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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16 views

Group action on closed subschemes

Let $G$ be a connected, linear, semi-simple algebraic group and $P \subset G$ the maximal parabolic subgroup. We know that $Z=G/P$ is a projective variety. Then, 1) Does $Z$ contain a line? 2) In ...
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0answers
41 views

Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
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3answers
170 views

Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
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1answer
57 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
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0answers
13 views

Radical of reductive but not connected linear algebraic groups

Let $G$ be a linear algebraic group over a field $k$ of characteristic zero. A definition of $G$ being reductive is that the radical of $G^0$ (the connected component of the identity of $G$) over ...
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1answer
15 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
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2answers
139 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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0answers
40 views

Definition of regular functions on a projective variety

I'm trying to read Algebraic Geometry : a First Course by Joe Harris. In Lecture 2, p. 20, he defines a regular function on an open set $U$ of quasi-projective variety $X$ as a function such that if ...
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0answers
45 views

Isolated points of fibers of regular morphism

Let $X,Y$ be affine varieties and $\varphi:X\to Y$ be regular morphism. I want to prove that isolated points of fibers of $\varphi$ form open subset in $X$. Can you give me advice how to do it?
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1answer
70 views

How to show that a map is finite

Let $V = Z\left(X^3 - Y^2\right)\in \mathbb{k}^2$. How to show that $f \colon t \in \mathbb{C} \mapsto \left(t^2, t^3\right) \in V$ is a finite map? Thanks in advance!
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0answers
44 views

Automorphism of $\mathbb{A}^2$ which maps the finite set of points to the finite set of points

Let $\mathrm{k}$ be infinite field. $P_1,\dots,P_n, Q_1,\dots,Q_n \in \mathbb{A}^2$ and $P_i \neq P_j, Q_i \neq Q_j$. I want to find automorphism(in a.g. sense) which maps $P_i$ to $Q_i$. I have tried ...
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1answer
196 views

A $X \subseteq \mathbb{A}^n$ such that $I(X) \neq I(V(I(X)))$?

Let $\mathbb{A}^n$ be the affine $n$-space over a field $K$. Denote by $V(S)$ the zero locus of a $S \subseteq K[x_1, \dots, x_n]$ and let $I(X)$ be the ideal of a $X \subseteq \mathbb{A}^n$. Is there ...
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0answers
57 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
1
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1answer
32 views

Representable morphism for algebraic spaces

I'm trying to understand the definition of algebraic spaces, but there is a notion of representable morphism that is a little confusing to me. Let $S$ be a scheme and let $Sch/S$ denote the category ...
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2answers
50 views

On the definition of degree of a hypersurface

Let $f \in \mathbb{C}[x_1, ..., x_n]$ be a homogeneous polynomial of degree $d$. I was trying to understand the definition of degree of hypersurfaces. It says on Wikipedia ...
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1answer
41 views

What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
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1answer
46 views

Separated Schemes and Intersection

Let $X$ be a separated scheme. I am trying to show that if $U$ and $V$ are affine open sets then $U\cap V$ is also. I can see that $U\cap V$ is homeomorphic to $d(X)\cap (U\times V)$. Where $d$ is the ...
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0answers
48 views

Why is the codimension of an algebraic set defined by $r$ equations at most $r$?

Suppose I have $r$ polynomials $g_1, ..., g_r$ in $\mathbb{Z}[x_1, ..., x_n]$. And let $H = \{ \mathbf{x} \in \mathbb{C}^n : g_i(\mathbf{x}) = 0 (1 \leq i \leq r) \}$. I was wondering why it then ...
6
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0answers
109 views

Higher direct image of morphism with generic fiber $\mathbb{P}^1$

Let $f:X\to Y$ be the morphism of smooth varieties over $\mathbb{C}$ with generic fiber equal to $\mathbb{P}^1$. How to prove that $R^if_*\mathcal{O}_X=0$ for $i>0$? (I do not need the complete ...
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1answer
41 views

Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
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1answer
52 views

Addition of points on elliptic curves over a finite field

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$: $$P + Q ...
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0answers
13 views

Birational map between a conic and an affine line (related to the classical formula of Pythagorean triples)

Could someone please explain me the following? It says in the notes I am reading that $$ Spec \ (\mathbb{Q}[x,y] / (x^2 + y^2-1) ) \rightarrow Spec \ \mathbb{Q}[m] $$ given by $$ f : (x,y) ...
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2answers
31 views

Basic question related to the definition of affine $k$- variety

The definition of affine $k$- variety $X$, I have is that $X$ is an affine scheme that is reduced and of finite type over $k$ ($k$ is a field here). The definition of finite type I have is that $X$ ...
2
votes
2answers
43 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
5
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1answer
33 views

Why $R^q(\Gamma \circ \eta_{*}) (\Bbb G_{m, \eta}) = H^q(\eta_{ét}, \Bbb G_{m, \eta})$?

Let $X$ be a smooth, projective and connected curve over an algebraically closed field, and let $\eta \rightarrow X$ be its generic point (we also call the inclusion as $\eta$). I want to understand ...
3
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2answers
104 views

How is the multiplicative group an algebraic variety?

