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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2
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22 views

How many points at infinity in Artin-Schreier type curve

Let $Y$ be an affine curve over a perfect (yet not necessarily algebraically closed) field $k$ given by $$y^p+a(x)y=b(x)$$ (abs. irreducible) with $p$ a prime number. Now one can normalize $k[1/x]$ in ...
0
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0answers
37 views

Extensions $\operatorname{Ext}^1(F, \omega)$ via Serre duality

Let $C$ be a smooth curve over a field $k$, and let $\omega$ be its dualizing sheaf. Serre duality tell us that for coherent sheaf $F$ the natural pairing $$ \operatorname{Ext}^1(F, \omega) \otimes ...
1
vote
1answer
17 views

Regular functions extension to normal points of varieties

I am doing the exercise 3.20 in Robin Hartshorne's Algebraic Geometry, Chapter 1. Let $Y$ be a variety of dimension $\geq2$, and let $P\in Y$ be a normal point. Let $f$ be a regular function on ...
1
vote
1answer
35 views

Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
1
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1answer
22 views

Finding algebraic curve satisfying given parameterization

Is there an easy way to find an algebraic curve that satisfies a given parameterization? Specifically, I am talking about the following parameterization: $$ x=z(1-z),\hspace{10pt} y=\sum_{n=1}^r ...
2
votes
1answer
28 views

$Z(y^2-x^3) \subset \mathbb{A}_{\mathbb{R}}^2$ is not isomorphic to $\mathbb{A}_{\mathbb{R}}^1$

Prove that the algebraic variety $Z(y^2-x^3) = \{(x, y)\in\mathbb{A}_{\mathbb{R}}^2\,\,|\,\,y^2-x^3=0\}$ is not isomorphic to the affine space $\mathbb{A}_{\mathbb{R}}^1$. [i.e., there are no ...
0
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1answer
62 views

About alternative ways of computing $H^1(X,\mathcal{O}_{\mathbb{P}^n}(m))$.

This is a follow up to my question : Applications of $Ext^n$ in algebraic geometry In the case of $\mathcal{O}_X$-Modules it is clear that $Ext^i(\mathcal{O}_X, \mathcal{F}) \cong H^i(X,\mathcal{F})$ ...
1
vote
1answer
31 views

Affine Zariski topology is normal

Let $C,D$ be two disjoint Zariski-closed subsets of $\mathbb{C}^n$, and let $f,g$ be polynomial functions on $C,D$ correspondingly. Then there is a polynomial function $h$ on $\mathbb{C}^n$ that ...
1
vote
1answer
36 views

Foliation dense if $G = \textbf{R}$, where $G$ is a subgroup of a Lie group $G'$.

I have the following statement: Let $G$ be a subgroup of a lie group $G'$, and the action is left multiplication. The leaves are then the left cosets of $G$ in $G'$. If for example, we let $G = ...
1
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0answers
32 views

Is this criterion for projective normality missing a hypothesis?

I proved the following: Let $S=S(X)$ be the homogeneous coordinate ring of a connected, normal closed subscheme of $\mathbf P^r_A$, where $A$ is a ring. Then $S$ is a domain, and ...
0
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0answers
18 views

Moving from polynomials into radical functions: does algebraic geometry still work?

Say I have a polynomial: $x + y = 0$, but subject to the constraints $x = \sqrt{1-u^2}$, $y = \sqrt{1-w^2}$, $u\in[-1,1],w\in[-1,1]$. I can re-write my original polynomial as an inequality: ...
1
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1answer
21 views

Dimension of cone of projective variety

Let $X \subset \mathbb{P}^n$ be a nonempty projective variety. Show that the dimension of the cone $C(X):=\{0\} \cup \{(x_0,...,x_n)\in \mathbb{A}^{n+1}:(x_0:...:x_n)\in X\}$ is dim$X+1$. I know how ...
0
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1answer
18 views

Finding solution of a system of polynomials through intersecting sub solutions?

I have a conjecture but I'm not sure if it's true. Intuitively, it seems correct, but... here it is: Conjecture: Let $S$ be a set of polynomials in $n$ variables over $\mathbb{C}^n$ (*see below). ...
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0answers
21 views

Deformation of a point on a Quot scheme

Let $\mathcal{H}$ be a coherent sheaf on a projective variety $X$. We say that a sheaf $\mathcal{E}$ is of $\textit{pure dimension}$ if for all non-trivial coherent subsheaves $\mathcal{E'} \subset ...
0
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0answers
18 views

Singular affine monomial Curve.

