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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Definition of the order of a meromorphic function

Let $X$ be a complex manifold and $Y \subset X$ a hypersurface. Let $x \in Y$ and $f$ a meromorphic function on $X$ near $x$. In Huybrecht's Complex Geometry the order of $f$ along $Y$ at $x$ is ...
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407 views

transcendental base extension

An exercise in Hartshorne claims that a scheme $X$ of finite type over a field $k$ is geometrically irreducible (respectively geometrically reduced) if and only if $X \times_k K$ is irreducible ...
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177 views

Etale maps are covering maps

I am looking for a reference for the following fact (I hope it is a fact, if not, can someone give a more precise formulation)? Suppose $V$ and $W$ are two algebraic varieties over an algebraically ...
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78 views

The 2 Charts of “Blowing up the Origin in $\mathbb{C}^2$ ”

Consider the algebraic curve $\mathcal{C}$ given by $f(x,y)=0$, where $(x,y)\in\mathbb{C}^2$. Suppose that the singular point of $f$ is $p=(x,y)=(0,0)$. The blow-up of $p$ is given by ...
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100 views

When a holomorphy ring is a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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123 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...
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157 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 ...
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90 views

Elementary algebraic geometry

Let $p(z,w)=z^2+w^2-zw+1,$and $Z(p)=\{(z,w)\in\mathbb{C}\times\mathbb{C}|\,p(z,w)=0\}.$ Is this variety irreducible? Is $Z(p)$ a connected subset of $\mathbb{C}\times\mathbb{C}$ ? (in usual topology ...
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129 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
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139 views

Notation: Tensor of a sheaf and residue field

This is a simple question of notation. Let $k(p)$ be the residue field of a point $p$ on $\mathbb{P}^{N}$. How is defined (and where is) the sheaf $\mathcal{O}_{\mathbb{P}^{N}}(1) \otimes k(p)$? Also, ...
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111 views

Hyperplane not containing a given set of points over a Noetherian scheme

This is related to the answer in this question: Showing that a power of an ample sheaf is equivalent to an effective Cartier divisor Let $X$ be a quasiprojective scheme over a Noetherian ring A and ...
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203 views

Moduli space of isogeny classes of elliptic curves

The modular curve $Y(1)$ classifies isomorphism classes of elliptic curves, namely its $K$-points for any field $\mathbb Q\subseteq K\subseteq \mathbb C$ correspond via the $j$-invariant to $\mathbb ...
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109 views

On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ...
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168 views

The Uniformization Theorem and Elliptic Curves

In the theory of elliptic curves, I have read that the elliptic curves is topologically equivalent to a torus, given by $\mathbb{C}$/ $\Lambda$, where $\Lambda$ is a lattice. The proof appears to ...
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Decomposing an Affine transformation

An affine transformation is composed of rotations, translations, scaling and shearing. In 2D, such a transformation can be represented using an augmented matrix by $$ \begin{bmatrix} \vec{y} \\ 1 ...
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184 views

Underlying set of the scheme theoretic fiber

Categorical constructions in the category of schemes usually do not preserve the underlying sets. For example, the underlying topological space of the product of schemes is not the topological product ...
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63 views

Poles of abelian differentials

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have ...
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1answer
191 views

Reference for Deligne-Mumford

What is a good reference for someone new to the theory of Deligne-Mumford stacks, other than the original Deligne-Mumford paper itself? The paper itself seems readable with some effort; but the fear ...
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268 views

Picard group of a Affine scheme

How do we define a Picard group of an Affine scheme? Is there way to define as for commutative ring? Thanks
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160 views

Rank one sheaves and ideal sheaves

For a coherent sheaf $\mathcal F$ on a smooth irreducible projective variety $X/k$, it makes sense to define the rank $\textrm{rk }\mathcal F$ as the rank of the vector bundle $\mathcal F|_U$, where ...
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144 views

Are the algebraic-valued points of a variety dense in complex-valued points?

Let $X$ be a variety (integral scheme of finite type) over $\overline{\mathbb Q}$. We may endow the sets $X(\overline{\mathbb Q})$ and $X(\mathbb C)$ of $\overline{\mathbb Q}$- resp. $\mathbb ...
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83 views

Why is the map: $GL_n(K)\times GL_n(K) \to GL_n(K)$ regular?

Let $K$ be a field and $GL_n(K)$ the set of all invertible $n$ by $n$ matrices over $K$. Let $m: GL_n(K)\times GL_n(K) \to GL_n(K)$ be the usual multiplication of matrices. Why the map $m$ is regular? ...
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93 views

Definition of equalizer for $\textbf{Sh}(X)$

Let $\textbf{Sh}(X)$ denote the category of all (set - valued) sheaves on a topological space $X$. My question is: Given sheaves $F,G \in \textbf{Sh}(X)$ and morphisms $\varphi : F \to G$ ...
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168 views

Example of a variety that is not toric

My question is simple, but I haven't seen it to be addressed anywhere: What would be a simple example of an affine variety that is not a toric variety? Toric varieties (the ones I have ...
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154 views

what does project away mean?

