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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Help with $\sqrt{I}$, where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$

$a)$ $\sqrt{I}$ where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$. first it's clear $y \in \sqrt{I}$ then $x=(x+yz)-yz \in \sqrt{I}$ because $yz \in \sqrt{I}$ is it $\sqrt{I}=(x,y)$ ? $b)$ $\...
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73 views

Question about the restriction of a bimeromorphic map to divisors

I'm studying some stuff about Picard groups and divisors and I started wondering something, here it is: Let $X$, $Y$ be algebraic varieties and $f:X\rightarrow Y$ a bimeromorphic map between them. ...
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68 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
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201 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; $I\cap(J+K)=(I\...
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73 views

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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121 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
36 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 $k(x)$-...
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271 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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39 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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155 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 $U\...
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1answer
58 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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1answer
33 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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1answer
113 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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260 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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99 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
69 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 = \mathrm{Spec}(k(\...
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93 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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51 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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74 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
131 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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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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1answer
72 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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1answer
73 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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1answer
48 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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1answer
85 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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256 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 f}{\...
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2answers
1k 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
56 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
142 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 $a_0,a_1,\ldots,a_{N-1}\in\{...
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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 ...
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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?
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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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1answer
195 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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499 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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102 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 $A=k[x_1,\dots,...
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1answer
166 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 $y_1,\dots,...
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119 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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2answers
56 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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2answers
115 views

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
100 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
89 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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44 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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38 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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108 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
77 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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135 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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1answer
109 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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1answer
88 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
51 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 prime ...