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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22 views

Finite groups and manifolds

my question is: can we connect finite groups with algebraic manifolds as: Take for example Togliatti surface $X$ with set of 31 singular points $P$. Then consider action $Aut(X)$ on $P$, then group ...
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2answers
29 views

Cremona group of $\mathbb{P}^n$

I know that the complex conjugation $\tau: \mathbb{P}^n \mapsto \mathbb{P}^n$ that sends any point $x$ to the point with complex conjugate coordinates $\tau(x)$ is a homeomorphism. In order to show ...
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1answer
19 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
3
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0answers
10 views

Coordinates on a Richardson variety

I'm looking for convenient coordinates to use to describe the intersection of a Schubert cell $X^\circ_\lambda$ and an opposite Schubert cell $\Omega^\circ_\mu$ in a Grassmannian $G(k,n)$. Describing ...
2
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0answers
22 views

Rational functions over curves define regular maps?

I've a rather simple question that is strucking me, perhaps because I'm a newbie in algebraic geometry. If we have a projective irreducible non-singular curve $C\subseteq \mathbf{P}^n_k$ where $k$ is ...
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0answers
32 views

Their product is a cubic of a rational number $x$ minus $x$

It is given the integer $6$. Analyze it into two parts such that their product is a cubic of a rational number $x$ minus $x$. $$$$ Let $y$ be the one factor. The other one is $6-y$. We have ...
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0answers
24 views

Abelian torsion group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3+8$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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1answer
21 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field k, and take it's maximal ideal $m = (X-\alpha,Y-\beta,Z-\sqrt{ f(\alpha)g(\beta)})$. How might one prove that a localization at ...
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4answers
46 views

Show that $X$ is not an affine variety

I need some help proving that $X=\{(x,x)~|~x \in \mathbb{R}, x \neq 1\}$ is not an affine variety. I would like to proceed by supposing it is an affine variety and then finding a contradiction. So ...
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0answers
34 views

solution to general quintic [on hold]

I am an amateur.I am very good with advanced algebra.I have put 11 parameters into the general quintic equation and apparently can put this equation into a quintic form soluble by radicals.to do this ...
2
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0answers
31 views

Dessins d'Enfants and Real Algebraic Curves

I wrote a thesis on the Grothendieck theory of Dessins d'Enfants (after some articles by Leila Schneps). In Shafarevich, vol.2, there's a section on real algebraic curves. Is it possible to formulate ...
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0answers
24 views

Complete intersection of algebraic varieties

Suppose that $X$ is a fixed geometrically integral hypersurface in $\mathbb{P}^n(K)$,say, for a field $K$ of characteristic zero. Suppose that $Y_1, Y_2$ are two distinct irreducible hypersurfaces. Is ...
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1answer
37 views

Abelian group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3-2$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
3
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1answer
48 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in ...
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0answers
36 views

line bundle descents?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
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1answer
35 views

$\dim N_1 X$ and $\mathbb{P}^2$ and $\mathbb{F}_n$

If $X$ is a smooth projective rational surface such that $-K_X$ is big, then why is it that if $\dim N_1X \leq 2$, then $X \cong \mathbb{P}^2$ or $X \cong \mathbb{F}_n$, a Hirzebruch surface. I ...
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3answers
43 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that ...
2
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1answer
41 views

What does the spectrum of the Grothendieck ring of varieties look like?

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation ...
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1answer
34 views

How to prove $V(5x^2+6xy+2y^2-2yz-z^2)$ is empty

Let $V/\mathbb{Q}$ be the projective variety $V:5x^2+6xy+2y^2=2yz+z^2$. I want to prove $V(\mathbb{Q})$ is empty. Given $[x,y,z]$ in $V$, WLOG assume $x,y,z\in \mathbb{Z}$ and $\gcd(x,y,z)=1$. ...
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1answer
47 views

Presheaf image of a monomorphism of sheaves is a sheaf

EDIT: The original version regarding coker is false. Consider a morphism of sheaves of abelian groups $\Phi:\mathscr{F}\to\mathscr{G}$. It is not in general true that Im$_{pre}\Phi$ is a sheaf. ...
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1answer
33 views

Open in Zariski but not product topology

Let $X \subset k^m , Y \subset k^n$ be algebraic sets ($k$ an algebraically closed field). Then $X\times Y \subset k^{m+n}$ is an algebraic set whose Zariski topology is finer than the product ...
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1answer
25 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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1answer
39 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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0answers
59 views

Closed sets flat morphism

Consider the sets $Y_{k}:=\{y\in Y\mid |f^{-1}(y)|\leq k\}$, where $f:X\to Y$ is a flat quasi-finite morphism between smooth irreducible affine varieties and $k$ a natural number. Are these sets ...
2
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1answer
50 views

First encounters with sheaves: could you tell me if my thoughts are correct?

Let $X=\mathbb R$ be the reals and $\mathbb Z$ the integers (with the discrete topology). Do I understand correctly the definition of sheaf? Here is how I understand it (illustrated by a ...
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0answers
27 views

Non-Singular Abstract Curve

Let $X$ be an affine non-singular curve, over the closed field $k$, with function field $K$. We define $C_K$ to be the collection of all discrete valuation rings of $K$ which contain $k$. For ...
1
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1answer
24 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
2
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0answers
56 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
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2answers
23 views

Equation of a non-singular cubic curve

The equation of a non-singular cubic curve in affine coordinates is $$y^2+a_1 xy+a_3 y=x^3+a_2x^2+a_4x+a_6 .$$ If $\text{ch } K \neq 2, 3$ then it is written $$y^2=x^3+ax+b .$$ Why do we write it ...
3
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1answer
38 views

When is the symmetric algebra of a vector bundle finitely-generated?

