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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Hartshorne Prop I.4.3 Proof

$\textit{The proposition}$: On any variety $Y$, there is a base for the topology consisting of open affine sets. $\textit{The proof}$: Assume $Y$ is quasi-affine in $\mathbb{A}^n$ and let $Z=\...
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1answer
41 views

Steps inbetween? Weil's zeta function

Why is it that, that is "just" what the Zeta function is? What happened in between? I messed around with it for roughly an hour and couldn't get it to come out right. The second photo is just for ...
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1answer
55 views

Why is the zero locus of a polynomial ideal a variety?

I'm sorry if this is a stupid question, but some books define a variety of an ideal as it's zero locus, what means that it's irreducible. I'm just new to algebra and wasn't able to prove it... Can ...
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1answer
46 views

Simple exact sequences of vector bundles

I've come across some simple exact sequences of vector bundles that make manifest some basic confusions I have. These questions may be quite intertwined, in ways that my limited understanding obscures,...
1
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1answer
33 views

Hartshorne: Definition of $K^*$ where $K$ a function field of scheme.

Let $X$ be a noetherian integral separated scheme which is regular of codimension one. Let $K$ be the function field of $X$. Now let $f \in K^*$, (I am interpreting $K^*$ to be the set of field ...
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1answer
74 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
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1answer
30 views

The affine coordinate ring of twisted cubic curve, $Y$ is $A(Y)=k[x,y,z]/(z-x^3, y-x^2)$?

I am working on the following problem: Let $Y \subset A^3$ be the set $Y={(t,t^2,t^3)|t\in k}$ ($A^3$ is the affine 3-space over $k$ an algebraically closed field.) Show that $Y$ is an affine variety ...
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1k views

Zariski Topology question

Could you please give a hint how to show that the zariski topology on $\mathbb{A}^2$ is not the product topology on $\mathbb{A}^1\times\mathbb{A}^1$
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1answer
827 views

Tensor product of reduced $k$-algebras must be reduced?

Let $A$, $B$ be two reduced $k$-algebras. Then if an element of the form $$\sum a_{i}\otimes b_{j}$$ is nilpotent, we can compose it with any $k$-homomorphism $f$ from $A$ to $k$ to get a homomorphism ...
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1answer
36 views

Positivity of a quartic form

If I have a quartic form that I can write as $$P(x,y)=(x^2/2,y^2/2,xy)M(x^2/2,y^2/2,xy)$$ where $M$ a a $n \times n$ symmetric matrix, what is the simplest way to derive whether the form is positive ...
3
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1answer
61 views

Counting the Number of Points in an Algebraic Variety

How can we count the number of points in $$S = \{(x,y) \in \mathbb{Z_m}^2: x^2+ky^2 = c\}$$ where $k,c$ are some positive integers?
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1answer
98 views

What is $f_!$ in the context of commutative rings?

Given a morphism of schemes $f:X \to Y$ there is a functor $f_!:Sh(X) \to Sh(Y)$ where $$ f_!\mathcal{F}(U) = \{ s \in \mathcal{F}(f^{-1}(U)) : f:\text{supp}(s) \to U \text{ is proper} \} $$ How do I ...
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0answers
50 views

Cohomology Group basis

I'm reading a text on Complex Torus and Abelian Variety and at a time is written as follows: The cohomology group $H^{1}(T,\mathcal O_{T})$ has a basis $w_{j}=d\overline{z}_{j}, j=1,2,...,g,$ as ...
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2answers
66 views

elementary algebraic geometry

Let $f$ and $g$ are positive-degree polynomials in $k[X,Y]$, $k$ is an infinite field, and $f,g$ have no common-factor. I want to prove: Intersection of two curves $f=0$ and $g=0$ is finitely-set. ...
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0answers
39 views

Dimension estimate for fibre products.

Let $X$ and $Y$ be smooth projective varieties, $f:X\rightarrow Y$ a birational morphism. Is there some way to compute the dimension of $X\times_Y X$ in Terms of the dimensions of $X$ and $Y$?
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35 views

Is there a name for an affine $\Bbbk$-scheme whose $\Bbbk$-algebra is finite dimensional as a vector space?

Is there a name for an affine $\Bbbk$-scheme whose $\Bbbk$-algebra is finite dimensional as a vector space? From what I understand 'finite type' is reserved for finite generation as algebras, not only ...
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0answers
35 views

Problem of Apollonius with 3 circles of equal radius

I want to find the circle which exclusively touches 3 other circles. This is essentially the classic problem of Apollonius. I use the following equation to find the center of that circle and its ...
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2answers
33 views

Difference between Ordering and Order?

