# Tagged Questions

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. ...

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### Examples of the primitive decomposition of a form

Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like ...
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### Classification of subschemes of $\mathbb{A}_K^2$ of dim $0$, deg $4$, $5$ with support at origin

What is the classification up to isomorphism of subschemes of $\mathbb{A}_K^2$ of dimenion $0$ and degrees $4$ and $5$ with support at the origin? Which are isomorphic as schemes over $\text{Spec}\,K$?...
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### What's the difference between $\mathbb{A}^n$ and $\mathbb{A}^{n+1}$?

Besides the obvious difference in topological dimension. If you want to distinguish between $\mathbb{R}$ and $\mathbb{R}^2$, take an open set in the plane, remove a point, then it's still connected. ...
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### Morphism of finite type schemes not surjective. Is there a closed point in the complement of the image?

I have the following proplem: Let $k$ be an algebraically closed field and let $X,Y$ be schemes of finte type over $k$. Now let $f:X\to Y$ be a morphism of schemes that is not surjective. Question: ...
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### Why is $Supp D=Supp(\pi^*(\pi(Supp D)))$?

I'm trying to understand the proof of the theorem at page 163 of Mumford, Abelian Varieties. At some point we have the following situation: $X$ is an abelian variety, $D$ is an effective Weil divisor ...
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### Seesaw principle

Let $k$ be an algebraically closed field of characteristic 0. The seesaw principle in algebraic geometry usually goes like this: let $T$ be a complete variety, let $X$ be an integral scheme of finite ...
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### Asymptotes of $(x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg)$, collinear points, …

Consider the curve: $$(x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg)$$ Question 1: What are his asymptotes? Answer: In projective space: $[(2+t^3,1+t^2,t)]$...
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### Need a counterexample to show that Cl$(X\times Y)$ is not always same as Cl$(X) \oplus$Cl$(Y)$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by $\operatorname{Cl}(X)$. Suppose $X$ and $Y$ be two quasi projective varieties.What is the ...
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### Number of points of Jacobian of hypereeliptic curve of genus 2

I have a question about the exercise I found here on page 11: http://www.maths.bris.ac.uk/~matyd/DE/Stoll.pdf If we have a hyperelliptic curve $y^2=f(x)$ of genus $2$ over field $K$ we can prove ...
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### Riemann-Roch Space for Quotient Curve

Let $C$ be a curve defined over a finite field $\mathbb{F}_q$. Let $\{f_1,..f_m\}$ be a basis for the riemann-roch space of functions, L(D), for the divisor $D= t\infty$. Suppose you have a subgroup ...
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### Center and axis of Quadratic Surface

Given a Quadratic Surface in the form: $ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0$ I know how to decide which kind of surface is represented (http://mathworld.wolfram.com/QuadraticSurface.html). ...
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### Algebro-geometric proof of Cayley Hamilton theorem

I am looking for a reference on the algebro-geometric proof of C.H. Theorem: Every square matrix satisfies its characteristic polynomial. There are several points I don`t understand reading my ...
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### Vanishing set of a pullback section

Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$. If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and ...
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### How do I show the tangent to an elliptic curve over the complex numbers meets the elliptic curve at another point?

If $E(\mathbb{C})$ is an elliptic curve given by $y^2=ax^3+bx+c$ for $a,b,c\in \mathbb{C}$, and $\ell$ is a line tangent to $E(\mathbb{C})$ at some point $p$, then why does $\ell$ meet $E(\mathbb{C})$ ...
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### Connected component of $0$: why is it an abelian variety?
With "Abelian variety" I mean a integral scheme $X$, proper over an algebraically closed field (complete variety) with a group law $m: X\times X \rightarrow X$ such that $m$ and the inverse map are ...
I've found in Hartshorne's book on Algebraic Geometry a categorical definition of a presheaf (Ch. 2.1, just after the first definition). He there defines for a topological space $X$ the category \$\...