Let $X$ be a projective variety over a field $k$, and $\mathcal L$ a vector bundle on $X$, i.e. a locally free $\mathcal O_X$-module of finite rank. For each $n\geq 0$, $\text{Sym}^n \mathcal L$ is a ...
2
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1answer
59 views

Inverse image of a line bundle on $\mathbb{P}^1$ and Euler-like exact sequence

Let $E=\mathcal{O}_{\mathbb{P}^1}(-1)$. Then we have the following exact sequence $$0\rightarrow E\rightarrow\mathcal{O}_{\mathbb{P}^1}^{\oplus 2}\rightarrow E^{-1}\rightarrow0.$$ This sequence can ...
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0answers
55 views

Curves that don't have lines as components

In my lecture notes we have the following: A point $P=\left [x, y, z\right ]$ of an algebraic curve $C_F=V(F)$ is called an inflection point of $C_F$ when $P$ is not a singular point of $C_F$. ...
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0answers
36 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
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1answer
35 views

how would i answer this question [on hold]

Please help me in this question. Let $$Y=m² - 4n²$$ $$m= 2x + 3$$ and $$n = x-1$$ Find $y$? I have tried it so many times but is not working. I would be very thankful...!!!
3
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1answer
43 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
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0answers
23 views

equivalent definitions

If $\pi :C^{'} \rightarrow C$ is a double unramified cover of a complex Riemann surface named $C$, we can define the involution sheet exchange $\tau: C^{'} \rightarrow C$. We say that a meromorphic ...
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2answers
62 views

Number of elements in fiber

My Question: If we have $f:X\to Y$ an etale morphism and we assume $X,Y$ smooth affine Varieties, why is it true, that $|f^{-1}(y)|\leq deg(f)$ ? Why isn´t there any point of $Y$, which has more ...
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0answers
34 views

Principal divisor of rational functions over nonsingular curves and pullback

I'm studying the theory of divisor over algebraic varieties for a seminar and I came across a problem that I think I solved almost completely except for a point that I'm missing. Let be $k$ an ...
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0answers
24 views

The order of the composition

Suppose that we have a meromorphic function $f$ defined on a complex riemann surface $C$ and $g$ a holomorphic function from $C$ to $C$ such that the composition is well defined. Write $ord_{p}(f ...
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0answers
33 views

Genus 2 Elliptic curves & their periods

The first part of my question is just a check of my knowledge on elliptic curves. I'm fairly happy with the number theory side of things (torsions, rank, whatever) but is my understanding of the more ...
1
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0answers
37 views

An order relation

Suppose that $\pi:C^{'} \rightarrow C$ is a double unramified cover of a riemann surface of genus $g>0$. Let $\tau:C^{'} \rightarrow C$ the involution sheet exanche and suppose that $f$ is a ...
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2answers
47 views

A doubly ruled surface which is not a plane must be quadratic

I want to show that a doubly ruled surface which is not plane must be quadratic. Any help will be appreciated.
2
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1answer
55 views

Colimit preserves monomorphisms under certain conditions

I know that colimit preserves epimorphisms. Consider the special case where The diagrams are indexed by a directed set $I$, We are in the category of certain algebraic structures, such as ...
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1answer
39 views

Skyscraper sheaf in a s.e.s

On a curve $C$, if $\mathbb{C}_p$ is the skyscraper sheaf at a point $p \in C$, then we have the exact sequence $0 \to \mathcal{L}(-p) \to \mathcal{O}_C \to \mathbb{C}_p \to 0$. On a variety with ...
2
votes
2answers
43 views

Pullback commutes with dual for locally free sheaf of finite rank

Let $ f:X\rightarrow Y$ be a morphism of ringed spaces. Let $ \mathscr{E} $ be an $\mathcal{O}_Y$ module that is locally free of finite rank. I want to show that $ (f^{*}\mathscr{E})^\vee\cong ...
4
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2answers
45 views

affine algebraic subset of $\mathbb{A}_k^4$

How do I go about proving the subset $V = \{(s^3, s^2t, st^2, t^3)\text{ }|\text{ }s, t \in k\}$ is an affine algebraic subset of $\mathbb{A}_k^4$ and find $\mathbb{I}(V) \subset k[x_0, x_1, x_2, ...
2
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2answers
47 views

Lines in the projective plane

In my lecture notes we have the following: The set $$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$ is called projective plane over $K$. There are the following cases: $z \neq ...
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0answers
26 views

Show that the set of unitary matrices is not an affine algebraic variety in complex space $C^{n^2}$.

This is an exercise from An Invitation to Algebraic Geometry by Karen Smith. It asks to show that the set of unitary matrices $U_n$ is not an affine algebraic variety in complex space $C^{n^2}$. ...
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0answers
29 views

involution of a riemann surface

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if $\pi: C^{'}\rightarrow C$ it's a double unramified cover of a complex riemann surface named $C$ that ...
2
votes
1answer
33 views

The classes are lines of $K^3$ that passes through $(0, 0, 0)$.

In my lecture notes we have the following: We consider $(K^3)^{\star}=K^3 \setminus \{(0, 0, 0)\}$ and we define the relation $$(a_1, b_1 , c_1) \sim (a_2, b_2, c_2) \Leftrightarrow (\exists ...