# 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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### classifying closed subschemes of an affine scheme

Looking at the Stacks Project proof that every closed subscheme of an affine scheme $\operatorname{Spec} A$ is of the form $\operatorname{Spec} A / I$, where $I$ is an ideal of $A$. It uses the fact ...
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### Exceptional divisor of blow-up in a non-rational point

Let $X$ be an prjective variety over $k$ and $Spec(L) \to X$ be a point of degree $n$ on $X$. Is there any description of $Bl_{Spec(L)}X$ in terms of $L$? Is there any connection between them in the ...
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### definition of closed immersions of schemes

Qing Liu's algebraic geometry and arithmetic curves defines closed immersion as follows(definition 2.22): We say that a morphism $(f, f^{\#}): (X, O_{X}) \to (Y, O_{Y})$ is an open immersion (resp. ...
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### Pullback is not exact

Let $f: X \to Y$ be a map of schemes. Then we have the "quasicoherent" pullback which takes $F \in Qcoh(Y)$ and gives $f^* F = \mathcal{O}_{X} \otimes_{f^{-1} \mathcal{O}_{Y}} F$. This ...
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### Decomposing a big and nef divisor into ample + effective

Let $\pi: X \to \mathbf{P}^2$ be the blow-up of the projective plane at one point. Write $H$ for a hyperplane divisor of $\mathbf{P}^2$. The pullback $\pi^* H$ is big and nef, so it can be written in ...
### What is the right adjoint to the functor $\sf{Psh}\to\sf{Set}$ which evaluates the presheaf on the whole space?
$\newcommand{\O}{\mathcal{O}}\newcommand{\T}{\mathcal{T}}\newcommand{\op}{^{\sf{op}}}\newcommand{\set}{\sf{Set}}\newcommand{\ps}{\sf{Psh}_{\T}}$Let $\T$ be a topological space and $\O(\T)$ the poset ...