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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What is an intuitive meaning of genus?

I read from the Finnish version of the book "Fermat's last theorem, Unlocking the Secret of an Ancient Mathematical Problem", written by Amir D. Aczel, that genus describes how many handles there are ...
17
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3answers
1k views

Polynomial map is surjective if it is injective

A friend of mine told me the following fact: If $k$ is any algebraically closed field, then a polynomial map $f\colon k^n\to k^n$ of affine space $k^n$ is surjective if it is injective. The ...
15
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3answers
1k views

Hensel's Lemma and Implicit Function Theorem

In the literature and on the web happened to me several times to read confused or simply cryptic assertions regarding the fact that Hensel's Lemma is the algebraic version of Implicit Function ...
14
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2answers
1k 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 ...
19
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3answers
1k views

Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?

I know that there are isomorphic projective varieties which have nonisomorphic coordinate rings, but I'm a little mystified as to "why" this is the case. Why doesn't a usual functoriality proof go ...
13
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2answers
569 views

Usefulness of completion in commutative algebra

After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions: (i) What is the usefulness of the concept of completion in ...
24
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1answer
712 views

Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
22
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0answers
2k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where ...
21
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3answers
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What is a local parameter in algebraic geometry?

Shafarevich offers the following theorem-definition: "At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
10
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3answers
449 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 ...
9
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3answers
751 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
8
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1answer
1k 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 ...
23
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1answer
384 views

Homotopy invariance of the Picard group

Is the Picard group of a scheme homotopy invariant in the sense that the projection $\pi : X \times \mathbb{A}^1 \to X$ induces an isomorphism $\mathrm{Pic}(X) \cong \mathrm{Pic}(X \times ...
22
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0answers
421 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
22
votes
1answer
286 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
16
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4answers
3k 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
1answer
248 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
2answers
763 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 ...
12
votes
1answer
711 views

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 ...
10
votes
2answers
639 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 ...
10
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1answer
961 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
votes
1answer
891 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? ...
9
votes
2answers
249 views

Why is it so difficult to find beginner books in Algebraic Geometry?

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
8
votes
2answers
416 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
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1answer
273 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 ...
5
votes
2answers
333 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
435 views

Reference for Gauss-Manin connection

I wish to understand the notion of ``Gass-Manin connection''. I have some understanding of differential geometry, topology and algebraic geometry. Where should I begin? IF the sources are freely ...
14
votes
1answer
352 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 $\mathbb{C}^{n}$. ...
11
votes
3answers
410 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 ...
10
votes
3answers
326 views

When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the ...
10
votes
1answer
729 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 ...
9
votes
1answer
66 views

An element of $f$ of a function field such that $P$ is the only pole of $f$.

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of all places of $F$. Prove that if $P \in S$, there is an element $f$ of $F$ such that $P$ is the ...
9
votes
1answer
627 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
1answer
505 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} ...
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
206 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
181 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
420 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
408 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?
24
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2answers
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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 ...
15
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3answers
907 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
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1answer
619 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
308 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
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1answer
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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 ...
11
votes
2answers
570 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
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2answers
271 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
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2answers
578 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 ...
9
votes
4answers
917 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
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1answer
70 views

Poles of a sum of functions

The other question is here. Let $F$ be a function field in one variable over a field $k$. Let $S$ a nonempty finite subset of the set of all places of $F$. Prove that if $P \in S$, there is an ...