The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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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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1answer
96 views

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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1answer
16 views

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 ...
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1answer
66 views

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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27 views

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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87 views

geometric motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...
3
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1answer
47 views

Kernels of power surjective maps

Suppose $k$ is an algebraically closed field, and $A$ and $B$ are finitely generated, commutative, graded $k$-algebras. Suppose $\varphi:A\to B$ is a map of $k$-algebras. Notice if $B$ is a domain, ...
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25 views

Fiber product of schemes: when is $X \times_S Y \cong X \times_U i^{-1}(U)$?

Let $j \colon X \rightarrow S$ and $i \colon Y \rightarrow S$ be morphisms of schemes. Let $U \subset S$ be an open subscheme of $S$ such that $j(X) \subset U$. Under which assumptions do we get an ...
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48 views

Algebraic varieties that are isomorphic after a base change

Let $k$ be a field, $\overline{k}$ its algebraic closure. Suppose $X$ is an algebraic variety over $\overline{k}$. This means that $X$ is a scheme with a finite covering by open affine varieties over ...
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205 views

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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68 views

Hartshorne Exersice 1.17 Skyscraper sheaf Chapter II Schemes

I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf. Observation- If $P\notin U$, then $U\cap ...
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1answer
52 views

Chow motives of quadratic fields

Let us write $CM_k$ for the category of effective Chow motives up to rational equivalence over $k$. Let $k = \mathbb{Q}$. We consider for different primes $p,q$ the Varieties $X = ...
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66 views

$SU(2)$ 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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39 views

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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62 views

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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1answer
75 views

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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48 views

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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50 views

A system of nonlinear equations with sine and cosine functions

Let $n$ be a natural number and $f:{\mathbb R}\rightarrow {\mathbb R}$ be defined such that $f(x) = (1+\cos x)\sin x$. How can I prove that the equations $$ f(x_1) = f(x_2) = \ldots = f(x_n) $$ and ...
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1answer
54 views

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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27 views

Just a definition of an algebraic bundle, bundles on $\mathbb{P}_n$

I have just realized the notion of an algebraic vector bundle. I have some questions. In particular, I'd like to understand wheather I understand it correctly. Let $$\pi:E\longrightarrow X$$ be a ...
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1answer
34 views

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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37 views

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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53 views

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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81 views

multiplicity of a singular point of a plane curve

First Notation: $f$ is a polynomial in $\mathbb{C}[x,y]$ such that $f=f_1\cdot...\cdot f_s$ is the decomposition of $f$ into relatively prime irreducible polynomials. $f_x:=\frac{\partial ...
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2answers
165 views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
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1answer
29 views

The structure morphism of a projective variety induces a morphism of $k$-algeras

Suppose that $k$ is an algebraically closed field and that $X=\textrm{Proj}{\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}}$ is a projective variety with a structural morphism $p:X\rightarrow\textrm{Spec} ...
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1answer
51 views

Counting solutions mod p of a polynomial equation

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 ...
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44 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 ...
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72 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?
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70 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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60 views

rank of quadrics

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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72 views

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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89 views

on the coordinate ring of $\mathbb{A}^n \times \mathbb{P}^{m}$

Consider the product $\mathbb{A}^n \times \mathbb{P}^{m}$. Let $x_i$ be affine coordinates on $\mathbb{A}^n$ and $y_j$ homogeneous coordinates on $\mathbb{P}^{m}$. Question: Is ...
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1answer
119 views

a topological question regarding the blowing-up of a line

The context of this question is argument (3) in the blow-up section p.28 in Hartshorne. All necessary details are given. Let $x_1,\dots,x_n$ be affine coordinates for $\mathbb{A}^n$ and ...
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29 views

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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39 views

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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60 views

Resolution three noncollinear points [closed]

Let $p_1$, $p_2$ and $p_3$ three noncolinear points, and let $R$ be the homogeneous coordinate ring. Show that $R$ have a resolution $$0 \to S^{ \oplus 2}( - 3) \to S^{ \oplus 3}( - 2) \to S \to R ...
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Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
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1answer
72 views

Generalisation of a result on Kahler differentials

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} ...
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1answer
59 views

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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30 views

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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35 views

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 ...
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50 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
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55 views

An extension of line bundles splits locally

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 ...
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2answers
56 views

Smoothness and field of fractions

If $k$ is an integral domain and $A$ is a Noetherian finitely presented $k$-algebra for which $A \otimes_k Q(k)$ is a smooth $Q(k)$ algebra, then can it be deduced that $A$ was initially smooth over ...
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61 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
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57 views

Isomorphism between Ext groups in Huybrechts and Lehn's book Geometry of Moduli Spaces of Sheaves

On p.46 (or p. 43 in the 1st edition) of Huybrechts and Lehn book Geometry of Moduli Spaces of Sheaves, 2nd ed., they write: Since $K$ is $A$-flat and $I \otimes_k F_0$ is annilated by $m_A$, ...
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58 views

Hilbert Nullstellensatz and ring of continuous functions

Is there any relation between Hilbert's Nullstellensatz and the fact that the maximal ideals in $\mathcal C([0,1])$ correspond to a point in $[0,1]$ (which can be generalized to compact hausdorff ...
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1answer
44 views

What is $\overline{Y}$ in $\text{Spec}A$?

Consider a subset $Y$ of $\text{Spec}(A)$. (Here $A$ is a commutative ring.) What is the closure of $Y$ (or $\overline{Y}$)? I have been under the impression that $\overline{Y}$ is the set of ...
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2answers
177 views

1-form on Riemann Surface

Good evening, I can not prove the following result: Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show ...