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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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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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
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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34 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 ...
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171 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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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$E \to S$ 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 ...
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1answer
67 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
38 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
86 views

Is $\mathbb{P}^{1}$ a fine moduli scheme?

I want to show that $\mathbb{P}^{1}_{\mathbb{C}}$ is a fine moduli scheme for the families of lines through the origin of the affine plane. I took a flat family $\mathcal{D}\rightarrow B$ and I tried ...
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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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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 ...
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1answer
34 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
45 views

Harnack's curve theorem for curves in $\textit{complex}$ projective plane?

The wikipedia page gives the statement for algebraic curves in real projective plane. Is the statement also true in $\textit{complex}$ projective plane? If not, is there a similar statement about ...
2
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1answer
46 views

Restriction and extension of scalars between flat algebras and their completion over a DVR and ideals.

So, in a proof I am currently reading I have stumbled upon the following. Let $R$ be a discrete valuation ring, $\hat{R}$ its completion and $t$ a uniformizing parameter for $R.$ Let $A$ be a flat ...
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1answer
41 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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26 views

$\{(x,y)\in \mathbb C^2|y^2=\sin x\}$ 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 ...
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81 views

Computing $f^{*}\mathscr{O}_X$ directly via colimit

I want to prove that, for affine schemes $X = \text{Spec} (A)$, $Y = \text{Spec} (B)$ and $f: Y \rightarrow X$ morphism of schemes ($\varphi: A \rightarrow B$), $f^{*}\mathscr{O}_X \cong ...
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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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38 views

intersection property holdds for every Veronese embedding?

The Veronese Embedding Doesn't the nice intersection property listed near the end here hold for every Veronese embedding? If so/not, where I could find a proof/counterexample? Thanks.
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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 ...
2
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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 ...
5
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2answers
93 views

Spectrum of $\mathcal{O}(U)$

Let $X=\operatorname{Spec}(A)$ be the spectrum of the comm. ring $A$ and let $\mathcal{O}$ be the associated sheaf of rings, i.e. for $U \subseteq X$ open, $\mathcal{O}(U)$ is the ring of all ...
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30 views

Precomposing a rational function $f$ with a birational isomorphism $g$ to make $f\circ g$ regular?

I think I have a gap in my understanding, Suppose $Y$ and $Z$ are quasi-projective varieties, and that $Y$ is irreducible. Suppose we have a rational map $f\colon Y\to Z$. Then we know there ...
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1answer
60 views

Question on exercise of ideal of a point

The question was to find the ideal of a point $(\sqrt{2},\sqrt{3})$ in $\mathbb{Q}[X,Y]$ and its conjugates in $\mathbb{C}^2$. Is is correct to say that the ideal of a point is ...
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54 views

Picard group of affine scheme of a UFD

In which book/notes can I find proofs of the following facts? 1) Pic(Spec$A)$ is $0$ where $A$ is a UFD. 'Pic' is the Picard group. 2) The invertible sheaves on projective space P$^n(k)$ for $k$ a ...
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26 views

connected fibers of a partial desingularization

I had been told this once, but cannot deduce it from the standard references. Is it correct? I'm working over $\mathbb{C}$. Let $f : X \to Y$ be a birational proper morphism where $X$ is normal with ...
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1answer
69 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq ...
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43 views

Does globas sections form an exact sequence?

Let $(X,\mathscr A)$ a ringed space and $0\to\mathscr F\overset{u_X}\to\mathscr G\overset{v_X}\to\mathscr H$ an exact sequence of $\mathscr A$-module homomorphisms. Is the sequence ...
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42 views

An alternative description of an holomorphic map associated to a complete linear system

I need an help with an exercise in Miranda's book "Algebraic curves and Riemann surfaces". More precisely is the exercise in Problems V.4 I. Given a Riemann surface $X$ and a divisor $D$ on $X$ ...
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1answer
71 views

Roadmap to reach Arithmetic Geometry for a Physics Major

I am a physics major but I self-study mathematics. my interests are number theory and geometry. it seems that due to the works of Grothendieck, algebraic geometry have to be used to study deepest ...
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1answer
81 views

Fiber as vector space over residue field.

