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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13 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
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20 views

A prime ideal with the algebraic set reducible

In "Algebraic Curves" by Fulton, section 1.7, page 11, there is the following corollary of the nullstellensatz: Corollary 2: If $I$ is a prime ideal, then $V(I)$ is irreducible. There is a ...
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26 views

27 Lines on a Cubic

In Ravi Vakil's notes, there is a proof (in section 27, of course) of the famous result that every nonsingular cubic hypersurface in $\mathbb{P}^3_k$ over an algebraically closed field $k$ has exactly ...
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26 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. $\textbf{Shafarevich definition}$ (pg 128) - A variety is $\textit{non-singular in codimension one}$ if the singular locus has ...
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18 views

When using the square method , how do you know what level equation you are working with? [on hold]

When using the square method , how do you know what level equation you are working with?
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1answer
32 views

Is $\mathbb{C}[T \times_T T] = (\mathbb{C}[T] \otimes \mathbb{C}[T])^T$?

Let $T$ be an algebraic group. There is a left $T$ action on $T$ given by left multiplication and a right action on $T$ given by right multiplication. Let $T$ acts in the middle of $T \times T$ by the ...
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2answers
22 views

Counting the number of $\mathbb{F}_q$ points on a homogeneous polynomial

This is an area of number theory that I am not too familiar with and I would appreciate any assistance! Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic not 2 or 3. I have the ...
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21 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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10 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
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1answer
66 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
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31 views

Twisting sheaf of Serre on projective schemes

Given $A$ a ring and $P=Proj(A[X_0,\cdots,X_n])$, I know that $\oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]$. This equality is not true in general for every graded algebra $B$ for which ...
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1answer
31 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
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1answer
16 views

Is $\mathbb{C}[T \times_T T] \cong \mathbb{C}[T]$?

Let $T$ be an algebraic group. Do we have the following result: The algebra $\mathbb{C}[T \times_T T]$ is equal to the image of the co-multiplication $\mathbb{C}[T] \to \mathbb{C}[T] \otimes ...
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0answers
18 views

functions on Riemann sphere

Let $C$ a complex riemann surface (compact). Take $f$ a meromorphic funtion on $C$. This function defines a function $\pi:C \rightarrow \mathbb{P}^1_{\mathbb{C}}$. How can i define a function $f^{'} ...
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33 views

About the smoothness of a non-reduced variety.

Consider a $k$-scheme $X$ with the following properties: $k$ is algebraically closed, $X$ is irreducible, non-reduced, separated and of finite type; moreover $X$ has dimension $1$ ($X$ is a non ...
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1answer
31 views

How to show that a polynomial maps an algebraic set to an algebraic set?

Sorry if this is an ignorant question. I am studying algebraic geometry. This isn't an exercise problem. It is an assumption I can use to prove something else. I think it must be obvious, but I don't ...
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1answer
23 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
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1answer
31 views

Do we have $\mathbb{C}[T] = \mathbb{C}[\Lambda] = \oplus_{\lambda \in \Lambda} \mathbb{C}[\lambda]$?

Let $\mathbb{C}[T]$ be the coordinate ring of a torus $T$. Suppose that $T$ acts on some variety $X$. Then $T$ acts on $\mathbb{C}[X]$: $t(f) = \lambda(t)f$ ($f$ is a homogenous function on ...
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49 views

Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
2
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1answer
36 views

Universal property of sheafification

Given a presheaf $\mathcal{F}$ there is a sheaf $\mathcal{F}^+$ and a morphism $\theta: \mathcal{F}\to\mathcal{F}^+$ with the property that for any sheaf $\mathcal{G}$ and any morphism $\varphi: ...
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0answers
35 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 $\mathrm{Ext}^*(\mathbb{Q}(0),\mathbb{Q}(n))$? Another question: Is this a notion which ...
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32 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 ...
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2answers
35 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 ...
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1answer
39 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 $\pi^{*}(E)$?
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1answer
34 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 ...
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62 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
48 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
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 ...
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19 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 ...
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36 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
41 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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1answer
55 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
49 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 $k$ with char $\not=2$ and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals such that the ...
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5answers
71 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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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
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0answers
35 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
27 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
45 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 ...
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1answer
51 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
38 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
41 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
44 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 ...
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1answer
43 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
49 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
34 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
27 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
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1answer
41 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
69 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
51 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 ...