# 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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### Bijective map between two definition for extension for sheaf on base

Let $F$ be sheaf on base $\mathscr B =${$B_i,i \in I$}, there are two equivalent definitions for the extention for $F$, denoted by $\mathscr F$: Def1:$\mathscr F(U)$ := {$(f_p \in F_p)_{p \in U}:$ ...
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### Proving the Relative Nullstellensatz

Let $X \subseteq \mathbb{A}^n$ be an affine subvariety and let $A(X)$ be the corresponding coordinate ring. For $Y \subseteq X$ and $S \subseteq A(X)$ we define the following relative analogues to the ...
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### Inverse and direct limits and cohomology

Suppose $X_n$ is a tower (an inverse system), indexed by the positive integers, of smooth projective complex varieties, and call $X = \varprojlim_n X_n$ as schemes over $\mathbf{C}$. Fix $\ell$ a ...
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### Dominant morphisms between projective varieties

Suppose $f : X\to Y$ is a finite surjective morphism of projective integral complex varieties, and let $g : X'\to X$ and $h : Y'\to Y$ be surjective birational morphisms from smooth projective ...
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### What is the relation between an affine variety and an affine space?

I was learning about algebraic varieties and the wikipedia page presents affine varieties as the "conceptually easiest" type. Having read about affine spaces and affine varieties, I am ...
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### Figure interpretation: $p$ in the affine plane is replaced by a projective line in the blow up.

In my text book the point in the affine plane is supposed to be replaced with "all the lines through $p$, a copy of $\mathbb{P}^{n-1}$." When I examine the figure of the blowup, there is one ...
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### height of regular sequence modulo minimal prime

Liu Qing's book Algebraic Geometry and Arithmetic Curves has the following statement on 6.3.11: Let $Y$ be a locally Noetherian scheme and $i \colon X \to Y$ a regular immersion of codimension $n$. ...
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### The difference of the vector bundle's definitions between the Hartshorne and the Stacks Project

I currently learn scheme theory by self-learning racking my brains. In that time, a difference give me a headache: The difference of the vector bundle's definitions between Exercise II.5.18. from the ...
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### Question on smooth affine curves over finite fields and Hasse-Weil

I don't have a strong background in algebraic geometry, so my question could seems trivial, but I would like to know more about it. So if you have a book to recomend, thanks in advance! Suppose you ...
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### Elimination property on scheme morphisms locally of finite type

I was trying to prove the following scheme morphism exercise. If $f: X \rightarrow Y, g: Y \rightarrow Z$ are such that $g \circ f$ is locally of finite type then $f$ is locally of finite type. I ...
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### Motivations for smooth morphisms, unratified morphisms and étale morphisms

I have learnt basic definitions and some properties by reading stacks project which is good reference. However, in my stupid view, I can not understand why we define them in this way. I have read two ...
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### Elements in projective space

Consider the projective line for simplicity. I was wondering was it means to look at the stalk at $0$. What is $0$ even referring to? The ideal generated by $0$ in one of the copies of the affine line?...
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### Curves homeomorphic under Zariski topology

Prove any two curves over some field $k$ are homeomorphic, where $k$ might not be algebraically closed. Curves are defined to be varieties (integral separated scheme of finite type) of dimension $1$. ...
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### Geometric interpretation of Minimal Prime Ideals

I'm doing Gathmann's commutative algebra notes Exercise 2.23: Let $I$ be an ideal in a ring $R$ with $I\neq R$. We say that a prime ideal $P\unlhd R$ is minimal over $I$ if $I ⊂ P$ and there is no ...
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### Classify U(1) bundle over $\mathbf{P}^3$, and its topological invariants

I am interested in knowing the classification of the U(1) bundle over the complex projective space $\mathbf{P}^3$. This is effectively a U(1) bundle over the 6-manifold $M^6$. What are the possible ...
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Mazur's theorem completely classifies the possibilities of $E_{tors}(\mathbb{Q})$ for an elliptic curve $E/\mathbb{Q}$, including the fact that only finitely many groups occur. What happens with $E_{... • 24.1k 0 votes 0 answers 24 views ### Rational points on fiber product I am trying to figure how rational points in fiber products look like. Let$K$be a field,$A=K[x_1, \ldots, x_n], B=K[y_1, \ldots, y_m]$. Rational points in SpecA and SpecB look like$(x_1-a_1, \...
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Neukrich defines a finite extension $L / K$ of henselian fields to be unramified if the extension of residue fields $\lambda / \kappa$ is separable and $[L : K] = [\lambda : \kappa]$ (emphasis added). ...