Let $A$ be a commutative ring with identity and let $M$ be an $A$-module. The fiber of $M$ at $P \in \text{Spec}A$ is the module $M(P):=M_P / PM_P$, which is a vector space over the residue field ...
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1answer
44 views

Birational proper morphism and global sections

Let $f:X \to Y$ be a proper, surjective, birational morphism of noetherian (connected) projective schemes. Assume that $X$ is non-singular. Let $D$ be a Cartier divisor on $Y$ and $L$ the line bundle ...
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19 views

Polynomial-time algorithm to decide polytope equality $\{x \mid Ax \leq b\}=\{x \mid Bx \leq d \} $

This is a homework. Given two sets of inequalities $$Ax \leq b ,\,Cx \leq d ,\,x \in \mathbb{R}^n$$ where the coefficients are all rationals. Give a polynomial time algorithm to decide if: $$\{x ...
2
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1answer
90 views

Proof on page 215 of Miranda's book

At the page 215, Miranda says that the dimension of the fiber of the map: $$ \gamma: \{(X,D_{2g-1})\} \mapsto \{X_g\} $$ where $\{(X,D_{2g-1})\}$ is the space of the pairs with $X$ an algebraic curve ...
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32 views

Reference about elliptic integral and Jacobi Inversion Problem

I read the section about Abel's theorem and the Jacobi Inversion Problem on the book of Forster, "Lectures on Riemann Surfaces". I would like if there were some books which treats more in detail this ...
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36 views

Miranda's Exercise on the Jacobian of a complex torus

I have to prove that the Jacobian of a complex torus $X=\mathbb{C}/L$ is isomorphic to $X$ by explicity showing that the subgroups of periods $\Lambda \subset \mathbb{C}$ is a lattice which is ...
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29 views

Composition of homomorphism $d^p$ is 0

Let $(X,\mathscr A)$ be ringed space, $\mathscr F$ an $\mathscr A$-module and $\mathfrak U=(U_i)_{i\in I}$ an open covering of $X$. Let $\overline C^p(\mathfrak U,\mathscr ...
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29 views

A question to lemma 4.14 chapter VII “ An Invitation to Arithemtic Geometry” Lorenzini

Lemma 4.14: (constant field extension) Let $k$ be perfect field. Let $k(X)/k$ be a function field. If $k'/k$ is any extension of $k$ in $\bar{k}$, then $k'(X)/k'$ is a function field. Moreover, if ...
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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 ...
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26 views

What does the Gamma means in local ringed space?

I found the following problem from an algebraic geometry course hold in 2003. Let $(X,\mathscr A)$ locally ringed space and $f\in \Gamma(X,\mathscr A)$. Prove that $$X_f=\{x\in X|f(x)\ne 0\}$$ is an ...
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32 views

Basic algebraic geometry

Given an algebraic variety $X$ and two $Q$-Cartier divisors $D_1$ and $D_2$. Given $f \in H^0(X, \mathcal{O}_X(D_1))$ and $g\in H^0(X, \mathcal{O}_X(D_2))$. It is always true that $\frac{g}{f}$ is a ...
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1answer
35 views

Siegel's theorem, best possible bound?

I am studying classic Siegel's theorem, and I was wondering if that result is the best possible bound, meaning: is there a compact connected complex manifold $X$ of dimension $n$ in which ...
4
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2answers
85 views

A question on Mumford's drawing of $\text{Spec}\,\mathbb{Z}[x]$

This might seem like a really silly question, but what are those weird curves connecting $(x^2 + 1)$ and $(5, x+2)$ in Mumford's picture of $\text{Spec}\,\mathbb{Z}[x]$?
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2answers
67 views

Elements of the zero-th Čech cohomology group versus global holomorphic sections

Something that is confusing (well, to me) has come up in the course of asking other questions. Let $\pi:V\to X$ be a holomorphic vector bundle of rank $r$ on a complex manifold $X$, such that $V$ is ...
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1answer
34 views

Closed immersion of separated scheme is separated

I'm working on exercise II.4.4 in Hartshorne. I'm stuck on a step and would appreciate some help. Here is the situation I have. $ S $ is a Noetherian scheme. $Y$ is a separated $ S $-scheme of finite ...