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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343 views

Example I.4.9.1 in Hartshorne (blowing-up)

Let $Y$ be the irreducible curve of $\mathbb{A}^2$ given by $y^2 = x^2(x+1)$. Let $t,u$ be homogeneous coordinates of $\mathbb{P}^1$. Then the total inverse image of $Y$ under the blowing-up $\phi: X ...
8
votes
2answers
388 views

Trying to understand the use of the “word” pullback/pushforward.

Essentially, my question is the following : Is everything we call "pullback" or "pushforward" an actual categorical pullback/pushout? I have seen tons of pullbacks in differential geometry but we ...
8
votes
1answer
264 views

Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is ...
8
votes
1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
7
votes
1answer
847 views

What is the definition of surjective morphism of schemes?

Let $f: X \to Y$ be a morphism of schemes, when talking about the surjectivity of $f$, there are at least several possibilities. (1) $f$ is surjective at the level of sets, that is $\forall \ y \in ...
5
votes
2answers
252 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
5
votes
1answer
330 views

Localizations of quotients of polynomial rings (2) and Zariski tangent space

I am sorry, in the whole text below $k$ is just meant to be $\mathbb{C}$. This question is closely related to my previous one here. I am considering the two rings $k[X]=k[x,y,z]/\langle ...
15
votes
1answer
199 views

What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
13
votes
1answer
329 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 $\mathbb{C}^{n}$. ...
12
votes
2answers
716 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 ...
10
votes
1answer
655 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 ...
8
votes
1answer
434 views

Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?

Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
7
votes
1answer
203 views

Does a section that vanishes at every point vanish?

Let $R$ be the coordinate ring of an affine complex variety (i.e. finitely generated, commutative, reduced $\mathbb{C}$ algebra) and $M$ be an $R$ module. Let $s\in M$ be an element, such that $s\in ...
5
votes
1answer
146 views

Hartshorne proposition II(2.6)

I am studying the proof of the following proposition in Hartshorne - Let $k$ be an algebraically closed field. There is a natural fully faithful functor $Var(k)\longrightarrow Sch(k)$ from the ...
4
votes
1answer
368 views

Global sections of Serre's twisting sheaf

Let $I_1$ and $I_2$ be homogenous ideals in $A:=\mathbb{C}[X_0,\ldots,X_n]$. Assume that $I_1 \subset I_2$. Let $X=\mathrm{Proj} (A/I_1)$ and $Y=\mathrm{Proj}(A/I_2)$. Then, 1) Is it true that the ...
4
votes
1answer
346 views

Image of a morphism of varieties

Suppose $A$ and $B$ are two algebraic varieties, and $f:A\to B$ is a morphism of algebraic varieties. I guess it is true that $\text{im}(f)$ is itself an algebraic variety. But how to prove it?
23
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2answers
2k views

How many cubic curves are there?

It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant"). It is noteworthy ...
16
votes
4answers
2k views

“Real”-life applications of algebraic geometry

Before you tell me that this question has been asked, give me a bit of your time please to read this question because it is not as simple as it sounds. I did my undergraduate degree in mathematics, ...
15
votes
3answers
818 views

How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
15
votes
1answer
593 views

When does variété mean manifold?

Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English. French-English dictionaries online ...
12
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1answer
267 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?
11
votes
2answers
492 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
11
votes
2answers
262 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 ...
10
votes
2answers
483 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 ...
10
votes
1answer
817 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 $U_1 = U_2 =\mathbb{A}^1_k - (x)$, ...
9
votes
2answers
480 views

How to think of the Zariski tangent space

The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given ...
9
votes
1answer
509 views

Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
8
votes
2answers
323 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
8
votes
1answer
1k views

The preimage of a maximal ideal is maximal

I'm having some difficulty with this homework problem: Let $A, B$ be reduced finitely generated $\mathbb{C}$-algebras, and $\psi : A \longrightarrow B$ a $\mathbb{C}$-algebra homomorphism. Let ...
7
votes
1answer
591 views

Meaning of holomorphic Euler characteristics?

