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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3
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43 views

Axiom of glueing: direct limit of sheaves in a noetherian topological space. [duplicate]

I'm trying to prove that in a noetherian topological space the following property is satisfied: Consider a direct system of sheaves and morphisms $\{ \cal{F}_t, f_{ij} \}_t$. Consider the presheaf ...
1
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0answers
22 views

Dévissage for complex manifolds

In algebraic geometry one has the following result: Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} ...
0
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0answers
28 views

Commutative diagram of surfaces of general type

Suppose that $X$ is a complex projective surface of general type. Let $\phi_{|K_X|}$ the canonical map of $X$ and assume that the image of $X$ via the canonical map $\Sigma=\phi_{|K_X|}(X)$ is a ...
-1
votes
0answers
25 views

Talking about varieties

hi I was recently reading ideals varieties and algorithms. I ah having problems showing things are not affine varieties. Previously with problems like. $V= \{ (a,a) | a \in R^* \}$ it was much easier ...
-1
votes
0answers
75 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
0
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0answers
25 views

I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
2
votes
0answers
52 views

Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
0
votes
0answers
39 views

Proj of some ring.

Let $R= \mathbb C[x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5]$ be the polynomial ring and let $S$ be the subalgebra generated by $x_1x_2x_3x_4x_5, x_1x_2x_3x_4y_5, \cdots ,y_1y_2y_3y_4y_5$ (the ...
2
votes
1answer
74 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
2
votes
0answers
43 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
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0answers
43 views

Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
0
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0answers
43 views

Questions about the function fields of complex algebraic surfaces

Let $X$, $Y$ be complex algebraic surfaces(Of course, they are smooth). Suppose that $X$ is normal. Let $K(X)$ and $K(Y)$ be the function fields of $X$ and $Y$, respectively. And we have a ...
2
votes
1answer
46 views

Show that a variety is irreducible

How do I show that the variety $V = \{(x,y)\in k^2 \mid x-y=0\}$ is irreducible for an algebraically closed field $k$? One approach, I think, is to view $f(x) = x-y$ as an element in $R[x]$, where ...
1
vote
1answer
44 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
6
votes
1answer
157 views

Gap in Hartshorne I can't fill

Page 142, Example 6.11.4. I've been trying to go through the details of the sentence The proof of (6.10) shows that if $f \in K$ is invertible at $Z$, then the principal divisor $(f)$ on $X - Z$ ...
4
votes
1answer
65 views

Does there have to be a point on elliptic curve over $\mathbb{C}(t)$

Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation. $$ y^2 = x^3 + A x + B$$ Where $A, B \in \mathbb{C} (t)$. Question: ...
3
votes
1answer
96 views

pullabck of rational normal curve under Segre map

Let $\nu:P^1 \rightarrow P^2$ be the veronese map of degree $2$, i.e. $[Y_0 : Y_1] \mapsto [Y_0^2 : Y_0 Y_1 : Y_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Consider the ...
2
votes
0answers
53 views

First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.

Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a ...
3
votes
1answer
50 views

Proof of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is ample $\iff$ $a,b >0$.

I would like some help understanding the proof in $(\impliedby)$ direction. Hartshorne on page 156, Example 7.6.2 says: If $\mathcal{L}$ is an invertible sheaf on $\mathbb{P}^1 \times \mathbb{P}^1$ ...
0
votes
2answers
39 views

Is the Godeaux surface irrational?

Studying examples of surfaces of general type, I've found the Godeaux surface. Here is a link for the definition of Godeaux's surface How can I see directly that this surface is not ruled?
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0answers
18 views

Number of fibres under finite non-flat morphism

Let $f:X \to Y$ be a finite, surjective morphism of projective, irreducible varieties over $\mathbb{C}$. We know that there is an open dense subset $U \subseteq Y$ such that every $y \in U$ has ...
5
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0answers
54 views

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
0
votes
2answers
24 views

Nullspace of a linear mapping from higher to lower dimension

It is a well-known fact that a linear mapping $A: \mathbb{R}^n \to \mathbb{R}^m$ , where $m < n$, has a non-trivial null space. This follows from the "rank-nullity" theorem of linear algebra. This ...
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votes
1answer
30 views

Surface constructed using curves

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$. Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i ...
2
votes
1answer
65 views

How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
0
votes
0answers
30 views

Subvarieties and finding ideals

Hi guys I am stuck working on this problem. I have a surface $W= V(xz-y^2)$ and we are trying to find an ideal $J \in K[W]$ so that the $V_w(J)=V(y-x^2,z-x^3)$ I showed that the second thing which is ...
3
votes
2answers
67 views

A curve has infinitely many points

Let $f\in k[x,y]$, where $k$ is an algebraically closed field. I would like to prove the curve $f(x,y)=0$ has infinitely many points. What I know is $k$ is infinite, but I don't know how to use this ...
3
votes
0answers
45 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
0
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0answers
32 views

What is the best Shafarevich's algebraic geometry book for beginners?

