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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Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
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30 views

$f\in k(\mathbb{A^2})$ not regular at the origin implies it is not regular at points of a curve passing through the origin.

This is Exercise 4.12 (a) in Undergraduate Algebraic Geometry by Reid. Prove that any $f \in k(\mathbb{A}^2)$ which is not regular at the origin $(0, 0)$ also fails to be regular at points of a ...
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1answer
28 views

Induced automorphism on a blow-up

Let $X$ be a surface, $p$ a point on $X$. If $\phi$ is an automorphism on $X$ that fixes $p$ than $\phi$ extends to an automorphism $\tilde \phi$on the blow-up $\widetilde X$ of $X$ in $p$. Then how ...
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30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
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1answer
40 views

Complex Hopf Fibration

The Hopf construction gives a circle bundle $p$ : $S^{3}$ → $\mathbb{CP}^1$. The equation of a 3-sphere in $\mathbb{R}^4$ is $X^2+Y^2+V^2+W^2=R^2$, where $R$ is the radius of the 3-sphere. We may ...
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1answer
41 views

Cech cohomology commuting with colimits? (Non noetherian confusion.)

Suppose that $X$ is a quasicompact, separated $A$-scheme, and $I$ is some directed poset. Suppose that $F_i$ is a system of sheaves on $X$ over $I$. I am having difficulties proving the claim that ...
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33 views

Question about the Hessian Criterion on a curve with singularity

So in class we have this theorem we call the Hessian Criterion: If we have a singular point in an affine curve in $\mathbb{C}^2$. Then $\frac{ \partial ^2 f}{\partial x^2}\frac{ \partial ^2 ...
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29 views

Singularities in affine and projective space.

Sorry to bother you guys I am trying to read a text that is a bit out of my league. I am doing some of the problems in the book to understand it better. Specifically the singularities and the tangent ...
2
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0answers
39 views

Manifolds or Complex Analysis for Algebraic Geometry? [closed]

I'm an undergraduate and I have one year left to take some courses at the graduate level to prepare myself for graduate school. I go to a quarter school (U. Washington) so I only have time to take two ...
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4answers
75 views

affine variety definition

I had this very elementary question which baffles me. Most introductions to the topic define an affine variety as a subset of affine space that is the zero-locus of a set of polynomials. Now, ...
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17 views

Action on Flag manifold

When $G$ is of type A,D,E and $B_4$ then the group of Dynkin diagram automorphisms is non-trivial. If $B$ is a Borel subgroup of $G$, then is there a nice action of the Dynkin diagram automorphism ...
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1answer
39 views

What's so special about hyperbolic curves?

This is really a two-part question, but I would be happy to get an answer for either bit. By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an ...
2
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1answer
58 views

Motivation for studying rational curves

Why do we study rational curves? A curve $f(x,y)=0$ is called a rational curve if there exists two rational functions $\chi(t)$ and $\psi(t)$ such that $f(\chi(t),\psi(t))=0$ for all $t$. Why is it ...
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1answer
25 views

Morphisms of varieties equal over algebraic closure

Let $X$ and $Y$ be two schemes of finite type over a field $k$. Fix an algebraic closure $\overline{k}$ of $k$. Under which conditions on $X$, $Y$ and $k$ is it true that for any two given ...
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1answer
40 views

Dimension of a finite irreducible algebraic group

Let $G$ be an irreducible algebraic group over the field $K$ of characterstic 0. Let $A=K[x_1,...,x_n]/I(G)$ be the coordinate ring and $K(X)=Q(R)$ be the quotient field of $A$. (Since $G$ is ...
3
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1answer
40 views

The map $\phi: k\rightarrow C: (y^2=x^3) \subset k^2$ over a finite field

On page 76 of Reid's book Undergraduate Algebraic Geometry, he says that Over an infinite field $k$, the polynomial map $\phi: k\rightarrow C: (y^2=x^3) \subset k^2$ given by $\phi(t)=(t^2,t^3)$ ...
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1answer
37 views

Proving a set is open in a locally ringed space $(X,\mathscr O_X)$

Let $(X, \mathscr O_X)$ be a locally ringed space and let $A = \Gamma(X,\mathscr O_X)$ be the global sections. For $f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) ...
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2answers
53 views

Open covering of a scheme and global sections

Let $X$ be a scheme. For a global section $f\in\Gamma(X,\mathcal O_X)$, let $X_f=\{x\in X\mid f_x\not\in\mathfrak m_x\}$. For $f_1,...,f_n\in\Gamma(X,\mathcal O_X)$, I wish to know if the following ...
2
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1answer
55 views

