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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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 ...
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51 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
59 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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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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29 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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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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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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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
28 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 ...
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66 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 $V(f)$ ...
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31 views

$G$ equivariant quasicoherent sheaves on $X$ as compatible $G$ actions on the total spaces?

Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$. Let $F$ be a quasicoherent sheaf on $X$. There is a notion ...
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62 views

Divisor on curve of genus 2

Let $C$ be a smooth, projective curve of genus 2. I want to show that there exists a non-constant rational function $f \in k(C)$ having divisor of the form $$(f) = P_1 + P_2 - P_3 - P_4 $$for points ...
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1answer
60 views

Rank 2 vector bundle

$E$ is a rank $2$ vector bundle. Why is $E\simeq E^*\otimes \det E$? Any generalization (arbitrary rank, $E$ non locally free etc.)?
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45 views

A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols ...
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1answer
34 views

Show that $Y$ is constructible

I'm stuck on the following problem: Let $X$ be a Noetherian space, and let $Y \subseteq X$ have the property that for every irreducible closed set $Z \subseteq X$, $Y \cap Z$ contains an open dense ...
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1answer
45 views

A combinatorial problem arising from an exercise on the weil conjectures

I am trying to learn a little bit about the Weil conjectures and specifically i'm trying to do exercise 5.5 in appendix C of Hartshorne. I'll try to explain quickly the problem so you don't have to ...
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2answers
46 views

Normalization of the projective closure of affine plane curve over $\mathbb{C}$

I am trying to understand how to do explicit calculations for finding the normalization of a plane curve. The intuition is somewhat clear to me: "separate" the singularities or smooth them out (for ...
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21 views

Parametrization of $K$-rational points of the hyperbola

Let $K$ be a perfect field of characteristic $\neq 2$ and consider its algebraic closure $\overline K$. Moreover define $$C=\{(x,y)\in \mathbb A^2(\overline K)\,:X^2-Y^2=1\}.$$ How can I get the ...
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70 views

The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continous map $f: (X,x)\rightarrow (Y,y)$ induces a ...
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94 views

Noether normalization in algebraically closed field

The Noether normalization lemma states that if $k$ is a field, and $A$ a finitely generated $k$-algebra, then there exist elements $y_1,...,y_m\in A$ such that $y_1,...,y_m$ are algebraically ...
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28 views

One-One correspondence between elements of G (algebraic group) and maximal ideals of K[G]

We are given that $G$ is an algebraic group over $K$ and $K[G]= K[x_1,...,x_n]/I(G)$ where $I(G)$ is the ideal consisting of all polynomials of which elements of $G$ are the common zeroes. Now,if we ...
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29 views

Why do we need to take the closure in the definition of projective closure?

Let $X=Z(I)$ be an algebraic set in $\mathbb{A}^n$. Given the standard covering $\{U_i\}$ of $\mathbb{P}^n$ and the homeomorphism $\varphi_0:U_0\to\mathbb{A}^n$, the projective closure of $X$ is ...
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31 views

Groebner basis and projective closures

New to algebraic geometry and Groebner basis, so I just wanted to bounce my argument off of somebody. I have a zero set defined by one polynomial, $Y=Z(S) = \{p(x)\}$ in affine space, I am interested ...
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11 views

Non-convex subdivisions of newton polygon of a tropical plane curve

This is probably an elementary question, but how come the Newton polygon of a tropical plane curve can't have non-convex subdivisions? Or can it?
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36 views

Jacobian criterion algebraic independence

I have two polynomials $f_{1}(x,y)$ and $f_{2}(x,y)$ and I want to know if they are algebraically independent. I am using the Jacobian criterion which says that $f_{1}$ and $f_{2}$ are algebraically ...
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35 views

Why are none of the $\overline{Y_i}$ contained in one another?

Let $X$ be a topological space which is a union of finitely many irreducible closed sets $X_1, ... , X_n$. Lemma: if none of the $X_i$ are contained in one another, then $X_1, ... , X_n$ are the ...
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1answer
43 views

Example of variety with big automorphism group? (Big as in, not a variety in a reasonable way.)

Let $X$ be some algebraic variety over $k$. In some situations, like $X = P^1_k$ (where it some quasi-projective variety), the set of automorphisms has a natural algebraic structure. (If $X$ is a ...
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Finite groups as intersection of algebraic groups

Well known that any finite number of points can be seen as intersection of two algebraic curves. Is it true that any finite group $G$ can be seen as intersection of two (connected) one dimensional ...
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48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola [on hold]

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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1answer
102 views

Grothendieck's “Relative” Point of View

I have often read that Grothendieck's insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why ...
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56 views

Why is the identity function not the exponential of a holomorphic function on $\mathbb{C}\backslash\{0\}$?

In chapter 2 of Qing Liu's book Algebraic Geometry and Arithmetic Curves he states that the identity function on $X=\mathbb{C}\backslash\{0\}$ is not the exponential of a holomorphic function on $X$. ...
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41 views

If $Y_i$ are the irr. components of $Y$, then $\overline{Y_i}$ are the irr. componens of $\overline{Y}$.

Got a real dumb question for ya. Suppose $Y$ is a subset of a topological space with irreducible components $Y_1, ... , Y_n$. Then $\overline{Y_1}, ... , \overline{Y_n}$ should be the irreducible ...
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342 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
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47 views

Algebraic K-theory of the cotangent bundle

Below, always let $A$ be the coordinate ring of a smooth affine variety over $\mathbb C$. What can be said about the (non)-triviality of the module of Kahler differentials $\Omega_{A/\mathbb C}^1$? ...
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64 views

Nonstandard construction of sheafification

Let $F$ be a presheaf on a topological space $X$ of some category of "sets with structure." In Borel's Linear Algebraic Groups, he gives the following explanation for how to construct the associated ...
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1answer
52 views

Hartshorne proposition 1.2 e)

We want to prove that $ Z(I(Y)) \subseteq \overline Y$. Let $W$ be any closed set containing $Y$. Then $W=Z(a)$ for some ideal $a$. So $Y \subseteq Z(a)$ and $I(Z(a)) \subseteq I(Y)$. Clearly we have ...
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38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
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2answers
65 views

Radical ideal of $\langle x^2+y^2+z^2, xy+yz+xz\rangle$

The following is exercise 3.7 from Undergraduate Algebraic Geometry by Reid. Let $J=\langle x^2+y^2+z^2, xy+yz+xz\rangle$; identify $V(J)$ and $I(V(J))$. The question did not specify the field. ...
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67 views

If $X$ is a closed set in $\mathbb{A}^n$, is $\mathcal O_X$ the inverse image sheaf of $\mathcal O_{\mathbb{A}^n}$?

Let $k$ be algebraically closed, and let $X$ be a closed subset in $k^n$ with corresponding radical ideal $I$. Let $\mathcal O$ be the standard sheaf associated with the space $k^n$, and let ...
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29 views

Class group of the cone

This is from exercise 6.3a of Hartshorne. Let $V$ be a projective variety over a field $k$ of dimension $\geq 1$ that is non-singular in codimension 1. Let $X = C(V)$ be the affine cone over $V$ in ...