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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75 views

Universal property of the Abel map

In the book Algebraic Geometry I edited by Safarevich, the following universal property of the Jacobian variety of an algebraic curve is given page 158 (with no more details): The Abel mapping $a: ...
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1answer
244 views

Regular functions on the punctured plane $\mathbb{A}^2\backslash\{(0,0)\}$

It is Exercise 4.3.2 in the book An Invitation to Algebraic Geometry. Show that the ring $\mathcal{O}_V(U)$ of regular functions on the punctured plane $U=\mathbb{A}^2\backslash\{(0,0)\}$ is the ...
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1answer
50 views

Why that a quasi-projective variety being isomorphic to a projective variety must itself be projective as well

On page 54 in the Book An Invitation to Algebraic Geometry, the author said that A quasi-projective variety in $\mathbb{P}^n$ is isomorphic to a Zariski-closed subset of some $\mathbb{P}^m$ (i.e. ...
5
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1answer
125 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
3
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2answers
64 views

Find side length of son-polygon.

Take a regular n sided polygon having side length 1, where n is odd. Draw all diagonals of this polygon. Around the center, you will find a smallest regular polygon similar to bigger one. Tell this ...
4
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1answer
98 views

Fermat Quartic Tiling

I have been reading about the Fermat quartic $Q \subset \mathbb{P}^{2}$, defined in homogeneous coordinates as $X^{4}+Y^{4}+Z^{4}=0$. This is the second most symmetric non-hyperelliptic surface of ...
3
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0answers
224 views

Frobenius on projective variety is not an isomorphism?

This is exercise 1.8 from Arithmetic of Elliptic Curves (Silverman). Part 3 confuses me because isn't $\phi$ the identity, thus an isomorphism? Let $\mathbb{F}_q$ be a finite field with $q$ ...
2
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3answers
114 views

Proving the area of an equilateral triangle

How do you prove that How do you prove that for any equilateral triangle with side length s, area is $\frac{s^2 √3}{4}$ ? I tried using an equilateral triangle in a square, but I keep coming up with a ...
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1answer
67 views

A question of algebraic geometry applied to field theory

I’ve come across this question in a coding theory course, and it has stumped me. Any hints and/or suggestions would be appreciated. Let $F$ be a field (for our purposes, assumed to be finite of ...
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0answers
55 views

Quadratic transformation with only ordinary singularities

I have to find the quadratic transformation of the following curve $\ C : y^2z^2 - x^3z - x^4 - y^4 = 0$ with only ordinary singularities. Any idea about how to solve this exercise?
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2answers
71 views

Topological space underlying this curve

I have to solve this exercise but I have really no clue even how to start with it: Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the ...
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1answer
52 views

Irreducible Components of $GL(n,K)$

View $M(n,K)$ as $n^2$-dimensional affine space, and $GL(n,K)$ as the principal open subset defined by the nonvanishing of the polynomial $det$. Then $GL(n,K)$ is an affine variety. What are its ...
4
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1answer
208 views

Question on Intersection Theory of Effective Divisors

I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...
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1answer
59 views

How to prove maxSpec $\mathbb{C}[x,y]/(x^2)$ is homeomorphic to $\mathbb{A}^1$ as topological spaces?

I am new to algebraic geometry, and am reading the book An Invitation to Algebraic Geometry. Exercise 2.6.4 in this book asks you to prove the topological space ...
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0answers
159 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
3
votes
1answer
80 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
3
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1answer
78 views

Can Bezout's theorem be generalized to non algebraically closed fields?

The Bezout's theorem says that the intersection of two curves in $\mathbb{P}^2_k$, (counting multiplicity, $k$ is algebraically closed) is equal to the product of their degrees. Can the theorem be ...
2
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1answer
56 views

Bounding the number of nodes of a trigonometric polynomial using Bézout's inequality

The zero set of a trigonometric polynomial $P$ in 3 variables $x,y,z$ is a two-dimensional manifold which we view as being inside the torus $\mathbb T^3$. For example, here is the zero set of ...
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1answer
61 views

why complitcated construction of PD-defferential operator in Berthelot and Ogus's book

In the book " Notes on Crystalline Cohomology" by P. Berthelot and A. Ogus, they introduced the cencept of PD-defferential operators in a complicate way, i.e. using dividied power hull. However if I ...
3
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1answer
297 views

A doubt in the proof of Prop. 1.10 of Hartshorne's Algebraic Geometry

I have a doubt in the proof of Proposition 1.10 of Hartshorne's book Algebraic Geometry, which states that if $Y$ is a quasi-affine variety, then its dimension is the dimension of its closure. In ...
1
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1answer
94 views

