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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Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
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34 views

Dual of etale fundamental group, field functoriality

Suppose $k$ is an arbitrary field and $X/k$ is a finite-type and integral scheme (hence connected). Let $K$ be the fraction field of $X$. Via the canonical morphism $$\phi: \rm{Spec}(K)\rightarrow ...
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70 views

Tensor product of structure sheaves of subvarieties

Suppose I have a complex variety $X$ and two closed subvarieties $A$ and $B$, with closed immersions $i:A\to X$ and $j:B\to X$. Then we have two $\mathcal O_X$-modules $i_*\mathcal O_A$ and ...
3
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2answers
223 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
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34 views

finite morphism (algebraic) vs finite morphism (analytic)

Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding ...
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1answer
102 views

Change of coordinates (referential system) mistake? Doesn't seem to yield the proper coordinates.

Let $\varepsilon$ be an affine space with referential system $R$ characterized by $O=(1,1,1)$ as origin and $B=(c_1,c_2,c_3)$ as its basis, which is the canonical. Now, lets define a new ...
2
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59 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
3
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1answer
105 views

Integral Domain with exactly two Prime Ideals

I am not looking for someone to give me an explicit example. I want to work this out myself if possible. Trying to learn schemes by reading The Geometry of Schemes by Eisenbud and Harris. Problem I-5 ...
4
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1answer
116 views

Question on morphism locally of finite type

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open ...
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311 views

What is an imaginary line?

An imaginary point is defined as an ordered pair of values, at least one of which is complex. My text says that if $h^2 < ab$ in the equation : $$ ax^2 + 2hxy + by^2 $$ it represents two ...
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135 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
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43 views

Differentials on a curve

Say I have an algebraic curve $C$ over a field $k$ and a group $G$ acting on $C$. Under what conditions on $C$ and/or the action of $G$ on $C$ can one conclude that $H^0(C,\Omega^1_C)^G = ...
3
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61 views

$h^{p,q}$ of projective space

How can we calculate the Hodge number $h^{p,q}= \dim H^p(\mathbb{P^n},\Omega^q_{\mathbb{P}^n})$ of projective space? Is there a reference for that?
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57 views

Covering a closed disk in a rigid analytic space by residue classes

Recently I have been reading through the PhD thesis of Dr. Louis Brewis, "Ramification theory of the p-adic open disc and the lifting problem", which is available free here: ...
3
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1answer
110 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
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60 views

Points lying over a closed point in a separable extension of the base field are rationnal

At the end of the proof of Proposition 4.3.30 In Liu's book we have the following situation: $X$ is an algebraic variety over a field $k$, $x\in X$ is a regular closed point of $X$ with $k'=k(x)$ is a ...
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1answer
144 views

How to calculate $\operatorname{Spec} \mathbb{C}[x,y]/(y^2-x^3)$

Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ? Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of ...
3
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159 views

Why Elliptic Curves have so many nice properties

As the definition referred from Silverman's book: An elliptic curve is a pair $(E,O)$, where $E$ is a nonsingular curve of genus one and $O\in E$. (We generally denote the elliptic curve by $E$, the ...
2
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1answer
110 views

Cubic curve in projective space

Is it true that every cubic curve in $\mathbb{P}^3$, which is not contained in a plane, can be parametrized by polynomials? $\\\\\\\\$
3
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1answer
46 views

Normalisation of curve

I am probably having a lot of confusion with the terminologies in shafarevich. In page 131, Normal varieties, it states a corollary. An irreducible algebraic curve is birational to a nonsingular ...
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2answers
89 views

Reid's UAG problem 4.7: isomorphism of affine line with a curve

Let $C:$ $(Y^2=X^3+X^2)\subset \mathbb{A}^2$; the familiar parametrization $$ \varphi\colon \mathbb{A}^1 \to C,$$ given by $$ T \mapsto (T^2-1,T^3 -T)$$ is a polynomial map, but is not an ...
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74 views

