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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Morphism of a linear system contracting curves

Let $X$ be a compact complex surface and $L$ a line bundle such that the linear system $|L|$ has no basepoints and $h^0(X,L)>0$. Denote by $\phi:X\rightarrow\Bbb{P}(H^0(X,L)^\vee)$ the morphism ...
2
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1answer
42 views

Comparing vector bundle degrees coming from different embeddings into projective space

This question is a follow-up to this recent question of mine: Comparing notions of degree of vector bundle In that question, the definition of the degree of a vector bundle is discussed — in ...
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0answers
29 views

why a projective transormation of a non singular curve gives a non singular cuve?

I was reading a book which has the following exersise.Suppose that V is a nonsingular projective curve and T is a projective transformation. Prove that T(V ) is also a nonsingular projective curve. ...
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1answer
26 views

Zariski Topology: Prove that $V(I(X))=\bar{X}$

I am studying Zariski Topology. Here is a problem I am trying to work on: Let $X\subset A^n$ be an arbitrary subset. Prove that $V(I(X))=\bar{X}$. This is my work so far: (1) Since $\bar{X}$ is ...
4
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1answer
39 views

$L$-Zariski closure of subgroup $SL_n(F)$ as subset of $M_n(F)$ also a subgroup of $SL_n(F)$

Let $F$ be a field, and $SL_n(F)$ be the group of $n \times n$ matrices with determinant $1$. Let $\Gamma \subset SL_n(F)$ be a subgroup. We can consider $\Gamma$ to be a subset of $M_n(F) \cong ...
2
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1answer
82 views

About Betti Numbers

I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in ...
0
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1answer
31 views

What is the fundamental theorem on discrete groups of Euclidean spaces?

I have been reading the book Using Algebraic Geometry by David A. Cox, John Little, Donal O'Shea for a university project. I am not clear as to what exactly in meant by the phrase "the fundamental ...
2
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1answer
29 views

Morphisms of the varieties and coordinate rings

I'm thinking something very wrong but I can't find what the flaw in my thinking is. Well, here goes. First, if a contravariant functor $\mathsf{F}: \mathscr{C}\to \mathscr{D}$ is a category ...
2
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1answer
43 views

Isomorphism of Homs

How can I show the existence of tensor product mappings. Namely, in Liu there is a problems to show that there exists a unique $A$-linear map ...
2
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2answers
111 views

“Closure” and “neighborhoods” in Spec(A)

While trying to work through the sequence of problems in Atiyah-Macdonald's first chapter regarding the prime spectrum of a ring, I've run across a small point of confusion. Namely: In the point ...
3
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2answers
118 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
2
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1answer
47 views

Every affine variety in $A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials

I am reading Gathmann's free online notes on Algebraic Geometry. One exercise asks to show that "Every affine variety in $A^n$ consisting of finitely many points can be written as the zero locus of ...
2
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1answer
30 views

Determine the radical of an ideal

Determine the radical of the ideal $(x^3-y^6,xy-y^3)$ in $C[x,y]$. I used Nullstellensatz theorem $\sqrt{I}=I(V(I))$. Factorization gives: $$x^3-y^6=(x-y^2)(x+(\frac{1}{2}+\frac{\sqrt{3}}{2}i) ...
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0answers
80 views

What would be the most rigorous book to stydy algebraic geometry and arithmetic curves on my own?

I would like to study algebraic geometry and arithmetic curves on my own but are there suggestions where to start? Namely, I like very rigorous way to do mathematics and I was suggested Liu's book ...
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0answers
49 views

Image of a maximal torus via epimorphism

Let $\phi \colon G \rightarrow H$ be a surjective morphism of linear algebraic groups. Let $T \subset G$ be a maximal torus: how can I prove that $\phi(T)$ is also a maximal torus? To show that ...
3
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1answer
73 views

Which polynomial level sets are bounded?

