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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Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.

In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that ...
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Material in a first course in algebraic geometry?

First I would like to say that my question is not about what books to use in algebraic geometry; for this there are many threads that discuss this on Math.SE and on MO. My question is about what ...
22
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576 views

An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
22
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1answer
642 views

Why is the Hessian of an irreducible polynomial not zero?

Let $k$ be an algebraically closed field, $\operatorname{char}k=0$, $F$ be an irreducible homogeneous polynomial of degree$>1$ in $k[X,Y,Z]$, and ...
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geometric motivation for negative self-intersection

consider the blow-up of the plane in one point. Let $E$ the exceptional divisor. We know that $(E,E)=-1$. Which is the geometrical reason for which the auto-intersection of $E$ is $-1$? In general ...
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Interesting implicit surfaces in $\mathbb{R}^3$

I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement! Some ...
21
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Why is the hard Lefschetz theorem “hard”?

Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear ...
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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 ...
21
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405 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
21
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325 views

Is there a complex surface into which every Riemann surface embeds?

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the ...
21
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506 views

functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
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What is Spec of the Adeles?

Let $K$ be a global field and $A_K$ the ring of adeles. What are the prime ideals of $A_K$? I have been told that a full proof of this is quite subtle, but have been unable to find a reference ...
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227 views

What geometrical obstructions to $M$ being flat do elements which map to 0 in $M \otimes I$ represent?

I'm trying to get geometric intuition for the notion of a flat module over a ring, and am running into some problems with my intuition. I am comfortable with flat modules and tensor products from the ...
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3answers
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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 ...
20
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602 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 ...
20
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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 ...
20
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Precise connection between Poincare Duality and Serre Duality

The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from ...
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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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630 views

cones in the derived category

If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' ...
20
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1answer
255 views

Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
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What does $H^0(Y',f^*N_Y)$ measure?

Let $X$ be a smooth variety and let $Y\subset X$ be a smooth subvariety. Let $f:Y'\to Y$ be a (say, finite surjective) morphism. When $f$ is the identity, the cohomology group $H^0(Y',f^*N_Y)$ ...
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Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
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Is there a way of working with the Zariski topology in terms of convergence/limits?

As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...
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875 views

Who first called the Grothendieck's schéma scheme?

Grothendieck called "schemes" schémas in French. I find it strange we call them schemes. In fact, Grothendieck called them (pre-) schemas(this is an English word) in his talk(in English) at Proceeding ...
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How should I think about very ample sheaves?

Definition. [Hartshorne] If $X$ is any scheme over $Y$, an invertible sheaf $\mathcal{L}$ is very ample relative to $Y$, if there is an imersion $i:X \to \mathbb{P}_Y^r$ for some $r$ such that ...
19
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1answer
642 views

Is there an algebraic reason why a torus can't contain a projective space?

Let $X$ be an abelian variety. As abelian varieties are projective then $X$ contains lots and lots of subvarieties. Why can't one of them be a projective space? If $X$ is defined over the complex ...
19
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1answer
510 views

Can an integral scheme have closed points of both positive and zero characteristic?

Background Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset. Given any point $p$, ...
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What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?

Let $X$ be a non-singular projective variety over $\mathbb{Q}$. Consider on the one hand $H^i_B(X(\mathbb{C}),\mathbb{Z}_\ell)$ the singular cohomology with value in $\mathbb{Z}_\ell$, and on the ...
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When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
19
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Hartshorne exercise II.5.12(b)

I've been working on the Hartshorne exercise in the title for quite a while, which goes like this: let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes, $\mathscr{L}$ a very ample invertible ...
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Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
19
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366 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
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Is there a proof of Bézout's theorem via residue theory?

Let's define intersection numbers as follows. Consider a collection $f_1,\dots, f_n$ of holomorphic functions on some neighborhood of zero in $\mathbb C^N$ cutting out divisors $D_1$, all of which ...
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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 ...
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4answers
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Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in ...
18
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3answers
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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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Slick proof the determinant is an irreducible polynomial

A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in ...
18
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2answers
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Did Zariski really define the Zariski topology on the prime spectrum of a ring?

The question is not: “Did Zariski really define the Zariski topology?” It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?” Here is the motivation. --- On page ...
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So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
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What are some applications outside of mathematics for algebraic geometry?

Are there any results from algebraic geometry that have led to an interesting "real world" application?
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“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, ...
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2answers
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What are normal schemes intuitively?

A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed. Some ...
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Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
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Reference for Algebraic Geometry

I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is ...
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Elliptic Curves and Points at Infinity

My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ...
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4answers
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Why do all parabolas have an axis of symmetry?

And if that's just part of the definition of a parabola, I guess my question becomes why is the graph of any quadratic a parabola? My attempt at explaining: The way I understand it after some ...
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1answer
695 views

Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?

Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields. I'm ...
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2answers
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Intuition for Blow-up.

If I blow up a complex manifold along a submanifold, can you give me a picture to have in mind for the blown-up manifold? Can you also tell me why this is the right picture?
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Walking on a torus

Everyone knows that when you walk on a sphere along a straight line, you eventually get back to the point you started from. I'm wondering about the same question for the torus. Obviously there are ...
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What problems, ideas or questions first got you interested in algebraic geometry?

Obviously, a lot of people are very interested in algebraic geometry. I suppose this means it is a fascinating area. However the few times I have tried to read introductory books and/or articles in ...