# 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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### Invariant of ramification under Mobius transformations

Is there some way of quantifying ramification of a cover up to Mobius transformations? i.e. If $K$ is an algebraically closed field of characteristic $p>0$, we can consider the Artin-Schreier ...
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### A problem I met when reading Griffiths'Periods of Integrals on Algebraic Manifolds I

I am reading Griffiths' paper Periods of Integrals on Algebraic Manifolds I, and in section 2 I met some problems. I wish that I could get some help here. My problem is that I cannot understand ...
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### Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
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### The quotient of the maximal ideal at the point $x$, $\mathcal{M}_x/\mathcal{M}^2_x$ is a $k(x)$-vector space

The question on a previous final was "Consider a scheme $X$, for any point $x \in X$, show that the quotient of the maximal ideal at the point $x$, $\mathcal{M}_x/\mathcal{M}^2_x$ is a $k(x)$-...
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### Finite pushforward commute with taking cohomology

Let $f: X \to Y$ be a finite morphism of schemes. How one can show that $f_*H^i(G) \cong H^i(f_* G)$ for any $G \in D(X)$ and any $i \in \mathbb{Z}$? In english, $G$ is a complex of quasi-coherent ...
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### Confusion over cross-ratio

I just learned about the cross-ratio and that it is a projective invariant. I would like to use it to look at the curve defined over some algebraically closed field $k$ of characteristic $p>0$ ...
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### What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?

Let $k$ be a field. Let $A$ and $B$ be two $k$-algebras, ie. two rings that are also $k$-vector spaces and their multiplication is $k$-bilinear. Any isomorphism of $k$-algebras is also a ring ...
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### $SU(2)$ as an algebraic group

The $\mathbb R$-valued points of the algebraic group $SU(2)$ can be identified with the real 3-sphere. But how does one define $SU(2)$ over the base field $\mathbb R$ as an algebraic group? What are ...
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### theorems that depend on the embedding of an affine variety into the affine space

Let $\mathcal{T}$ be a theorem regarding an affine variety $Y$ of $\mathbb{A}^n$. Question 1: What does the phrase "$\mathcal{T}$ does not depend on the embedding of $Y$ in $\mathbb{A}^n$" mean? ...
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### function meromorphic on C

Good evening I have a doubt: let $f$ and $g$ are two functions meromorphic on $\mathbb{C}$ such that $g(w) =f(\frac{1}{w})$. Now g is defined for $w = 0$ (because of all meromorphic $\mathbb{C}$).Can ...
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### Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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### Quotient of smooth variety is smooth if fixed point set is a divisor?

I've heard (a variant of) the following result being mentioned , but haven't been able to find a reference. I would like to know if the following is true, and if so, I'd very much appreciate a good ...
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### Why a cubic plane curve meets a line three times?

Can someone explain to me why a cubic curve in a projective plane always meets a line three times?
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### a general definition of the volume of a high dimensional polytope

I would like to find a general definition of the volume for a full dimensional polytope in $R^n$. Could anyone give me a hint please! Thank a lot
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### an equivalent statement of “morphisms of projective varieties are closed”

I am interested in seeing why the statement (1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed." implies the statement ...
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### Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
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Hello: Does somebody know if the following is true?: Let $f\in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Then there exists a positive integer $N$ and $a_0,a_1,\ldots,a_{N-1}\in\{... 0answers 67 views ### Hochschild dimension I'm curious; if$A$ia a commutative$k$-algebra over a field$k$of global dimension$n$, then is its$A^e$-projective dimension$2n$(this is also sometimes called the Hochschild cohomological ... 0answers 108 views ### Is there any visual animation to show the basic concept of algebraic geometry? [closed] Is there any visual animation to show the basic concept of algebraic geometry? There are rarely pictures in textbooks, so are there any animation to show basic but important concepts? 1answer 89 views ### What is$\operatorname{Pic}(\mathbb{P}^n_{\mathbb{Z}})$? I would like to know the Picard group of the projective spaces over the integers$\mathbb{Z}$. I know that the projective space over a field$k$has$\operatorname{Pic}(\mathbb{P}^n_{\mathbb{k}}) \...
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Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
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### Is there a coherent sheaf which is not a quotient of locally free sheaf?

Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf？ (Hartshorne II Cor 5.18 showed that on every projective variety, ...
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### Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
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### Parabolas and axis of symmetry?

I have the parabola $$(x+y)^2 = 8(x−y)$$ and know that the axis of symmetry is $$x+y=0$$ but I know when this is the case the left hand side equals 0 but apart from that I can't see how this equation ...
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### Disconnected Algebraic Set over non-Algebraically Closed Field

I'm trying to find an algebraic set $V$ that can be written as the disjoint union of two proper algebraic sets, such that the coordinate ring $k[V]$, where $k$ is NOT algebraically closed, is NOT the ...
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### Criterion for disconnectedness of affine algebraic set.

I'm trying to prove that, if $V$ is an affine algebraic set, then $V$ is connected in the Zariski topology iff $k[V]$ is not the direct sum of two ideals. Note that $k$ is algebraically closed here. ...
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### Do the zeros of a prime ideal on closed points of the Zariski topology uniquely determine it?

That is, if two prime ideals share the exact same zeros on maximal ideals, are they the same ideal? Or at least is there a result with other assumptions that shows this? Learning algebraic ...
Consider an extension $0\rightarrow L \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} L' \rightarrow 0$ of bundles and bundle homomorphisms, where $L$ and $L'$ are line bundles. (Let's ...