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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31 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
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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 $O(X)$ ...
1
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1answer
107 views
+100

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: ...
1
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0answers
20 views

$k-$defectivity of the Veronese variety $v_{2}(\mathbb{P}^{n})$

Let $2\leq k \leq n$ and $N=\binom{n+2}{n}-1$. The image of the Veronese map $$ \begin{array}{cccc} v_{2}: & \mathbb{P}^{n} & \rightarrow & \mathbb{P}^{N} \\ & (a_{0}:\ldots:a_{n}) ...
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0answers
29 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
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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
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1answer
17 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
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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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0answers
29 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
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1answer
35 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 ...
5
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0answers
32 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 ...
2
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1answer
44 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, ...
0
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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
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0answers
27 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
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1answer
64 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
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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
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0answers
65 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
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1answer
33 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
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0answers
24 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
118 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\{ ...
3
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1answer
55 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
0
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1answer
29 views

Quotient of line bundles is flat

Let $k$ be a field, $C$ a smooth, projective, connected curve over $k$ and $T$ a $k$-scheme. Let $\mathcal{F}$ be a line bundle on $C\times_{speck}T$. Let $0\rightarrow \mathcal{O}_{C\times ...
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0answers
31 views

Intersections and intersection numbers [closed]

If $$f=x^3 y+2 x^3 z+6 x^2 y^2+3 x^2 y z+4 x^2 z^2+10 x y^3+3 x y^2 z+7 x y z^2+8 y^3 z+11 y^2 z^2+10 y z^3 \mod 5$$ is a curve and $L= y+2 z$ is a line . What is the intersections between $f$ and ...
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0answers
27 views

Elliptical Triangle

As per my proposal of a new geometric shape as outlined at: http://ellipticaltriangle.blogspot.com/ I tentatively think the figure could be reduced to something like this in polar coordinates: r = ...
-1
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0answers
33 views

Let $R$ be a PID, let $P$ be a nonzero, proper, prime ideal in $R$. Show that $P$ is generated by an irreducible element. Show that $P$ is maximal [closed]

The question comes from Fulton's book, Algebraic Curves, Problem 1.3. Let $R$ be a PID, let $P$ be a nonzero, proper, prime ideal in $R$. Show that $P$ is generated by an irreducible element. Show ...
2
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0answers
58 views

Understanding the isomorphism of Picard group with the first cohomology group

I am learning the subject for the first time, and the material has not yet settled inside me. I would like to get some intuitive understanding of the following: Let $X$ be a complex manifold. The ...
5
votes
1answer
146 views
+100

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
1
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1answer
54 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
1
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1answer
59 views

$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I ...
1
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1answer
43 views

Find and show that the residues of the meromorphic differential $dx$ for Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ is zero

Find the residues of the meromorphic differential $dx$ of Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ at its poles. Check that their sum is zero, as it must be. Attempt: Let $\xi_2\not=0$. Then ...
0
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1answer
37 views

Does cubic equation always have analytical solution?

I saw this blog : http://www-old.me.gatech.edu/energy/andy_phd/appA.htm that in the bottom of the page about the analytical solution of cubic equation said " there are cases when this analytical ...
2
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0answers
55 views

Base-free proof: Set of generators is Zariski-open

Let $V$ be a $d$-dimensional vector space over a field $\mathbb{k}$ (algebraically closed for simplicity), and let $G$ be a finite group acting on $V$. Then the set of elements $v \in V$ such that $\{ ...
2
votes
1answer
35 views

Inductive limit of sheaves over noetherian topological space

Let $X$ be a topological space. Let $I$ be a poset and let $\mathcal F_i$ for $i\in I$ be sheaves on $X$, and $\{\pi_{ij}\colon \mathcal F_i \rightarrow \mathcal F_j\}_{i,j\in I}$ be an inductive ...
1
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0answers
20 views

Convolution product in Borel-Moore homology

I have a question about Exampla 2.7.10 from the book "Representation theory and complex geometry" by N. Chriss and V. Ginzburg. It concerns the convolution product. In the example we have $M_1 = M_2 ...
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votes
0answers
38 views

