# Questions tagged [algebraic-geometry]

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. This tag should not be used for elementary problems which involve both algebra and geometry.

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### $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
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### Undergraduate algebraic geometry textbook recommendations

What are the best algebraic geometry textbooks for undergraduate students?
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### If $\operatorname{Spec} A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\operatorname{Spec} A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ is an ...
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### Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
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### A sufficient and necessary condition for $\mathbb{C}(f(x),g(x))=\mathbb{C}(x)$?

Let $f=f(x),g=g(x) \in \mathbb{C}[x]$. Is there a sufficient and necessary condition for $\mathbb{C}(f(x),g(x))=\mathbb{C}(x)$? This paper is perhaps relevant, although it deals with polynomials ...
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### What is an algebraic variety?

I'm studying abelian varieties from Milne's book, but I'm having difficulty juggling different conventions and definitions of basic concepts, like those of algebraic and projective varieties. First, ...
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### Irreducible polynomials and affine variety

Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
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### Use irreducible fibers to show $X$ is irreducible

Let $\pi:X\rightarrow Y$ be a proper morphism to an irreducible variety and all fibers of $\pi$ are nonempty, irreducible, and of the same dimension. Show $X$ must also be irreducible. Thanks (Any ...
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### on the adjointness of the global section functor and the Spec functor

In fact, it is an exercise on Hartshorne, Ex 2.4 to the second chapter of it (p.79): Let $A$ be a ring and $(X,\mathcal{O}_X)$ be a scheme. Given a morphism $f:X\longrightarrow \operatorname{Spec} A$,...
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### Number of points in the fibre and the degree of field extension

Let $X,Y$ be varieties over $\mathbb{C}$, $k(X), K(Y)$ be function fields of $X, Y$. Suppose $\pi: X \to Y$ is a dominant, $\textit{injective}\$ morphism, why the degree of the function field ...
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### Why is every Noetherian zero-dimensional scheme finite discrete?

In the book The geometry of schemes by Eisenbud and Harris, at page 27 we find the exercise asserting that Exercise I.XXXVI. The underlying space of a zero-dimensional scheme is discrete; if ...
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### Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain

I want to show that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw)\subset\mathbb{A}^4$, is not a unique factorization domain. Morally, all we need to do is find some nonzero element that can ...
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### Couple facts about fibers of a morphism of schemes

Let $f: X \to Y$ be a morphism of schemes, take $y \in Y$ and let $k$ be the residue field of $y$. We also have $i_y: \operatorname{Spec}{k} \to Y$. Then we can form a fiber product $Z$ which is the ...
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### Why Zariski topology?

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ...
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### Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
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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 $\mathcal{O}(-1)$...
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### Why is it "easier" to work with function fields than with algebraic number fields?

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes ...
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### 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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### Surjective morphism of affine varieties and dimension

I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry: Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ ...
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### Open affine neighborhood of points

$X$ is a variety and there are $m$ points $x_1,x_2,\cdots,x_m$ on $X$. Can we find an open affine set which contains all $x_i$s?
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### Direct Image by a Blow up

Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ a long of $Y \subset X$, with exceptional divisor $E$ and $\text{dim}Y > 0$, where $X$ and $Y$ are smooth projectives ...
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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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### 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 I....
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### Intersection of open affines can be covered by open sets distinguished in *both*affines

Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$. It's not true in general that $U \cap V$ is affine, so if we ...
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### Spectrum of $\mathbb{Z}^\mathbb{N}$

Is anything known about the spectrum of $\mathbb{Z}^{\mathbb{N}}$? Notice that the fiber of $\mathrm{Spec}(\mathbb{Z}^{\mathbb{N}}) \to \mathrm{Spec}(\mathbb{Z})$ at a non-zero prime ideal $(p)$ is ...
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