I am confused by the two terms order and ordering. I am learning on Ideals, Varieties and Algorithms by Cox et all. The context is monomial orderings and Gröbner basis on polynomial rings. How are ...
2
votes
1answer
66 views

Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD

I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as a ...
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0answers
56 views

Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
3
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0answers
48 views

Linear separability / Number of positive solutions of a random linear system

This one is on linear separability of cyclic patterns. The shorter geometric version: Take a ring of length $p$ of randomly assigned mean-free binary values $x_i = \pm 1$, $i = 1 \cdots p$. ...
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0answers
154 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
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0answers
97 views

Projection of a curve in $\mathbb{P}^3$

I've been reading the proof of Theorem IV.3.10 in Hartshorne (p. 313 - 314), which states the following: Given a curve $X \subset \mathbb{P}^3$, there is a point $O \notin X$ such that the ...
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1answer
71 views

An interesting open subset of a scheme X.

Let $(X,O_X)$ scheme and $f\in{O_X(X)}$ then $X_f$ is an open subset of $X$ where $X_f=\left\{{x\in{X}| f_x\in{u(O_{X,x})}}\right\}$ and $ u(O_{X,x})$is the set of invertible elements. proof: Let ...
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0answers
62 views

What kind of morphisms should I expect to be proper?

I'm trying to learn more about proper pushforward, but I'm stuck at coming up with interesting examples of proper morphisms of schemes. The only examples I can think of are inclusions of projective ...
3
votes
1answer
59 views

Is the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + t^2)$ irreducible over $k(t)$?

Let $k$ be an algebraically closed field of characteristic 2 and let $k(t)$ be rational function field of one variable. Consider the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + ...
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votes
1answer
13 views

If $B$ is a commutative domain, $Aut(B)$ acts on $Der(B)$ by conjugation

I'm reading Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenberg, and I don't understand what's meant on the line $Aut(B)$ acts on $Der(B)$ by conjugation: $\alpha \cdot D = \alpha ...
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1answer
38 views

Ampleness and Extensions of Line-Bundles

I believe I have a proof that any vector bundle $V$ of rank $n$ on a projective variety $X$ has a filtration by line bundles (that is, there is a filtration $V = L_n \subset ... \subset L_0$, where $...
2
votes
1answer
820 views

The cone over a projective variety

I'm trying to prove that $I(C(Y))=I(Y)$, where $C(Y)=\pi^{-1}(Y)\cup \{(0,\ldots,0)\}$ the cone over $Y$ and $\pi:\mathbb A^{n+1}-\{0,\ldots,0\}\to \mathbb P^n$ the projection which sends the point ...
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0answers
19 views

Max Noether's Fundamental Theorem proof in Fulton's book

In the end of the proof we get that $H=A'F+B'G$ and $A'=\sum A'_i$, $B'=\sum B'_i $ while $A'_i$ and $B'_i$ are forms of degree $i$. I don't understand how then he makes the conclusion that $H=A'_sF+B'...
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3answers
118 views

Why is $\mathbb{C}^2\setminus\{(0,0)\}$ not a basic open set?

Consider the affine variety $\mathbb{C}^2$ equipped with Zariski topology. By the question above, I mean why $X:=\mathbb{C}^2\setminus\{(0,0)\}$ cannot be written as $$ X:=U_f:= \{(x,y)\in \mathbb{C}...
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1answer
26 views

Kahler differential calculation

Take the conic $X: z^2-xy=0$ in $\mathbb P^2$. On the patch with $z=1$ and coordinates $x, y$, $X$ is cut out by $1-xy=0$. On the patch with $y=1$ and coordinates $u, v$, $X$ is cut out by $v^2-u=0$. ...
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0answers
34 views

branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
5
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2answers
119 views

What exactly is $\mathbb{P}_\mathbb{Z}^n$?

So, I have the following definition of $\mathbb{P}_A^n$ for an arbitrary (commutative) ring $A$, from Hartshorne: Set $S=A[x_0,\ldots,x_n]$, so that $S=\bigoplus_{d\geq 0}S_d$ as a graded ring, $S_+=\...
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1answer
23 views

Tracing the sides of an equilateral triangle

Is there any way I can get the points in 2D plane on the sides of an equilateral triangle for certain infinite animation sequence? For example in case of tracing the circumference of the circle, I ...
2
votes
0answers
48 views

When is a finite $R$-algebra isomorphic to $R$?

Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there ...
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1answer
42 views

A cartesian diagram?

let $k$ be a field, and $X$ and $Y$ varieties over $k$. Let $L$ be an extension of $k$, and $X_L=X\times_k L$. Is the diagram $$\require{AMScd} \begin{CD} X_L\times Y_L\times X_L @>>> X\...
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1answer
71 views

Loci of intersection points of two curves

There are two continuous, negatively-sloped curves,A and B. They intersect at least once ,say at $(x,y)$. If I introduce a third curve C, whose X axis intercept has a higher magnitude than that of B, ...
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votes
1answer
52 views

Quick question: Extension of vector bundles on a compact Riemann surface

Given the following short exact sequence of holomorphic vector bundles on a compact Riemann surface: $0\rightarrow M\rightarrow E \rightarrow N\rightarrow 0$ Fix a hermitian metric on $E$ and $n=...
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0answers
45 views

Making sense out of $F$-structures and the notion of $F$-variety

For almost two years I have been trying to make sense out of several claims about varieties over nonalgebraically closed fields made in the first chapter of the textbook Linear Algebraic Groups by T.A....
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1answer
55 views

About the definition of polynomials on vector spaces?

In the book Linear Systems Theory and Introductory Algebraic Geometry (R. Hermann) the author defines A polinomial on V (a $\mathbb K$-vector space) is an element of the smallest ...
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0answers
27 views

A question of terminology regarding exceptional curve or is it divisor.

So I kept on reading the book by Griffiths and Harris called Principles of Algebraic Geometry and I've seen a definition of exceptional divisor of the first kind. On page 487: A smooth rational ...
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0answers
29 views

must this extension of a DVR be unramified?

Let $A$ be a normal domain, and $P$ a height 1 prime, then $A_P$ is a DVR. Let $K$ be the fraction field of $A_P$, and let $L$ be a finite Galois extension of $K$ of degree $e$, let $B$ be the ...
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1answer
33 views

Multiplicity of Cartier divisor on locally noetherian scheme is only non-zero at generic point

I'm following chapter 7 in Qing Liu's book 'Algebraic Geometry and Arithmetic Curves' about 'Divisors and applications to curves'. My question concerns Definition 1.27: Let $A$ be a Noetherian ...
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0answers
56 views

Showing $\exp:\mathscr{O}_X\to\mathscr{O}_X^*$ is an epimorphism of sheaves

$\newcommand{\O}{\mathscr{O}}$Let $X=\Bbb{C}$. Define $\O_X$ to be the sheaf of holomorphic functions, and $\O_X^*$ to be the sheaf of invertible (i.e. nonvanishing) holomorphic functions, the latter ...
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1answer
70 views

What is the fundamental group of a modular curve $\mathcal{H}/\Gamma$?

Let $\Gamma$ be a finite index subgroup of $PSL_2(\mathbb{Z})$. What is the fundamental group of $\mathcal{H}/\Gamma$? By the Kurosh Subgroup theorem, $$\Gamma \cong F_n * C_2^{*r} * C_3^{*s}$$ ie, $...
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votes
1answer
54 views

Why does this homological lemma hold?

Let $A$ be a noetherian ring; let $C^{\boldsymbol\cdot}$ be a bounded above complex of flat $A$-modules in positive degrees, let $L^{\boldsymbol\cdot}$ be a bounded above complex of free $A$-modules ...
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1answer
37 views

Are local rings of non-singular curves noetherian integral domains?

Let $P$ be a point on a nonsingular curve $Y$, then the local ring $\mathcal{O}_P$ is a regular local ring of dimension one. Hartshorne gives the following theorem: Let $A$ be a noetherian ...
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votes
2answers
133 views

Why is $\mathrm{Spec}(\mathbb{Z})$ a terminal object in the category of affine schemes?

I've seen this claim repeated in many places (always without source or proof), that $\mathrm{Spec}(\mathbb{Z})$ is a terminal object – however, the most I've been able to prove myself is that for any ...
0
votes
1answer
29 views

Quotient field of ring of regular functions at some point in affine variety is the field of rational functions on the variety

I am reading Hartshorne's book of Algebraic Geometry. I am stuck in understanding why quotient field of the local domain O_p (where O_p denotes the ring of regular functions at a point p in affine ...