Geometric meaning of being integrally closed in some overring

The geometric counterpart of integrally closed rings (in their fraction fields) are normal varieties, as described in this MathOverflow post. Is their a similar notion in algebraic geometry for being ...
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0answers
91 views

Hironaka 1964 theorem in the context of S. Watanabe 2009 book

I am trying to read the following book of S. Watanabe: "Algebraic Geometry and Statistical Learning Theory". More particularly, I am currently interested in chapter 2 and Hironaka (1964) theorem on ...
3
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1answer
126 views

An example of ample sheaf with no global section

In viewing the tags about ample bundle with no global sections I found an example below: If $C$ is a curve of genus $2$, and $p,q,r$ are general points on $C$, then the bundle ...
1
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1answer
114 views

Pushforward of pullback of an etale sheaf

I hope this question is not too elementary, but I'm a bit lost : Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the ...
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0answers
67 views

Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
2
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1answer
91 views

An easy proof that $\mathrm{SL}(n,F)$ is irreducible in the Zariski topology

Let $F$ be an infinite field (that is not necessarily algebraically closed) and consider the algebraic variety $\mathrm{SL}(n,F)=\mathcal{V}(\det-1)$ of $F^{n^2}$, where $$\mathcal{V}(S)=\{\alpha\in ...
3
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0answers
59 views

Inequality involving multiplicities of points introduced via Quadratic Transformations of a Plane Curve

I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF). The question is: Let $F=F_1, ...
4
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1answer
140 views

meromorphic functions on proper varieties are rational

Suppose $X$ is a proper variety over $\mathbb{C}$, is every meromorphic function rational? In the case of projective variety, can this be derived from Chow lemma? How does the GAGA principal ...
1
vote
1answer
49 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
2
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1answer
140 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
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2answers
106 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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2answers
146 views

Looking for an introductory Algebraic Geometry book

I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the ...
3
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0answers
149 views

Computing the genus in positive characteristic

That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to ...
1
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1answer
54 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
3
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0answers
116 views

Zariski closures exercise.

Compute the Zariski clousures $\overline{S} \subset \mathbb{A}^2(\mathbb{Q})$ of the following subsets: (a) $S=\{(n^2,n^3):n \in \mathbb{N}\}\subset \mathbb{A}^2(\mathbb{Q})$; (b) $S=\{(x,y): ...
4
votes
1answer
77 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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0answers
75 views

Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
0
votes
1answer
57 views

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular
2
votes
1answer
161 views

how to show that $V( Y-X^2 )$ is irreducible?

show that $V( Y-X^2 )$ is irreducible. $Y-X^2$ is an irreducible polynomial ($Y-X^2$ cann't be factored into more irreducible components). Can we conclude that $V(Y-X^2)$ is irreducible??
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0answers
77 views

Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)$ into irreducible components

Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)\subset A^2(C)$ into irreducible components. I tried like this: $Y^4-X^2=(Y^2-X)(Y^2+X)$ and $Y^4-X^2Y^2+XY^2-X^3=(Y+X)(Y-X)(Y^2+X)$. What should I do from ...
0
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1answer
128 views

An affine plane curve and intersection

Suppose $C$ is an affine plane curve and $L$ is a line in $A^2(k)$, $L \not\subset C$. Suppose $C=V(F), F \in K[X,Y]$ a polynomial of degree $n$. Show that $ L \cap C$ is a finite set of no more than ...
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1answer
60 views

$f_*(O_X)=O_Y$ and connectedness of fibers

Suppose $X\to Y $ is a morphism , under what conditions we have direct image sheaf $f_*(O_X)=O_Y$? For example, suppose $\tilde{S}\to S$ is a blow up, do we have $f_*(O_{\tilde{S}})=O_S$? ...
2
votes
2answers
189 views

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big?

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ? Is there a formula relating the dimension of the Zariski tangent space and the order of ...
2
votes
1answer
109 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
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1answer
183 views

The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
1
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1answer
92 views

pullback of differential form by constant morphism

Let $G/S$ be a group scheme (i'm fine if you want to assume everything is affine). Let $$g : S \to G \in G(S)$$ be an $S$ point of $G$. We can define a morphism $$\phi_g : G \to G$$ defined at the ...
4
votes
2answers
73 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
4
votes
1answer
148 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ are considered the same. Is it true? Why? I'm a beginner so please answer in detail.
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0answers
39 views

Multiplicity of an affine curve at a point same as that of its projectivization

Consider the projective curve $C=V(P)$ in $\mathbb{P}^2$ where $P(x_0,x_1,x_2)$ is an homogeneous polynomial of degree $d$. At a point $[a,b,1]$, the multiplicity of $C$ is ...
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0answers
45 views

Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $ defined as follows: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ This map defines a Variety ($V$) of dimension $2$ in ...