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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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Irreducible components of an algebraic set

I am totally stack in how to do this exercise : How can I find the irreducible components of $ V(X^2 - XY - X^2 Y + X^3) $ in $A^2$(R)? Given an algebraic set, what is the consideration I have to make to find them? I know the definitions of…
User
  • 531
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Why is this map almost an étale covering?

Let $V$ and $W$ be absolutely irreducible quasi-projective varieties over $K$ with char $K=0$ with dim $V=$ dim $W$ and let $f: W \rightarrow V $ be a dominant morphism. Why is it possible to make $f$ into an étale covering by restricting $V$ ? I…
kalb
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Generalization of the statement that $\mathbb{A}^2\setminus (0,0)$ is not affine?

$\mathbb{A}^2\setminus (0,0)$ is often given as an example of a variety that is not affine. I am trying to understand this example better by seeing it as a special case of a natural general claim. Claim: Let $X$ be an affine variety over an…
Ben Blum-Smith
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Line bundle corresponding to the Segre embedding

I am trying to understand the theorem that characterizes morphisms to projective space as equivalent to the data of a line bundle together with global sections generating it. I tried to find the corresponding line bundle associated with the Serge…
user115940
  • 1,969
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Morphism into a Dedekind scheme

I am trying to solve the following exercise without using Zariski's main theorem. Let X be an integral scheme of dimension 1 and $f:X \rightarrow Y$ a birational, separated morphism of finite type where Y is a Dedekind scheme. Show that f is an open…
RogozjinRob
  • 151
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Where can I find details of the proof of Weil's theorem?

I heard that Weil proved the Riemann hypothesis for finite fields. Where can I found the details of the proof? I found the following sketch but I was unable to fill the details: Motivation: I try to understand the elementary theory of finite fields…
studying
  • 197
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What is a finitely generated sheaf?

I do exercise asking to show on Noetherian space $X$, any subsheaf $\mathscr{R} \subseteq \Bbb{Z}_U$ is finitely generated. $\Bbb{Z}_U$ is sheaf $i_{!}(\Bbb{Z}|_U)$ for $U \subseteq X$ open. What is the meaning of a finitely generated sheaf?…
Dylan B.
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Any finite algebraic subset of $\mathbb A^2_k$ can be determined by 2 equations

Please give me a hint for this exercise. Show that every finite subset of the affine plane, $\mathbb A^2_k$, over an algebraically closed field, $k$, can be determined by two equations. I am using Shafarevich's book.
Framate
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Doubt in a proof in algebraic geometry

I'm trying to understand this proof: Theorem If $\nu_{n,d}:\mathbb P^n\to \mathbb P^N$ is a Veronese embedding and $X$ a hypersurface of degree $d$, then $\nu(X)$ is a hyperplane in $\nu (\mathbb P^n)$. Proof We write the homogeneous coordinates in…
user108205
  • 604
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Hartshorne, exercise II.2.18: a ring morphism is surjective if it induces a homeomorphism into a closed subset, and the sheaf map is surjective

Let $\phi:A\to B$ be a ring morphism, and let $f:X=Spec(B)\to Y=Spec(A)$ be the induced map of affine schemes. I'm trying to show that if $f$ is a homeomorphism onto a closed subset of $Y$ and $f^\#:\mathcal{O}_Y\to f_*(\mathcal{O}_X)$ is…
Bruno Stonek
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Equation of a regulus

How does one compute the equation for the points on a regulus given the equations of 3 mutually skew lines that define it? I saw the definition of a regulus in a geometry book but I had trouble computing its equation given 3 skew lines, and…
tehjh
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Questions about the definition of a Cartier Divisor from Liu pg 256

I am reading Definition 1.17 of Liu on what a Cartier Divisor is: I have several questions concerning this definition. Question 1: It makes sense to say to me that an element $D \in H^0(X, \mathcal{K}_X^\ast/\mathcal{O}_X^\ast)$ can be…
user38268
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2 answers

stuck on computing points of ramification of a map of curves via sheaf of relative differentials

Let $C$ be the curve in $\mathbb{C}^2$ given by $f(x,y)=x^3+y^3-y=0$ and let g:$C \to \mathbb{C}$ be the projection $g(x,y)=x$. I want to find where this map $g$ is ramified, and do this by computing the sheaf of relative differentials. I have…
ymo
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Why is $H^n(\mathbb{P}_k^n, \omega) \cong k$?

Hartshorne makes this sound like a coincidence: we us Cech cohomology on the usual open affine cover $\mathcal{U}$ to get the chain complex $$\check{C}^\bullet\left(\mathcal{U}, \bigoplus_{n \in \mathbb{Z}} \mathcal{O}(n)\right): 0 \to \bigoplus_i…
Daniel McLaury
  • 24,656
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What does $\bigoplus_{n \in \mathbb{Z}} \mathcal{O}_X(n)$ represent geometrically?

For simplicity let's say $A$ is a noetherian ring, $S = A[x_0, \ldots, x_r]$, and $X = \operatorname{Proj} S = \mathbb{P}^r_A$. I want to understand what, if anything, the sheaf $$\mathscr{F} := \bigoplus_{n \in \mathbb{Z}} \mathcal{O}_X(n)$$ means…
Daniel McLaury
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