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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4
votes
5answers
58 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 ...
0
votes
2answers
30 views

Checking normality for quasi compact schemes

Let $X$ be a quasi compact scheme. We know that any point on $X$ is a generization of a close point. Could someone possibly explain me why it then follows that to check if $X$ is normal, it suffices ...
0
votes
1answer
22 views

Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
2
votes
0answers
14 views

Cohomology with coefficient $\mathbb{Q}(n)$

What is the definition of $$\mathrm{H}^i(X,\mathbb{Q}(n))$$ for a variety $X$? and What is its relation with $\mathrm{Ext}^*(\mathbb{Q}(0),\mathbb{Q}(n))$? A good reference is also very helpful. ...
0
votes
0answers
26 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
0
votes
1answer
34 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 ...
1
vote
1answer
28 views

Simple question about the definition of divisor

Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define the divisor $\pi^{*}(E)$?
6
votes
1answer
117 views

Fibers equal implies schemes equal in a neighborhood

Let $f:X \rightarrow Y $ be a morphism of locally Noetherian schemes. Let $Z$ be a closed subscheme of $X$ and suppose that there exists a point $y \in Y$ such that $Z_y=X_y$ as schemes. Show that if ...
0
votes
0answers
53 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 ...
4
votes
0answers
72 views
+50

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map ...
2
votes
0answers
31 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 ...
0
votes
2answers
43 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 ...
2
votes
0answers
65 views
+50

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on ...
1
vote
1answer
23 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
votes
0answers
13 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 ...
4
votes
1answer
126 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
4
votes
0answers
37 views
+50

$\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ as interior of compact Riemann Surface with Boundary

A takehome exam problem for my Riemann Surfaces class, which used Griffith's Introduction to Algebraic Curves, was the following: Show that $S=\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ is not interior ...
0
votes
0answers
39 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 ...
0
votes
1answer
29 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 ...
0
votes
1answer
42 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 ...
0
votes
0answers
26 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 ...
-4
votes
0answers
37 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
votes
0answers
33 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 ...
3
votes
1answer
45 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 ...
10
votes
2answers
168 views

What does $Tor^{R}_n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
3
votes
1answer
50 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 ...
1
vote
2answers
89 views

Cohen-Macaulay and regular rings

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
0
votes
1answer
38 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 ...
1
vote
0answers
37 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. ...
1
vote
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 ...
1
vote
1answer
468 views

Why does surjectivity of the induced map show that a morphism of affine varieties has closed image?

Let $\phi : X \rightarrow Y$ be a morphism of affine varieties and let $\phi^\ast : k[Y] \rightarrow k[X]$ be the induced map on coordinate rings. My text says that if $\phi^\ast$ is surjective then ...
2
votes
1answer
76 views

Pascal's theorem by Bezout's theorem

I need to prove the following theorem Let the hexagon $ABCDEF$ be inscribed in the nondegenerate conic $q=V(f)$. Assume that $A,B,C,D,E,F$ are distinct. Let $P=\overline{FA}\cap \overline{CD}, ...
2
votes
1answer
42 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 ...
2
votes
0answers
57 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 $\Phi$ ...
0
votes
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. ...
2
votes
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 ...
0
votes
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 $\gcd(x,y,z)=1$. ...
1
vote
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 ...
0
votes
1answer
27 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
1
vote
0answers
65 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 ...
0
votes
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 $Q=-P$ ...
3
votes
1answer
187 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
2
votes
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 ...
0
votes
0answers
35 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 ...
2
votes
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 ...
3
votes
1answer
70 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
0
votes
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
vote
1answer
25 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 ...
1
vote
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 ...
1
vote
1answer
108 views

Generalization of Bézout theorem in higher dimension.

I am wondering if there is a generalisation of Bézout theorem in higher dimension. By Bézout theorem I mean: If $f_1,f_2\in\mathbb{C}[X_1,X_2]$ such that $\dim V(f_1,f_2)=0$, then $\#V(f_1,f_2)\leq ...