Let $$\begin{align} \mathcal{P}^n&=\left\{p:\mathbb{R}^n\to\mathbb{R}, (x_1,\ldots,x_n)\mapsto\sum_{i_1,\ldots,i_k\in\mathbb{N}\cup\{0\}} a_{i_1\cdots i_k}x_1^{i_1}\cdots x_n^{i_n}\;\bigg|\; ...
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48 views

Component Group Neron Model Elliptic Curve Cyclic

I'm studying the chapter on Neron Models in Silverman's book "Advanced Topics in the Arithmetic of Elliptic Curves" at the moment, and I do not quite understand why in the split multiplicative case, ...
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65 views

Leray's theorem up to some degree

I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology. The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such ...
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1answer
83 views

Proved conjectures (now theorems) in algebra/algebraic number theory/algebraic geometry

I would like to collect some proved conjectures (not so non trivial) in algebra/algebriac number theory/algebraic geometry. For example, I consider Serre's conjecture on projective modules over ...
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1answer
15 views

Reduction of general conic

The given equation is - $$3x^2 + 2xy + 3y^2 - 32y +92=0$$ To get rid of xy term i used the substitutions - $$x=p+q , y=q-p$$ Then the equation becomes - $$(p-4)^2 + 2(q-2)^2=1$$ which is an ellipse ...
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128 views

The geometric interpretation of Gorenstein local ring

Many local rings have geometric interpretations. Cohen–Macaulay rings correspond to equi-dimensionality, and regular local rings correspond to non-singularity, but what is a geometric interpretation ...
2
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1answer
39 views

Zeros of this multivariate polynomial

I have an equation, wich is somewhat related to the doppler effect : $$ x_1^2x_3^2+x_2^2x_4^2+2x_1x_2x_3x_4-Cx_1^2-Cx_2^2=0 $$ Where C is a known real positive constant. My background in math isn't ...
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31 views

Questions about coordinate ring of a closed subvariety.

How to prove the following statement: If $L$ is a closed subvariety of $M$, then there is a surjection from $\mathbb{C}[M]$ to $\mathbb{C}[L]$? I think that maybe this follows from definitions. ...
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1answer
75 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
2
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1answer
73 views

prerequisites for serre's FAC?

Is the knowledge of undergraduate's basic algebra and general topology enough to reading FAC? Do I need learn some algebraic topology and homological algebra, commutative algebra, or several complex ...
2
votes
1answer
49 views

$K_X^*/O_X^*$ is a flasque sheaf for smooth variety over $\mathbb{C}$?

Suppose $X$ is a smooth variety over $\mathbb{C}$, why do we have $K_X^*/O_X^*$ is a flasque sheaf? (Beauville "Complex Algebraic Surface" p.28) (To show the surjection $K_X^*/O_X^*(X)\to ...
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1answer
109 views

Leray-Hirsch Using Kunneth Formula from “Differential form in Algebraic Topology” by Bott and Tu

Kunneth Formula: Let M and F are manifolds. If M has a finite good cover then $H^n(M\times F)=\bigoplus _{p+q=n} H^p(M)\bigotimes H^q (F)$ Bott and tu says One can prove Leray-Hirsch theorem by the ...
5
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1answer
76 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...
4
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2answers
146 views

The Zariski density for two given sets.

Let $A$ and $B$ be two subsets of $\mathbb{C}^n$: $ A = \mathbb{Z}^n$, and $B=\{ (z_1,z_2, \dots , z_n) \in A \text{ such that } z_1>z_2>\cdots> z_n\}$. My questions: Are these two subsets ...
6
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1answer
168 views

Fibrations with isomorphic fibers, but not Zariski locally trivial

I am looking for examples of fibrations $f:X\to Y$ where the fibers are all isomorphic, but $f$ is not Zariski locally trivial. In particular, I am interested in understanding how much such examples ...
4
votes
2answers
173 views

Prym variety associated to an étale cover of degree 2 of an hyperelliptic curve.

In view of this question, I have an additional question. The situation is as follows. Let $C$ be the hyperelliptic curve over $\mathbb{C}$, which is given on an affine by the equation $y^2 = x^5 +1 ...
4
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1answer
62 views

What's the connection between exceptional divisor and projectivized tangent space?

