# 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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### Proving representibility of some functor in Gortz's Algebraic Geometry book.

I want to understand proof of next proposition ( Gortz's Algebraic Geometry, Proposition 8.4. ) Proposition8.4. Let $S$ be a scheme and let $v : \mathcal{E} \to \mathcal{F}$ be a homomorphism of ...
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### Confusion on proof of "dimension of the fiber is an upper semicontinuous function on the source" (From Vakil FOAG, Theorem 12.4.3).

I'm confused on a step of the proof of Theorem 12.4.3(a) in Vakil's FOAG. The theorem states: Suppose $\pi: X \rightarrow Y$ is a morphism of finite type $k$-schemes. Then the dimension of the fiber ...
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### Hilbert polynomial of the union of two intersecting lines

I am trying to compute the Hilbert polynomial of the union of two intersecting lines $L_1$ and $L_2$ in $\mathbb{P}^3$. In order to get it, I supposed WLOG that $L_1=Z(x,y)$ and $L_2=Z(y,z)$. Thus, ...
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### Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the internet ...
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### The Lie algebra of an algebraic group

I have a question concerning different visions of the Lie algebra of an algebraic group. To be more specific, let $G = \operatorname{Spec}(H)$ be a smooth (finitely generated) algebraic group over a ...
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### Isomorphism $\mathbb{A}^2 \to \mathbb{A}^2$

This exercise is taken from the notes of Gathmann of Algebraic Geometry, in particular it is the point d) of the exercise 4.13 of the version 2021/2022. If $f:\mathbb{A}^2 \to \mathbb{A}^2$ is an ...
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### the infinite category of pullback squares in an infinite stable category is also stable

I am currently reading Lurie's paper infinite stable category, in the proof of proposition 4.4, to show that every pushout square in an infinite stable category $C$ is also a pullback, he considers ...
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### Two-dimensional unipotent algebraic groups

How to prove that two-dimensional unipotent algebraic group over the field of characteristic 0 is commutative?
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### Homogeneous spaces for unipotent groups are isomorphic to $\mathbb{A}^n$

I am looking for a proof of the following fact: Let $U$ be a unipotent algebraic group over $\mathbb{C}$. Then any $U$-homogeneous space is isomorphic to $\mathbb{A}^n$. Context: Let $G$ be a ...
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### Line Bundles on Elliptic Curves lift From Quotient

Suppose I have an elliptic curve $E$ and take its quotient by some finite group $G$. Is there some relation between the line bundles on $E$ and the line bundles on $E/G$? For example does every line ...
I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states: (a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $f:\mathbb A^1_\mathbb C\rightarrow \mathbb A^1_\mathbb C$ be induced by the ring homomorphism \$t\...