Genus formula $\;\;\;\;$

Can someone give me a reference in which I can find the following result Let $C$ be a curve, then $$g(C)=\frac{(n-1)(n-2)}{2}-s.$$ where $g=$ genus of $C,$ $n=$ degree of curve, $s=$ number of ...
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139 views

Books on algebraic surfaces

I am interested in learning about algebraic surfaces (e.g. their classification in characteristic 0), and I was wondering whether any knowledgeable people would be so kind as to give their thoughts ...
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25 views

Rank of $K_X^\perp$ on a surface $X$

Let $X$ be obtained by blowing up $9$ points on $\mathbb{P}^2$. So rank of $\text{Pic} X = 10$. I want to know the rank of $K_X^\perp$ in $\text{Pic} X$. Also, if $E$ is a $-1$-class in $\text{Pic} ...
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98 views

singular locus and jacobian matrix

Let $R=k[x_1, \cdots ,x_r] / I$ be an affine ring over a perfect field $k$ and suppose that $I$ has pure codimension $c$. Suppose that $I= (f_1, \cdots , f_s)$. If $J$ is the ideal of $R$ generated by ...
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1answer
55 views

Cubic surfaces not containing lines.

I can't understand the following assertion, from Shafarevic,Basic Algebraic Geometry,Vol 1,pag 80. It is easy to construct a cubic surface $X\subset\mathbb{A}^3$ not containing lines. For example, if ...
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1answer
45 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
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1answer
44 views

Lines through a point

Let V(f) be a plane curve of degree d over an algebraically closed field having a point of multiplicity d. Prove that V(f) consists of d distinct lines. If V(f) is d distinct lines then $V(f)=V(f_1) ...
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86 views

an Example of Elliptic Curve over finite field has no CM

I have known this property (from Silverman's The Arithmetic of Elliptic Curves): Let $\operatorname{char}(K)=p>0,$ and let $E/K$ be an elliptic curve with $j(E)~ \overline{\in}~ \overline{ ...
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1answer
231 views

Is the cotangent sheaf quasi-coherent?

Allow me to reconstruct what is written here, in order for me to present the question. Let $(\mathscr{M},\mathscr{O}_{\mathscr{M}})$ be a smooth manifold, and let $\mathscr{M\times M}$ be the ...
2
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1answer
98 views

Rational points on symmetric products and effective zero cycles

Let $X$ be a variety defined over the finite field $\mathbb{F}_p$. The following is "well known". Let $a_n$ be the number of $\mathbb{F}_{p^{n}}$-rational points of $X$, and let $b_n$ be the number of ...
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1answer
53 views

Does $f_*f^*(L)\cong L$ hold for a birational morphism?

Suppose $X\to Y$ be a birational morphism of non-singular varieties, do we have $f_*f^*(L)\cong L$ for $L$ is a quasi-coherent sheaf on $Y$? Especially when $L$ is a line bundle?
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1answer
91 views

Very ampleness of pullbacks of twisting sheaves

Suppose we are given $X =\mathbb{P}^1_k\times_{k}\mathbb{P}^1_k$ with its canonical projections $\pi_i$ to $\mathbb{P}^1_k$. I'd like to show that ...
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1answer
90 views

When the ring of regular functions is a UFD?

Let $X$ be an irreducible affine variety over $\mathbb{k}$. There is the following theorem in algebraic geometry: the algebra $\mathbb{k}[X]$ of regular functions is a UFD if and only if each ...
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1answer
189 views

Hartshorne notation in Theorem 3.4

I have a nitpicky question about notation in Hartshorne's proof of his Theorem 3.4 in his AG book; in particular, it is in his proof of part (b). The point in contention is the line One checks ...
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55 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
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1answer
66 views

Difference in surfaces described by equivalent quadratic forms

It is fairly straightforward to prove that a quadratic form $Q(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}$ can equivalently be written $Q(\mathbf{x})=\mathbf{x}^{T}M\mathbf{x}$ for ...
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1answer
38 views