This is one homework problem and hence I want some hint but not a whole answer. Let $P$ be a projective space and $X\subset P$ be a non-singular variety. Prove that the collection $L_p$ of lines ...
5
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2answers
164 views

function field of an integral scheme

Suppose $X$ is an integral scheme, and let $\eta \in X$ be its generic point. Then the local ring $\mathcal{O}_{X,\eta}$ is a field, called the function field of $X$ and denoted $K(X)$. Why is $K(X)$ ...
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0answers
35 views

Question regarding the definition of finitely generated graded ring

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring and $S_+ = \oplus_{n \geq 1} S_n$. The notes (Ravi Vakil's online notes on algebraic geometry) I am reading defines the graded ring $S$ is finitely ...
6
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0answers
51 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
0
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2answers
62 views

When a prime ideal is restricted to a basic open subset of projective space, is it still prime?

Suppose $I\subset k[x_0,\ldots,x_n]$ is a prime ideal. Now restricted on the basic open subset $\mathbb{P}^n_{x_i}$ of $\mathbb{P}^n$, is $I$ still prime? Note: 1. Here $\mathbb{P}^n_{x_i}$ is ...
2
votes
1answer
275 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
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1answer
85 views

The Theorem on Formal Funtions - Harthshorne Theorem 11.1

Let $f:X \rightarrow Y$ a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, and let $y\in Y$ be a point. For each $n\geq 1$ we define $X_n=X \times_Y Spec ...
3
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1answer
42 views

Basic computation for the degree of an isogeny

I am trying to compute the degree of the isogeny $\phi:E_{1} \to E_{2}$ where $\phi(x,y)=(\frac{y^2}{x^2},\frac{y(b-x^{2})}{x^2})$ with $E_{1} : y^{2} = x^{3} + ax^{2} + bx$, $E_{2} : Y^{2} = X^{3} - ...
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1answer
76 views

Module of differentials in the functorial approach to schemes and quasi-coherent modules

Recall that for a functor $X : \mathsf{CAlg}(R) \to \mathsf{Set}$ from commutative $R$-algebras to sets one can define quasi-coherent $\mathcal{O}_X$-modules as "compatible" families of $A$-modules ...
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1answer
147 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
4
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1answer
68 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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1answer
60 views

A rigorous characterization for a ringed spaces to be isomorphic to an affine scheme.

On page 21-22 of the book The Geometry of Schemes by Eisenbud and Harris there is a characterization for when a ringed spaces $(X,\mathcal{O}_X)$ is isomorphic to an affine scheme ...
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1answer
41 views

Equality of two $K$-valued points for reduced $K$

I'm reading "Red Book of varieties and schemes". There is definition 2, page 118. Let $f,g:K \to X $ be to $K-$valued points of scheme $X$, we say that they are equal at $x\in K$ $(f(x) \equiv g(x))$ ...
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1answer
60 views

Intersection of ideals of ring sheaf

Let $k$ be a field, $A=k[T]$, $X=\operatorname{Spec}A$, and $\mathfrak a_n=AT^n\subset A$ $(n\in \mathbb N)$. Why do the intersection of the ideals of sheaf of rings $\tilde A$, i.e. $\bigcap ...
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2answers
39 views

Why is the complement of an affine subset of projective space a hyperplane?

Let $P$ be a projective space of dimension $n$ and $Q$ a linear subspace of it. If the complement of $Q$ is affine, why must $Q$ be of dimension $n - 1$? The following is my thought: Take the ...
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1answer
35 views

Is taking the product of quasi-projective varieties associative?

I was reading a bit of Hartshorne, and I know that the product of quasi-projective varieties is again a quasi-projective variety. I should hope that taking products is associative, but I am unsure. ...
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1answer
86 views

Fano-ness of moduli space of stable vector bundles when determinant line bundle is *not* fixed…

According to Drezet-Narasimhan, Invent. Math. 97 (1989), no. 1, 53--94, the moduli space $\mathbb M$ of slope-stable holomorphic vector bundles with fixed rank $r$ and fixed determinant line bundle ...
2
votes
1answer
66 views

Moore space, induced map in homology

Let $A$ be a finitely generated abelian group and $n$ a positive integer. I have built a connected space $M(A,n)$ such that all its reduced homology groups are zero but the i-th reduced homology group ...
6
votes
1answer
156 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...