I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
7
votes
1answer
223 views

Cardinality and degrees of irreducible components of an affine variety

Let $Q_1,\ldots,Q_s \in k[x_1,\ldots,x_n]$, where $k$ is not necessarily algebraically closed (I'm thinking of $k$ as some field with positive characteristic $p$). I'm somewhat new to the world of ...
7
votes
1answer
231 views

the fundamental exact sequence associated to a closed space

Let $(X,\mathcal O_X)$ be an algebraic variety. If $Y\subseteq X$ is a closed subset, then we can equip $Y$ with a structure of algebraic variety $(Y,\mathcal O_Y)$. The function $i:Y\rightarrow X$ ...
7
votes
1answer
196 views

Why is the kernel of this strange polynomial homomorphism what it is?

I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious. Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ ...
6
votes
2answers
284 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
6
votes
1answer
194 views

$X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?

This question came from a proof in Algebraic Geometry by Hartshorne (Chapt3, Corollary 9.6) To be precise, Let $f:X \to Y$ be a flat morphism of schemes of finite type over a field $k$. Then is it ...
4
votes
3answers
316 views

Hartshorne's Exercise II.5.1 - Projection formula

I'm trying to solve Exercise 5.1 of Chapter II of Hartshorne - Algebraic Geometry. I'm fine with the first $3$ parts, but I'm having troubles with the very last part, which asks to prove the ...
3
votes
2answers
318 views

Sites or youtube videos to learn algebraic geometry

Is there any sites or free lecture videos to learn algebraic geometry? or should I call abstract algebra? I want to understand about rings, ideals, and real spectrum of rings but my understanding on ...
3
votes
3answers
352 views

What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
20
votes
1answer
529 views

What is $\operatorname{Spec}\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$?

Let $X = \operatorname{Spec} \mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$. I would like to describe $X$ set-theoretically. My questions are: Can one explicitly say what the elements in $X$ are? Is it ...
13
votes
3answers
723 views

Localization at a prime ideal is a reduced ring

Here is the question that I came up with, which I am having trouble proving or disproving: Let $A$ be a ring (commutative). Let $p \in Spec(A)$ such that $A_p$ is reduced. Then there exists an open ...
12
votes
3answers
1k views

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
12
votes
1answer
450 views

Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Serre's criterion for affineness (Hartshorne III.3.7) states that: Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine. ...
10
votes
2answers
362 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 $V$. ...
9
votes
4answers
752 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
8
votes
3answers
393 views

Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves

Let $X = \operatorname{Spec} k[x]_{(x)}$ which consists of two elements, the generic point $\zeta$ corresponding to the zero ideal and the closed point $(x)$. Define an $\mathcal{O}_X$-module ...
8
votes
1answer
364 views

Properties of quotient sheaves

I am reading Hartshorne's Algebraic Geometry, II.6 about Cartier Divisor. It is defined to be the global section of the sheaf $K^*/O^*$. Then it said: " thinking of the properties of the quotient ...
8
votes
0answers
225 views

Ideal of the pullback of a closed subscheme

Let $f : X \to Y$ be a morphism of schemes and $J \subseteq \mathcal{O}_Y$ a quasi-coherent ideal. Let $I$ denote the image of $f^* J \to f^* \mathcal{O}_Y = \mathcal{O}_X$. Then $I \subseteq ...
7
votes
2answers
291 views

Is this quotient ring $\mathbb{C}[z_{ij}]/\ker\phi$ integrally closed?

A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm ...
7
votes
2answers
834 views

Chern numbers of Projective Space

Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field ...
7
votes
4answers
519 views

Determining the generators of $I(X)$

Let $ X \subset \mathbb A^3 $ be the union of three coordinate axes. How do I determine the generators of $I(X)$? Also, how do I show it has at least 3 elements?