Graph of a morphism between two $K$-schemes: an open covering

Consider a separated morpshism $f:X\longrightarrow Y$ between two $K$-schemes ($K$ is a field). The graph of $f$ is the image of the morphism $(Id_X,f):X\longrightarrow X\times_{\operatorname {spec}K} ...
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17 views

Begin study of convex algebraic sets in complex projective space

Where should I begin the study of convexity of (semi-)algebraic sets? In other words, projective varieties defined by polynomials of complex variables. The long-term goal is to study optimization in ...
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2answers
110 views

How does $\text{Gal}(L/K)$ act on the automorphism group of an elliptic curve?

Let $L/K$ be a finite Galois extension of number fields; I'm interested mainly in the case $K = \Bbb{Q}$ and $L= \Bbb{Q}(\sqrt{d})$. Let $X$ be an elliptic curve over $K$ and $\text{Aut}(X_L)$ the ...
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38 views

$g^*f_*(F)\cong f'_*g'^*(F)?$

Suppose we have $g:S'\to S$, $f:X\to S$, $g':X'\to X$ and $f':X'\to S'$ forming a fiber product diagram. It is proved that when $f$ is affine, we have isomorphism $g^*f_*(F)\cong f'_*g'^*(F)$, where ...
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36 views

Calculating equations for image of variety under map

This is another qualifying exam practice problem that I'm having trouble with. I'm asked to compute polynomials which cut out the image of the variety $x^2+y^2-1=0$ in $\mathbb{C}^2$ under the map ...
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234 views

Definition of smooth (variety)

I don't understand the motivation for the definition of smoothness of a variety: A variety $V(f_1,...,f_m)$ in $n$-space is smooth iff $\mbox{rank}$ = $n-\mbox{dim} V$. Could you please give me ...
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65 views

Idea behind the concept of schemes

Having taken an introductory course on algebraic geometry (without introducing schemes), the notion of schemes seems to be quite unrelated to all we've done there. What are the most important reasons ...
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1answer
136 views

Closed points of a fibred product of k-schemes

This question comes from Shafarevich, Chapter V.4, Let $X$ and $Y$ be schemes over an algebraically closed field $k$. Show that the correspondence $ u \to (p_x(u),p_y(u)) $ establishes a 1-1 ...
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50 views

Parametric ideal decomposition

Let $x = \{x_{1},\dots, x_{n}\}$ be a set of variables and let $a = \{ a_{1}, \dots, a_{m}\}$ be a set of parameters. Let $\{f_{1}(a,x), \dots, f_{s}(a,x)\} \subset \mathbb{C}[a,x]$ be a set of ...
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61 views

-1-curves on a nonrational surface.

I would appreciate any guidance to help me prove the claim below. Let $X$ be a nonrational surface (say a blowup of points on a nonrational minimal model Y), the set of irreducible curves $C$ ...
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1answer
69 views

Reid, Undergraduate Algebraic Geometry, Problem 4.3, showing a polynomial map is an isomorphism

$\varphi_n \colon \mathbb{A}^1 \to \mathbb{A}^2$ is the polynomial map given by $X \mapsto (X^2,X^n)$; show that if $n$ is even, the image of $\varphi_n$ is isomorphic to $\mathbb{A}^1$, and ...
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1answer
139 views

A Question from Algebraic Geometry

For any two disjoint closed subsets $Y_1$ and $Y_2$ of $ \mathbb A ^n$ show that there exists $g \in\mathbb C [x_1, x_2, ..., x_n]$ such that $g(Y_1)=0$ and $g(Y_2)=1$.
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335 views

Exceptional Lines (Blowing Up)

Please refer to the image given below (Book: Singular Points of Plane Curves by C.T.C. Wall). Example 3.3.1: For the third blow up, how do we come to the conclusions that $E_{0}$ is $x_{3}=0$ and ...