I realise I should know this but I have no idea what people mean when they say "we project away from this point" (or replace point with line, plane or whatever in projective space). What does this ...
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78 views

is prime spectrum $Spec(R)$ countable?

Let $R$ commutative ring with identity, given $Spec(R)=\{I|\text{$I$ prime ideal of $R$}\}$, does the set $Spec(R)$ countable? Also, if $\{\langle p^n \rangle\}$ is closed in $P_{-}Spec(R) = \{I| ...
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207 views

Pole set of rational function on $V(WZ-XY)$

Let $V = V(WZ - XY)\subset \mathbb{A}(k)^4$ (k is algebraically closed). This is an irreducible algebraic set so the coordinate ring is an integral domain which allows us to form a field of fractions, ...
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673 views

Projective Normality

What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
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300 views

Intersection of open affines can be covered by open sets distinguished in *both*affines

Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$. It's not true in general that $U \cap V$ is affine, so if we ...
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211 views

Serre's Theorem for $\operatorname{Proj}$

Let $k$ be an algebraically closed field. If $S$ is a positively graded $k$-algebra which is finitely generated by $S_1$ over $S_0 = k$ then quasi-coherent sheaves on $\operatorname{Proj}S$ are ...
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94 views

Reference on a theorem of algebraic geometry

In the book GTM 52 by R.Hartshone, there is a theorem as following : Every variety of dimension $r$ is birational to a hypersurface in $\mathbb{P}^{r+1}$ Could you please tell me, who is the ...
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163 views

Tangent Space Exercise

Can you please describe how to use the intrinsic definition of tangent space to show that the tangent space of the curve $Z \left( y^2-x^3 \right)$ at the point $x = \left(1,1\right)$ is ...
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190 views

Open affine neighborhood of points

$X$ is a variety and there are $m$ points $x_1,x_2,\cdots,x_m$ on $X$. Can we find an open affine set which contains all $x_i$s?
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319 views

Extension of a short exact sequence of group schemes

Let $S$ be a Dedekind scheme with rational functions field $K$. Consider an exact sequence $$ 0 \to G'_K \to G_K \to G''_K \to 0$$ of smooth $K$-group schemes of finite type. Assume that $G'_K$ and ...
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372 views

Is locally free sheaf of finite rank coherent?

Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent? By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that ...
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149 views

Exposition on hyperelliptic curves

Is there any literature that introduces hyperelliptic curves without the view towards cryptography? Even better is if there are any books that talk about them with (about) the same amount of detail as ...
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1answer
107 views

Product of varieties

If we have two rational varieties (i.e varieties which are birational to some projective space) is their product also a rational variety? would this rely on the fact that the Zariski topology is finer ...
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248 views

Birational morphism

I have a question on rational and birational maps: Is the map $$\mathbb{P}^1\rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:1)$$ rational? Birational? If birational what is its inverse? Same questions ...
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352 views

What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
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694 views

Studying the envelope of a family of circles.

This is an exercise on page 150 of Cox/Little/O'Shea's Ideal, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra, 3rd ed. I get lost in this ...
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1answer
223 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
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130 views

Structure sheaf of a fiber

Let $\phi: Y\to X$ be an affine (finite & dominant) morphism of (smooth) $\Bbbk$-varieties. Let $Y_P$ be the scheme-theoretic fiber of $P\in X$, i.e. $Y_P=Y\times_X\mathrm{Spec}(\Bbbk(P))$. I was ...
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1k views

irreducibility of a polynomial in several variables over ANY field

The irreeducibility of a polynomial $f\!\in\!K[x_1,\ldots,x_n]$ in general depends on what the field $K$ is (for example, if $K=\mathbb{R}$, then $f=x_1^2+1$ is irreducible, but if $K=\mathbb{C}$, it ...
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457 views

Euler characteristic in Zariski vs. classical Topology

Let $X$ be a smooth projective, complex variety. Denote by $\bar X$ its analytification, i.e. $X$ with the "classical" topology of a complex manifold. Now do these spaces have the same Euler ...
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155 views

Bounding the Number of Zeros of Modular Forms

Given a (meromorphic or holomorphic) modular form $f$ of weight $k$ on some genus zero congruence subgroup $\Gamma$, are there known bounds for the number of zeros and poles that $f$ has on the ...
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311 views

Line bundles, line bundles on a homogeneous space, and sections of line bundles

I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section ...
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156 views

Algorithms to prove that polynomials don't have integer solutions

OK, I know that Matiyasevich's solution to Hilbert's 10th problem shows that there is no algorithm to decide whether or not a polynomial $p(x_1,\ldots,p_n)$ with integer coefficients has a solution ...
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82 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 ...
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69 views

Where in EGA is the result of Serre Vanishing located?

We recall Serre's Vanishing Theorem which states the following. Let $X$ be a closed subscheme of $\Bbb{P}^n_A$ with $A$ a Noetherian ring. Then for any $\mathcal{F} \in \operatorname{Coh}(X)$, there ...
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66 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...