According to various places, we define an algebraic group as a group that is also an algebraic variety (along with some compatibility conditions). Many places also list some examples, one of which is ...
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1answer
32 views

Given a point $P$ and a hyperplane $H$ in $\mathbb{P}^n$ such that $P \in H$, there is $T$ linear such that $T(P)=(0:\cdots:0:1)$ and $H:X_0=0$

Show that given a point $P$ and a hyperplane $H \subseteq \mathbb{P}^n$ such that $P \in H$, there is a linear transformation $T$ such that $T(P)=(0:\cdots:0:1)$ and $H$ is given by the equation ...
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0answers
53 views

Product of Schemes and Open Subsets

Let $X$ be a scheme and $U$ an open subset, view $U$ as a scheme also. Let $X\times X$ be the product in the category of schemes. Show that there exists an open subset $V$ of this product, such that ...
4
votes
2answers
132 views

Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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0answers
37 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
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0answers
32 views

Base of homology on a Riemann surface and holomorphic differentials

I have two questions: 1) Given a Riemann surface $X$ of genus $g$ and an holomorphic differential $\omega$ on $X$, is it always possible to find a base $\{\delta_i\}_{i=1,\dots 2g}$ of ...
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0answers
44 views

An integrally closed subdomain of a polynomial ring

Let $\mathbb{C} \subset R \subset \mathbb{C}[x,y]$ be a noetherian integral domain. Further assume that: (1) $\mathbb{C}[x,y]$ is separable over $R$. (2) $\mathbb{C}[x,y]$ is algebraic over $R$ ...
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1answer
42 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
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0answers
58 views

Computation of Riemann-Roch space L(kQ) to a specific Divisor D

I am trying to build a Reed-Solomon Code through a Goppa-Code Construction. I start with the projective line $\mathcal{X}$ $aX+bY+cZ=0$. The genus $g$ of this line is $0$. Futhermore, let ...
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1answer
49 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
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1answer
39 views

Morphisms induced by effective divisors on $\mathbf P^1$

This question is about the proof of Theorem V.2.17 in Hartshorne's Algebraic Geometry. Here everything is defined over some algebraically closed field $k$. Define $\mathcal O = \mathcal O_{\mathbf ...
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1answer
63 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
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1answer
46 views

Calculating the coordinate ring and irreducible components

Consider the graded ring $S=(R/I)\oplus (I/I^2)\oplus (I^2/I^3)\oplus\cdots$ Take $R=k[X,Y],I=(X^2Y,XY^2)$. Then $S=k[X,Y]/(X^2Y,XY^2)\oplus(X^2Y,XY^2)/(X^2Y,XY^2)^2\oplus\cdots$. I am not sure ...
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1answer
42 views

With regards to Theorem 3.2 in Hartshorne: Are regular functions on a variety simply polynomials?

I am reading Hartshorne's Algebraic Geometry for the first time and I am having some trouble understanding Proposition 3.2. The proposition implies that $\begin{array}{ccccc} \mathcal{O}(Y) ...
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1answer
40 views

base change of an equivalence relation of fppf sheaves

Let $S$ be a scheme, $R,U$ be $S$-schemes and $s,t : R \to U \times_S U$ be an equivalence relation i.e. it's a monomorphisme such that for every $S$-scheme $T$, $R(T) \to U(T) \times U(T)$ is and ...
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0answers
25 views

Applications of projective normality

Why study projective normality of a variety ? What are the applications ? How does it relate to non-singularilty, rationality etc of the variety ?
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2answers
134 views

Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

I want to read the proof of the following theorem: This is from p.35. But it is not complete there. There is written that: Can someone tell me where I can find the rest of the proof? Any other ...
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1answer
35 views

Closure of Set in Zariski Topology [closed]

If $U$ is a set in affine space, is the closure of $U$ simply $V(I(U))$? If so, how might one prove this?
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2answers
74 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
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0answers
27 views

filtration of vector bundle

Suppose I have a vector bundle on a variety. When is it true that my vector bundle admits a filtration with subquotients all line bundles? (I am most interested in the case of an integral projective ...
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0answers
30 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
4
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0answers
44 views

Birational map between reduced schemes.

I am reading Ravi Vakil's notes "Foundation of Algebraic Geometry" and in Proposition 6.5.5., it states the following: Suppose $X$ and $Y$ are reduced schemes. Then $X$ and $Y$ are birational if and ...
2
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2answers
39 views

Isomorphism of Varieties

Let $V=V(x^2+y^2-1) \subset \mathbb{R}^2$ be an affine variety. Show that $V$ is rational, but isn't isomorphic to $\mathbb{R}^1$. I could show that $V$ is rational, by parametrization ...
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1answer
41 views

Weierstrass normal form of an elliptic curve

without knowing any deeper theory, I am required to find the Weierstrass normal form of an elliptic curve, i.e. a representation of type $y^2z-x^3-axz-bz^3$ where $x,y,z $ are variables and $a,b$ are ...
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0answers
35 views

Connection between local freeness and the rank of matrices

I am reading ch.16 of Eisenbud's Commutative Algebra, more precisely it's the very first paragraph of 16.7, where he wants to prove: Suppose that $\mathcal{J}: R^t \longrightarrow R^r$ is a map of ...