What is the difference between these books from Shafarevich: Basic Algebraic Geometry and Basic Algebraic Geometry 1: Varieties in Projective Space? I suppose the second one is better for beginner ...
0
votes
1answer
29 views

Is there a field k, not necessarily algebraically closed, for which variety V(y) in k^2 is reducible?

I am looking for a field $k$ such that the variety V in $k^2$ given by $V =\{(x,y)\in k^2|y=0\}$ is reducible. Thanks
0
votes
1answer
29 views

Primary Decomposition Theorem applied to projections in $\mathbb{R}^2$

We have recently learned the Primary Decomposition Theorem in my Algebra course. I came up with what I think is an instance of this theorem but I haven't convinced myself. Supposed we have the map ...
1
vote
0answers
32 views

About the construction of Quot-Schemes

I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the ...
1
vote
2answers
38 views

Residue class field of coordinate ring

If $X$ is an irreducible affine curve over an algebraically closed field $k$, then its coordinate ring $O(X)$ is a Dedekind domain. Suppose $\mathfrak{p}$ is a prime (hence maximal) ideal in $O(X)$ ...
2
votes
1answer
164 views

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
0
votes
0answers
31 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
1
vote
1answer
25 views

Reference to line parametrization

Defining two lines in space, $\mathbb{R}^3$, as: $l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$ $l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$ The line to line intersection condition is: $\textbf{b}_1\cdot ...
0
votes
1answer
18 views

Find gradient of a equi-angular spiral (log spiral)

I encountered a problem in determining the gradient in cartesian coordinates (x,y) of a logarithmic spiral (or equi-angular spiral) profile. The log-spiral definintion is as shown below (similar to a ...
1
vote
0answers
28 views

When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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votes
0answers
32 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
2
votes
1answer
37 views

Question on the existence of a prime ideal contained in the $\ker$ of a homomorphism $\mathbb{C}[x,y]\rightarrow\mathbb{C}[t]$.

I found this exercise in a basic algebraic geometry book: Let $f:\mathbb{C}[x,y]\rightarrow \mathbb{C}[t]$ a non-zero homomorphism such that $\ker f$ strictly contains a prime ideal $P\neq0$. Is it ...
6
votes
0answers
35 views

Affine curve is union of $d$ lines through point of multiplicity $d$. [closed]

Let $C$ be an affine curve defined by a polynomial of $P(x, y)$ of degree $d$. Show that if $(a, b)$ is a point of multiplicity $d$ in $C$ then $P(x, y)$ is a product of $d$ linear factors, so $C$ is ...
3
votes
1answer
47 views

Isomorphism of varieties via coordinate rings

There is a result which I think is true but it's not written anywhere. Since I just began studying algebraic geometry, it's hard to figure out this by myself. Let $K$ be a field, ...
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votes
0answers
19 views

Pullbacks and Varieties

Hi guy I am given two varieties $V(x_1^4+x_2^4-1)$ and $V(y_1^2+y_2^2-1)$ and I have the morphism $\phi(a_1,a_2)=(a_1^2,a_2^2)$ We want to show that $\phi (V(x_1^4+x_2^4-1)) \subset V(y_1^2+y_2^2-1)$ ...
2
votes
0answers
30 views

Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
4
votes
1answer
65 views

A problem from Artin

This is a problem from old artin (11.3.10) or new artin (12.3.5): Consider the map $\varphi : \mathbb{C} [x,y] \rightarrow \mathbb{C} [t]$ defined by $f(x,y)\mapsto f(t^2-t,t^3-t^2)$. Prove that ...
1
vote
1answer
38 views

$D$ is effective iff $f^*D$ is effective?

Let $f: X\rightarrow Y$ be a proper birational morphism between normal varieties and let $D$ be a Cartier divisor on $Y$. Then is it always true that $D$ is effective $\Leftrightarrow$ $f^*D$ is ...
3
votes
0answers
68 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
0
votes
1answer
35 views

Describing ideal that vanishes at the variety

We have the following morphism $$\phi(a_1,..a_m;b_1,...,b_n)= \begin{pmatrix} a_1 b_1 & \ldots & a_1 b_n \\ \vdots & \ddots & \vdots \\ a_mb_1 & \ldots & a_m b_n ...
3
votes
0answers
25 views

How to calculate a reduced volume?

Let's say we have an irregular 3D shape with volume=V ( we know V but we don't know its equation= F). Now I want to calculate another 3D shape which is exactly the same shape but one size smaller, ...
3
votes
1answer
127 views

Classification of line bundles by Griffiths and Harris

I am reading pages $132$ and $133$ of Principles of Algebraic Geometry by Griffiths and Harris. They consider a holomorphic line bundle $L \to M$ over a manifold $M$ and an open cover $\left\{ ...