 views

Representability of diagonal of $\mathscr{M}_g$

Let $\mathscr{M}_g$ be the moduli stack of genus $g$ curves ($g \geq 2$). That is, $\mathscr{M}_g$ is the category whose objects are proper smooth morphisms $f: C \to S$ whose geometric fibers are ...
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192 views

Is a scheme with a single closed point affine?

Let $X$ be a quasi-compact, separated scheme with a single closed point. Is $X$ necessarily affine, and thus isomorphic to the spectrum of a local ring? I could not think of a counter-example; is ...
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Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
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Geometric Explanation of Tamagawa Numbers

Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view. My interest always lies with understanding the ...
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328 views

Which continuous functions are polynomials?

Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some ...
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What is the most influential work of Grothendieck in mathematics?

Recently Alexander Grothendieck has passed away but his mathematical wave is still alive and passes its growth ages. It is hard to describe the influence of such a great man in mathematics just in few ...
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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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843 views

An example of a scheme in the language of schemes

Somewhat related to this question, but almost infinitely more basic. A Confession I am, should classification prove essential, a differential geometer and a topologist by inclination and by ...
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Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...
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935 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
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What is the intuition behind the concept of Tate twists?

For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
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611 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
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977 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 ...
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660 views

Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff

In the book of Atiyah and MacDonald, I was doing exercise 3.11. One has to show that for a ring $A$, the following are equivalent: $A/\mathfrak{N}$ is absolute flat, where $\mathfrak{N}$ is the ...
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813 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
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Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
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927 views

tensor product of sheaves commutes with inverse image

Let $f : X \to Y$ be a morphism of ringed spaces and $\mathcal{M}$, $\mathcal{N}$ sheaves of $\mathcal{O}_Y$-modules. Then one has a canonical isomorphism $f^*(\mathcal{M} \otimes_{\mathcal{O}_Y} ...
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385 views

What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
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566 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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How do I teach university level mathematics to myself? [closed]

So here I go, I have enrolled myself in maths major this year but due to less marks in SSC I couldn't secure admission in a good university so I have to take admission wherever I could get with my ...
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1answer
281 views

How are $Spec \mathbb{Q}, Spec \mathbb{R}, Spec \mathbb{C}$ etc different?

By definition $Spec k$ is a point for any field $k$. So $Spec \mathbb{Q}, Spec \mathbb{R}, Spec \mathbb{C}$ etc are all the same as topological spaces. But according to the natural inclusion map $$ ...
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Existence of divisors of degree one on a curve over a finite field

Let $C$ be a smooth, geometrically irreducible projective curve defined over a finite field $\mathbb{F}_q$. Given a (scheme-theoretic) point $x \in C$, define the degree of $x$ to be the degree of ...
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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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What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
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391 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
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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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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 ...
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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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246 views

Real points of a complex curve

Since the "real points" of a complex curve can mean a couple of different things, bear with me while I'm annoyingly formal here. Consider first a cubic curve $y^2 = x^3 + a x + b$. Write $$S := \{ ...
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318 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
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Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
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The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...
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338 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
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386 views

Variety of pairs of product-zero matrices

Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components ...
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Irreducible components of topological space

Let $X$ be a topological space. Let $\Sigma$ be the set of irreducible components of $X$. Let $X=\cup_{i\in I} X_i=\cup_{j\in J} Y_j$, $X_i,Y_j\in \Sigma $ for some index set $I,J$. $X_i$'s are ...
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Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies? x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n ...
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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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Why is there no polynomial parametrization for the circle?

How does one show that the unit circle admits no polynomial parametrization? What is needed for this, are there general criteria? Thanks
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553 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
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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 ...