Quotient $G \to G/N$ induces quotient $H \to H/N$ by restriction?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. Consider closed subgroups $N \subseteq H \subseteq G$ such that $N$ is a normal subgroup of $G$. Then restricting the ...
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68 views

Simple logical proof of Fermat's Last Theorem [closed]

My interest in the Fermat Conjecture (FC,) began as an interest in the Pythagorean theorem. I wasn't looking for integer solutions of n>2. I was more interested in the fact that odd integer values of ...
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1answer
40 views

Show that Affine Curves are not compact

Hi guys I have a question and not sure how to connect the dots. I am suppose to show that over a algebraically closed field $K=\mathbb{C}$. The affine variety in $K \times K$ is never compact. There ...
2
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1answer
60 views

For homogenous polynomials of degree $d>1$, can $\sum x_i F_i(x)=0$?

Let $\{F_i(x)\}$ be homogeneous polynomials of degree $d>1$ in $n>1$ variables. Suppose also that the $F_i$ have no common zeros besides $0$. Prove the following relation cannot be ...
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37 views

Compatibility of isomorphisms between distinguished opens

Let $f\colon X\to Y$ be a morphism of schemes. Let $\operatorname{Spec}A,\operatorname{Spec}C$ be affine open subschemes of $Y$ such that $\operatorname{Spec}A_g=\operatorname{Spec}C_f$ for some $g ...
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1answer
47 views

Homogeneous coordinate rings of product of two projective varieties

In Ex 3.15, chapter I of Hartshorne's "Algebraic Geometry", we have shown that $ A(X \times Y) \cong A(X) \otimes A(Y)$, when X and Y are affine varieties. Is the same statement true for projective ...
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0answers
15 views

Inverse limit defining etale fundamental group.

Let $(S,s)$ be a connected scheme with geometric point $s$. In many places, I can find the etale fundamental group being defined as $$\varprojlim_{X \to S} \text{Aut}_S(X)$$ Where $X \to S$ ranges ...
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0answers
27 views

The degree of smooth projective curve included in $\mathbb P^n$ which is nondegenerate is more or equal the dimension of the projective space

I had a theorem during lecture, with proof which I don't understand. Theorem says: $ X \subset \mathbb{P}^n(\mathbb{C})$ smooth projective curve which is nondegenerate (not contained in a ...
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0answers
20 views

Method of counting constants / Krull's principal ideal theorem

In his book "Commutative algebra with a view towards algebraic geometry", Eisenbud cryptically remarks that Krull's principal ideal theorem (on the maximum codimension of components the zero set of a ...
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1answer
18 views

open set whose inverse image under a dominant morphism is contained in an open set

Let be $X$ and $Y$ two proiective irreducible varieties with positive dimension and $f:X\rightarrow Y$ a dominant morphism. If $A\subseteq X$ is open, there exist an open non empty set $B\subseteq Y$ ...
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1answer
32 views

fiber of a branched covering of curves over a branch point

Let $X$ be a smooth curve over $\mathbb{Q}$, though possibly geometrically disconnected. Let $f : X\rightarrow \mathbb{P}^1_\mathbb{Q}$ be a finite map of smooth curves over $\mathbb{Q}$. Let ...
3
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1answer
94 views

Adjoint to $\mathsf{Proj}$? - A quest to understand categories of graded objects.

I've been having a hard time with graded objects in algebraic geometry for some time. Lately I realized a lot of my difficulties come from not having any idea at all of where graded objects live. ...
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0answers
44 views

Why are transition functions of an algebraic vector bundle are maps of algebraic varieties?

This is from Le Potier's Lectures on Vector Bundle Definition: A complex linear fibration (or just fibration) over an algebraic variety is a pair $(E,p)$ where E is an algebraic variety and ...
2
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1answer
50 views

Extended ideals and algebraic sets

Let $L\subset k$ a field extension such that $k$ is algebraically closed. Now consider the algebraic set $Z(\mathfrak a)$ where $\mathfrak a$ is an ideal of $k[T_1,\ldots, T_n]$ but it is generated ...
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2answers
50 views

Function field on a non-singular algebraic curve as the field of meromorphic functions

Let $k$ be an algebraically closed field and let $X$ be a non-singular algebraic projective curve over $k$. Very often, when most books present the function field $k(X)$, they say that "it is the ...
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1answer
57 views

What is the general structure of the complex curve $xy=y^2$?

