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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.

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Global sections of a proper variety over an arbitrary field

While doing some research, I stumbled upon the following fact that I'd taken for granted. Theorem: Let $ X $ be a proper variety over a field $ k $ (variety = geometrically integral, separated, finite type). Then $ \Gamma(X, \mathcal{O}_X) $ is a…
Cranium Clamp
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Questions on Reduced Induced Closed Subscheme

I've just read the definition of a closed subscheme in Hartshorne's recently and I collected here and there (notes that people put online) the following statement. Claim. Suppose that $(X,\mathcal{O}_X)$ is a scheme, $Y \subset X$ is a closed…
mr.bigproblem
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Ring of Regular functions on $\Bbb{A}^2 - \{(0,0)\}$

Suppose I want to determine the ring of regular functions on $U = \Bbb{A}^2 - \{(0,0)\}$. Now I can do this assuming the following fact: Fact: If $f$ is regular on $U$, then we can write $f = g/h$ with $g,h$ polynomials in $k[x,y]$ such that…
user38268
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The analytic and the algebraic "small disc"

I would like to understand the relation between an analytic object (the so called "small disc") and an algebraic one (the spectrum of a DVR). The framework is that of one-parameter families of complex curves $X\to S$. Analysis: $S_0=\{z\in \mathbb…
Brenin
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Is the natural map $f^* f_* \mathcal{F} \to \mathcal{F}$ surjective?

I'm trying to solve Exercise III 12.4 from Hartshorne's Algebraic geometry. There is a flat projective morphism $f: X \to Y$ of schemes of finite type over an algebraically closed field $k$. Also $Y$ is assumed to be integral, and all fibers are…
red_trumpet
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exact sequence of ideal sheaves (Hartshorne Theorem III.3.7)

Let $X$ be a scheme and $Y$ a closed subscheme. Let $i : Y \hookrightarrow X$ be the inclusion. Then, we define the ideal sheaf of $Y$, denoted $\mathcal {I}_Y$to be the kernel of the morphism $i^{\sharp} : \mathcal{O}_X \rightarrow i_{*}…
Sang Cheol Lee
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An fpqc morphism is a "quotient" morphism (as a continuous map between topological spaces)

Let $f: X \to Y$ be a faithfully flat quasi-compact morphism. Then for a subset $V \subseteq Y$, $V$ is open in $Y$ iff $f^{-1}(V)$ is open in $X$? I know this is EGA IV2 2.3.12. But its proof is very complicated for me. (I'm not familiar with…
k.j.
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$X \cong \operatorname{Proj} \oplus_n H^0(X, O(n)) $

Let $ X \subset \mathbb{P}^n$ be a smooth closed subvariety, and $O(1)$ is the pull-back of the line bundle of $O(1)$ on $\mathbb{P}^n$. Then it is claimed: $$X \cong \operatorname{Proj}\left(\bigoplus_n H^0\big(X, O(n)\big)\right)\;.$$ This seems…
Li Yutong
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Algebraic de Rham Cohomology of Projective Space over $\mathbb{C}$

I am looking to understand algebraic de Rham cohomology a bit better, and I realized that I don't really understand something that should be quite elementary. Take for example $\mathbb{P}^1_{\mathbb{C}}$. We know that the algebraic de Rham…
baltazar
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How to prove that a mapping is rational/regular?

I'm in an algebraic geometry class and I've been asked, for homework, to prove that certain maps are regular/rational. I have no idea how to do this. We did prove that the only regular functions on $\mathbb{P}^n$ are constant functions, using the…
Julien Clancy
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Automorphisms of the Weil restriction

Given a complex variety, say $\mathbb{P}^1_\mathbb{C}$, I want to compute $$\text{Aut}_\mathbb{R}(\mathbb{P}^1_\mathbb{C}\vert_{\mathbb{R}}).$$ Using the definition of Weil restriction, one can show that for a complex variety $X$,…
Pulcinella
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What do we lose if we only consider quasi-projective varieties?

What do we lose if we only consider quasi-projective varieties? What are merits of considering varieties which are not quasi-projective?
Makoto Kato
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Concrete Example: Subsheaf is not Quasi-Coherent

In II.5.2.4 of Hartshorne (the example) Hartshorne remarks: If $Y$ is a closed subscheme of a scheme $X$, then the sheaf $\mathcal{O}_X| _Y$ is not in general quasi-coherent on $Y$. In fact, it is not even a sheaf of $\mathcal{O}_Y$ modules in…
A. Thomas Yerger
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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?
Summer
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What is the tangent space for intersection point of irreducible varieties.

Given a (real) plane, I know the definition of tangent space for irreducible varieities. (If $P \in V(f_1, \cdots, f_n)$ irreducible, then $T_P V = V(f^{(1)}_{1,P},\cdots, f^{(1)}_{n,P})$. Suppose $V=V_1\cup V_2$ is an irreducible decomposition.…
Gobi
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