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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.

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.

29074 questions
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Extension of a Line Bundle given on the generic fibre

Let $C$ be a regular projective integer curve over a field $k$ and $X\rightarrow C$ a projective $k$-Variety with generic fibre $X_\eta$. Why is there for every line bundle (invertible sheaf) $L$ on $X_\eta$ a nonempty open subset $U\subseteq C$ and…
Altinior
  • 143
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Prove that if $V$ and $W$ are affine varieties, then $V \times W$ is an affine variety.

I am working on the problem from "Ideas, Varieties and Algorithms" by David Cox, John Little and Donal O'Shea. Here is the homework problem for my course. Let $V \subset k^n$ and $W \subset k^m$ be two affine varieties, and let $$\begin{aligned} V…
NasuSama
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Quotient varieties

I have a variety $V$ given by polynomial equations. These equations admit a lot of symmetry. This means there are a lot of automorphisms on $V$. I want to get rid of this symmetry. So I somewhat want to form a quotient variety. However, googling for…
wood
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Solutions to "Vakil - Foundations of Algebraic Geometry" exercises

I think the notes of Professor Ravi Vakil are a great source to learn algebraic geometry. The exposition is very clear, and much effort was put to give an intuitive picture together with a flawless formal precision. Of course throughout the book…
Abramo
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Hilbert polynomial Twisted cubic

Let $C$ the twisted cubic in $\mathbb{P}^3$ defined as $V(XZ-Y^2, YW-Z^2,XW-YZ)$. I have to calculate Hilbert polynomial of $C$, that I denote $P_C(n)$. In order to calculate Hilber polynomials in general, I consider the exact twisted sequence $$ 0…
ArthurStuart
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About the existence of a generic point on an irreducible closed subset of a prescheme

This is Proposition 2 on page 81 of Mumford's The Red Book of Varieties and Schemes: Let $X$ be a prescheme, and $Z \subset X$ an irreducible closed subset. Then there is one and only one point $z \in Z$ such that $Z = \overline{\{ z \}}$. Proof.…
sunkist
  • 1,133
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Are "$n$ by $n$ matrices with rank $k$" an affine algebraic variety?

Identify the set of all complex $n$ by $n$ matrices with $\mathbb{C}^{n^2}$. We say a subset $S \subset \mathbb{C}^{n^2}$ is an affine algebraic variety if $S$ is the common zero set of a collection (possibly infinite or uncountable) of polynomials…
Elliott
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defining a dominant rational map from an algebra homomorphism (Theorem I 4.4 in Hartshorne)

The following shows up in the proof of theorem I 4.4 in Hartshorne. Let $X,Y$ be varieties and $\theta: K(Y) \rightarrow K(X)$ a homomorphism of $k$-algebras. We want to construct a dominant rational map $X \rightarrow Y$. We can assume that $Y$ is…
Manos
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Ruled Surfaces and Rational Fibrations

I have a few doubts about Ruled Surfaces. Edit: We're working on ground field $\Bbb{C}$ The term ruled stands here for birationally ruled Let $S$ be an algebraic smooth surface and $C$ a smooth projective curve. If there is a morphism…
Heitor Fontana
  • 3,676
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Notation in Hartshorne Exercise 1.2.6

I am now doing Hartshorne Problem 1.2.6. Hartshorne 1.2.6: Let $Y$ be a projective variety with homogeneous coordinate ring $S(Y)$, show that $\dim S(Y) = \dim Y + 1$. [Hint: Let $\varphi_i : U_i \to \Bbb{A}^n$ be the homeomorphism of (2.2), let…
user38268
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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 extension $[K(X) : K(Y)] =1$? If $\phi : X \to Y$ is a…
Li Yutong
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On an exercise from Hartshorne's Algebraic Geometry (Ex. I.1.8)

My question is in fact the exercise 1.8 page 8 in Algebraic Geometry by Robin Hartshone. Let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Let $H$ be a hypersurface in $\mathbb{A}^n$ and assume that $Y\nsubseteq H$. Then prove that…
Arsenaler
  • 3,930
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Why is a variety etale locally like affine space?

I remember from a talk somebody saying that ''a scheme is etale locally like affine space'' and I wonder what this could mean. Let $Var/K$ be the site of varieties over a field $K$ with the etale topology. My first guess for a meaning of the above…
Ronald Bernard
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Finite morphism that is not projective

There are two definitions of a projective morphism. Hartshorne: A morphism $f: X\to Y$ is projective if it factors as $f=gi$, where $i: X\to P_Y^n$ is closed imbedding and $g: P_Y^n\to Y$ is canonical. Ravi's notes: Morphisms of the form $Proj(L)\to…
minimax
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2 answers

Tangent space of a point of an algebraic variety

Let $V$ be a non-singular affine variety in $\mathbb{C}^n$. $V$ can be regarded as a complex manifold. Let $p = (a_1,\dots,a_n) $ be a point of $V$. Let $\mathcal{O}_p$ be the local ring of $V$ at $p$. A tangent vector $v$ at $p$ is a derivation…
Makoto Kato
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