How can you determine how a complex curve looks like in four dimensions, especially near singularities? In my example, the curve $xy=y^2$ consists of the lines $y=x$ and $y=0$ ($x,y$ complex). I think ...
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2answers
65 views

Constructing prime ideal of tensor product from two prime ideals

If $M,N$ are $R$-algebras and natural maps $m:M\to M\otimes_RN,n:N\to M\otimes_RN$, is there any way to construct a prime ideal $T$ of $M \otimes_R N$ given two prime ideals $A,B$ of $M,N$ such that ...
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1answer
51 views

Decompose some algebraic sets into irreducibles

Find the irreducible components of $$V(Y^2-XY-X^2Y+X^3),\ V(Y^2-X(X^2-1)),\ V(X^3+X-X^2Y-Y)$$ in $\Bbb A^2(\Bbb R)$ and also in $\Bbb A^2(\Bbb C)$. Now for $V(Y^2-XY-X^2Y+X^3)$, ...
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42 views

Normal Space of an orbit at a point

Let $X$ be a curve of genus $g$. If $d>n(2g-1)$,then for any vector bundle $E$ of rank $n$ and degree $d$ over $X$ has $H^1(X,E)=0$ and $E$ is generated by its global sections. For such a vector ...
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1answer
28 views

Show that every algebraic subset of $\Bbb A^2(\Bbb R)$ is equal to $V(F)$ for some $F∈\mathbb R[X,Y]$.

Show that every algebraic subset of $\Bbb A^2(\Bbb R)$ is equal to $V(F)$ for some $F∈\mathbb R[X,Y]$. Suppose $X=V(S)$ for some set $S \subseteq \Bbb R[X,Y]$. Now $V(S)=V(<S>)$ where ...
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0answers
38 views

Existence of a anywhere nonvanishing section

Let $\mathcal{S}$ be a globally generated vector bundle of rank $r$ on a projective variety $X$ with dimension $n$ and the rank of $\mathcal{S}$ shall be greater than the dimension of $X$. Now I ask ...
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1answer
42 views

moduli space of triangles

I found an article which seems to be aimed for general audience. I couldn't understand sentences about triangles. The link to the article is the following. ...
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23 views

A category of closed immersions

Fix a scheme $Z$, and consider a category whose objects are schemes $X$ equipped with a closed immersion $Z\to X$. Obviously, a morphism $f:X\to Y$ should commute with the respective closed ...
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49 views

Is $\Bbb A^n(k)$ irreducible if $k$ is finite?

Is $\Bbb A^n(k)$ irreducible if $k$ is finite? For finite field $\exists f(x)$ s.t $V(f)=\Bbb A^n(k)$ but this does not imply anything.. I think if I can show that corresponding any $a \in k$ there ...
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0answers
12 views

Topological nets and triangulations

How does one construct a net and triangulation for a space? For example the identification space of the unit square with these identifications $(0,y)$~$(1, 1-y)$ for all $0 \leq y \leq 1$ ...
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29 views

Any solutions to problems in Harris book moduli of curves

Is there any place where I can find solutions to problems of the book Moduli of Curves. I am learning the subject, and want to do some of the problems. Also if there is any good source for problems ...
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1answer
32 views

Line bundle with a nowhere vanishing global section is trivial.

Let $k$ be a field and $X$ be a projective variety over $k$. I think it should be true that if $L$ is a line bundle on $X$ such that exists $s \in \Gamma(X,L)$ with $s_x \neq 0$ for all $x \in X$, ...
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1answer
27 views

Number of points on a line in a finite projective plane

I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for ...
3
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2answers
63 views

Residue fields of schemes of finite type (over $\mathbb{Z}$)

Suppose $X$ a scheme of finite type over $\mathbb Z$. I want to prove that: (1) The residue fields of closed points of $X$ are finite; (2) For a given $q=p^n$ with $p$ prime, there is only a finite ...
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24 views

Computing the inverse to a rational map

The setup: say I have some rational projective variety $X$ of dimension $n$ over $\mathbb{C}$ such that the map $$ X \dashrightarrow \mathbb{P}^n $$ is given by some linear series $\mathcal{L}$. My ...
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1answer
48 views

Does every subvariety of $\Bbb C^n$ have a smooth point?

Let $X$ be an algebraic subvariety of $\Bbb C^n$. Is it true that $X$ always admit a smooth point and if it is, how can one prove it ?
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29 views

Characterizing the tangent line of a multivariable polynomial

I am trying to learn algebraic geometry at a basic level and came across this problem: Let $K$ be a field. For $f(x,y)\in K[x,y]$, a line $L$ in $K^2$ is called a tangent line of the curve $V(f)$ ...