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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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Underlying set of the scheme theoretic fiber

Categorical constructions in the category of schemes usually do not preserve the underlying sets. For example, the underlying topological space of the product of schemes is not the topological product of the underlying spaces. It is true that if…
user115940
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Projective space is not affine

I read a prove that the projective space $\mathbb P_{R}^{n}$ is not affine (n>0): (Remark 3.14 p72 Algebraic Geometry I by Wedhorn,Gortz). It said that the canonical ring homomorphism $R$ to $\Gamma(\mathbb P_{R}^{n}, \mathcal{O}_{\mathbb…
user93417
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3 answers

Understanding the support of a function

This is from exercise 5.5.A of Vakil's lecture notes. Consider $f$, a function on $A: = \mathrm{Spec}(k[x,y]/(y^2, xy))$. Show that its support either empty, the origin or the whole space. Now, I know that the support of any function $f$ must be…
Rodrigo
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Gluing schemes Hartshorne example

In example 2.3.5 Hartshorne says Let $X_1$ and $X_2$ be schemes. Let $U_1 \subseteq X_1$ and $U_2 \subseteq X_2$ be open subsets, and let $\varphi: ( U_1, \mathcal{O}_{X_1 \mid U_1}) \to ( U_2, \mathcal{O}_{X_2 \mid U_2})$ be an isomorphism of…
Grobber
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Finite implies Quasi-finite

Please help if you know a proof or a good reference for the following fact (exercise 3.5, Hartshorne's text). Fact. A finite morphism of schemes $f: X \rightarrow Y$ is quasi-finite. Here, the definition of quasi-finite is taken as $f^{-1}(q)$ is…
mr.bigproblem
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Do the pictures in Hartshorne Ex. 1.5.1 make sense?

I have done exercise 1 of section 1.5 of Hartshorne and am able to determine that the curves (a),(b),(c) and (d) are respectively those with a tacnode, node, cusp and triple point. Now when I did this exercise in the back of my mind these were…
user38268
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The definition of a smooth morphism is too abstract. Can we make it simpler in a special case?

The definition of a smooth morphism of schemes $f:X \rightarrow S \space$ given here on stacks project is so abstract that it is intractable to me. $f$ is called smooth if for every point $x$ of $X$, there exists an affine open neighborhood $U$…
D_S
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Why do we have to deal with constructible sets?

I'm a beginner in algebraic geometry. Recently, I learned about constructible sets and Chevalley's theorem. In a Noetherian space, constructible sets are some kind of finite union of open and closed sets, and Chevalley's theorem states that if $f :…
Seewoo Lee
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Hartshorne Exercise II. 6.3

I've been turning this exercise over for a while, and I appear to be stuck in particular on part (c). The question is: Let $V$ be a projective variety in $\mathbb{P}^n$ of dimension $\geq 1$ and nonsingular in codimension $1$. Let $X$ be the affine…
A. Thomas Yerger
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Why should moduli spaces be compact?

This is going to be a vague question. It seems like enumerative geometry people really like their moduli spaces to be compact (/projective). For example as soon as we mention $M_{0,n}$ we start talking about finding out a good choice for…
Rob Silversmith
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Questions about Serre duality

I've read the section "Serre duality" in Hartshorne's book and have several questions. 1) In Remark 7.1.1 it is claimed that on $X = \mathbb{P}^n$ $\alpha = \frac{x_0^n}{x_1 \cdot ... \cdot x_n} d(\frac{x_1}{x_0}) \wedge ... \wedge…
Martin Brandenburg
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Morphisms of finite type are stable under base change

I am trying to prove that morphisms of finite type are stable under base change, but I am having some trouble moving from the case where everything is affine to the general case. Suppose $f:X \rightarrow Y$ is a morphism of finite type and $Y'$ is…
Vitaly Lorman
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Does presheaf send the empty set to zero?

Hartshorne requires (in his Algebraic Geometry) a presheaf (of abelian groups) to send the empty set to the zero group. But Wikipedia's definition doesn't have that condition (just a contravariant functor from the category of open subsets to the…
ashpool
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Tensor product of $\mathscr{O}_X$-modules which results in a presheaf.

Background: Over a locally ringed space $X$, if we define the tensor product of two $\mathscr{O}_X$-modules $\mathscr{F}$ and $\scr{G}$ naively as $U \mapsto \mathscr{F}(U) \otimes \mathscr{G}(U)$, we won't necessarily get a sheaf and we sheafify…
beeflavor
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The local ring of the generic point of a prime divisor

Suppose $X$ is a noetherian integral separated scheme which is regular in codimension one, i.e. every local ring $O_x$ of dimension one is regular. Let $Y$ be a prime divisor of $X$, i.e. $Y$ is a closed integral subscheme of codimension one. The…
mez
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