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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1answer
24 views

Finding the Krull dimension of a quotient ring $\mathbb{C}[x,y,z]/I$

I am interested in finding the Krull dimension of the quotient ring $A$ defined as follows: $$ A = \mathbb{C}[x,y,z] / (f_1, f_2, f_3), $$ where $$ f_1 = \frac12 y^3 z - (z-1) - xy $$ $$ f_2 = y^2 z^2 ...
2
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0answers
16 views

On a sufficient condition for a closed morphism of schemes to be affine

Let $f \colon X \to Y$ be a closed morphism of schemes (i.e., the image of any closed subset of $X$ under $f$ is closed in $Y$). Let $y \in Y$. Consider the following assertions: (i) There is an ...
1
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1answer
22 views

Question about basic properties of degrees of algebraic sets

I am learning about degrees of algebraic sets at the moment, and in an article I am reading I came across the following: Let $V_i \subseteq \mathbb{C}^n$ be a hypersurface of degree at most $D$ for ...
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1answer
19 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
3
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0answers
49 views

Equation of 27 lines on a cubic surface

For a smooth cubic surface $S$ in $\mathbf{P}^3$, there're always $27$ lines on it, with the same configuration. We know the automorphism group of the lines is not solvable. How do we show the ...
4
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0answers
63 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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3answers
64 views

What is the convention for the codimension of an empty set?

I am learning about dimension and codimension of algebraic sets at the moment. I know that if $V \subseteq \mathbb{C}^n$ is an algebraic set defined by polynomials $f_1, ..., f_r \in \mathbb{C}[x_1, ...
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0answers
25 views

Proof of formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
2
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2answers
75 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
1
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0answers
23 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
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1answer
31 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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0answers
44 views

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curves second edition (J. H. Silverman) [on hold]

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curve second edition (J. H. Silverman)
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0answers
45 views

How to obtain the genus of the Riemann surface corresponding to an algebraic curve

I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve ...
1
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1answer
82 views

Basic question related to dimension of intersection of two varities

Let $V$ and $W$ be irreducible varieties in $\mathbb{C}^n$. I have learned that intersection $V \cap W$ satisfies the following: $$ codim \ V + codim \ W \geq codim \ V \cap W. $$ I was wondering if ...
3
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0answers
62 views
+200

Why is the morphism induced by this linear system birational?

I have seen (what seems to be) the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a ...
0
votes
1answer
83 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
2
votes
0answers
25 views

Basic question about the properties of dimension of an algebraic set

I am learning about the dimension of an algebraic set and I have a couple of questions I am hoping to resolve to have a better understanding. Let $V$ be an algebraic set in $\mathbb{C}^n$, defined ...
0
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0answers
36 views

How to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to$\sum_{P\in X} i_P(I_P)$

This a Hartshorne exercise (Ex II 1.21d) Let $X=\mathbb{P}^1$. Let $\mathcal{K}$ be the constant sheaf of the quotient field of X. Then we need to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to ...
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0answers
35 views

Show that exist point $w \in \mathbb{C}$ with $w^2 \in K$ and $z \in K(w)$

Let K is subfield of $\mathbb{C}$ with $K=\overline{K}$. $F(k)$ is a set with all circles in complex plane with midpoint in K and radi^us equal distance between two points from K. Let $z\in ...
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0answers
15 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
38 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, ...
2
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0answers
43 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 ...
5
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1answer
55 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
10 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
14 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
99 views
+50

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 ...
2
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0answers
34 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 ...
3
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0answers
41 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
68 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
43 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
184 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
54 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
31 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 ...
1
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2answers
46 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 ...
4
votes
1answer
33 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 ...
2
votes
1answer
41 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 ...
1
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0answers
46 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
105 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 ...
3
votes
1answer
34 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 $F$, $Y=Spec(Sym(F^{\vee}))$, ...
1
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1answer
49 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) ...
2
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2answers
30 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
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2answers
41 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
30 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
votes
2answers
99 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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0answers
49 views

Generalizing points on the x, y, and z planes

I am having a little trouble developing the intuition to understand where the points $P = (x,y,z)$ in $\mathbb{R}^3$ with planes only. For instance, the equation $ xyz = 0 $ represents just the ...
1
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1answer
31 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 ...
0
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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
110 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 ...
1
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0answers
36 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 ...