The second Chern class of line bundle

This is from the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle Ch2 Lemma 1. Let L be a line bundle on the effective divisor D$\to$X and j:D$\to$X be the inclusion. Then ...
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0answers
22 views

group scheme of prime order p is killed by p

In the article "Group Schemes of Prime Order" by Tate and Oort (see here) it is proved that a group scheme of prime order $p$ over the base $S$ is killed by $p$ (Theorem 1). The authors state that it ...
0
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0answers
13 views

Rank 1 Azumaya algebra

Let $X$ be a locally Noetherian scheme. Let the topology on $X$ be etale. Let $A$ be an Azumaya algebra over the scheme $X$ of finite rank. It can be thought of as an element of the Brauer group ...
2
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1answer
50 views

Show that the meromorphic differential of the homogeneous polynomial is holomorphic and not isomorphic to $\mathbb{P_1}$

Consider the elliptic curve i.e. non-singular cubic, $X$ given by the equation $\xi_0\xi_2^2=\xi_1^3-\xi_0^2\xi_1$ in projective coordinates $(\xi_0:\xi_1:\xi_2)$, or, equivalently, by the equation ...
2
votes
1answer
63 views

Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
1
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0answers
64 views

Analogue of locally constant sheaf in algebraic geometry

If I take just the definition of locally constant sheaves for algebraic varieties I get something pretty trivial (basically due to irreducibility); so how can one make a set up similar to what happen ...
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0answers
40 views

irreducible subvariety of a torus

Let $H$ be an irreducible subvariety which is also a closed subgroup of $(\mathbb{C}^*)^n=spm(k[x_1,x_2,...,x_n,y_1,...,y_n]/(x_iy_i-1,i=1,2...,n))$.How to show that $H$ is also isomorphic to a torus? ...
0
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1answer
29 views

Conditions of $f=a+bx+cz+dx^2+exz+fz^2+…$ such that its tangent line is $z=0$ and inflection point is at the origin.

Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as $$f=a+bx+cz+dx^2+exz+fz^2+...$$ Write down the conditions in terms of $a,b,c,...$ such that (a) $P=(0,0)\in C: (f=0)$; (b) the ...
1
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1answer
40 views

Direct limit sheaf.

Let $\{ \cal{F}_i, \mu_{ij}\}$ a direct system of sheaves and morphisms on a topological space $X$. Define the direct limit os the system $\{ \cal{F}_i, \mu_{ij}\}$ as the sheaf associated to the ...
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1answer
15 views

How to solve this parallelism problem? [closed]

How do you solve this problem step by step because I don't understand how to do it all...
0
votes
0answers
39 views

Picard group of projective space

There are many ways to prove that $\mathrm{Pic}(\Bbb{P}^n)=\Bbb{Z}$. One is certainly by means of invertible sheaves, however: What are the (most) elementary ways to see that any Cartier divisor $D$ ...
0
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0answers
28 views

global section of some divisor

Let $f:X\longrightarrow \mathbb{P}^2$ be the blow-up at $p=(1,0,0)$ and let $D:(x_0x_1x_2=0)$. Set $D'=$ strict transform of $D$. Then how can I compute $h^0(X,\mathcal{O}_X(n(K_X+D')))$ for any ...
1
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0answers
21 views

Analytically determining whether a laser beam will hit a moving target

I'm tinkering on a space-related computer game. The objects of the game are in 3D space and their motions are defined by 3 3D vectors: ${vector}\ V: \{X, Y, Z\} \\ {motion}\ M: \{V_{position}, ...
0
votes
1answer
29 views

$X_1,X_2$ disjoint closed in $Spec(R)$ properties

This is a problem in three parts, I managed to prove the first part, but the others I couldn't. Let $R$ be a ring and let $X_1,X_2\subset Spec(R)$ be closed (in Zariski topology) and disjoint such ...
2
votes
1answer
71 views

Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
7
votes
2answers
88 views

How can hypersurfaces “know” the degree of their defining polynomials?

I'm currently trying to learn some complex and projective geometry. There is one issue bugging me again and again, from different perspectives, and I just can't get my head around it. One incarnation ...