To me, an affine monomial curve $C$ is parametrized as $C:(t^{m_1},\cdots,t^{m_r})$ with $\textrm{G.C.D.}\{m_1,\cdots,m_r\}=1$ and $m_1<\cdots<m_r$. How show that $C$ is singular in ...
1
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1answer
36 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on ...
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2answers
86 views

Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
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0answers
23 views

Computing Intersection Product in a Cell Complex

Here's the information I have: I have an abstract cell complex that represents some space I am studying, and it is known that the space is orientable. I can identify the sub complexes which ...
2
votes
1answer
31 views

Correspondence between prime ideals and irreducible algebraic sets

Let $k$ be an algebraic closed field. The Nullstellensatz theorem prove that $$I(V(J))=\sqrt{J}$$ and we have $$V(J)\text{ irreducible }\iff I(V(J)) \text{ prime }$$ So if $J$ is prime, $I(V(J))=J$ is ...
1
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0answers
13 views

The Variety X(K) as a subset of the analytification of X

I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties. The article: ...
4
votes
1answer
54 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
2
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2answers
35 views

Dimension of irreducible affine variety is same as any open subset

Let $X$ be an irreducible affine variety. Let $U \subset X$ be a nonempty open subset. Show that dim $U=$ dim $X$. Since $U \subset X$, dim $U \leq$ dim $X$ is immediate. I also know that the result ...
1
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1answer
42 views

What is the ramification locus of $Spec\mathbb{Z}[x]\rightarrow Spec\mathbb{Z}[x]$ given by $x\mapsto x^2$?

$\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\Spec}{\text{Spec }}$ Let $A = B = \ZZ[x]$, and consider the map $B\rightarrow A$ given by $x\mapsto x^2$. Intuitively, $f : \Spec A\rightarrow\Spec B$ ...
0
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0answers
89 views

Proper morphism between non isomorphic objects with $f_*\mathcal{O}_X=\mathcal{O}_Y$ [closed]

I need an example of a proper morphism $f \colon X \to Y$ between nonisomorphic objects $X$ and $Y$ such that $f_*\mathcal{O}_X=\mathcal{O}_Y$.
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0answers
30 views

Non-proper intersections and singular points

Let $X$ be a smooth variety in $\mathbb{P^N}$. Consider an intersection $Z=X\cap\mathbb{P}^n\subset\mathbb{P}^N$ with some subspace $\mathbb{P}^n$ and assume that $\dim Z>\dim X+n-N$. Is it true ...
6
votes
1answer
83 views

How to tell if a system of polynomial equations has no real solutions

I have a system of $3n + 3$ polynomial equations in $6n$ variables, where $n$ is probably going to be less than about $5$. I can compute its Groebner basis and I see that it does not contain $\{1\}$, ...
0
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0answers
38 views

Subspace of topological space has lower dimension

The dimension of a topological space $X$ is defined to be the length of the maximal chain of closed irreducible subsets $\varnothing \neq X_0 \subsetneq X_1 \subsetneq ... \subsetneq X_n \subset X$. ...
0
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1answer
14 views

Space admitting an irreducible connected open covering is irreducible

Let $\{U_i:i \in I\}$ be an open covering of topological space $X$, where $U_i \cap U_j \neq \varnothing$ for every $i,j$. If $U_i$ is irreducible for all $i \in I$ then $X$ is irreducible. I am ...
1
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0answers
23 views

Constructing an algebraic surface with singularities on the unit circle.

I am currently doing a project on algebraic surfaces, and I want to construct an algebraic surface $\mathbb{V}(f(x, y, z))$ that exhibit its singularities on the unit circle $x^2 + y^2 - 1 = 0$. My ...
0
votes
0answers
17 views

Is the completion of Puiseux series spherical complete?

Let $S$ be a completion of the field of formal Puiseux series over $\mathbb C$. Is $S$ spherical complete? It should be not spherical complete. I tried some examples, but failed to find an right ...
1
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1answer
29 views

Algebraic function fields

I am trying to understand what an algebraic function field is, so i was looking for some examples. The example on Wiki says: Given a polynomial ring $k[X,Y]$. Consider the ideal generated by the ...
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votes
0answers
20 views

Maximal ideal of subalgebra over a field [duplicate]

Let $A$ a finite $k$-algebra (with $k$ a field) and $B$ a subalgebra of $A$. Prove that if $\mathfrak{m}$ is a maximal ideal of $A$ then $\mathfrak{m}\cap B$ is a maximal ideal of $B$. It is easy ...
1
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1answer
69 views

Algebraic geometry reference for people with limited background [closed]

My background in algebra is groups and rings from Herstein's Topics in Algebra and field theory from Gallian's Contemporary Abstract Algebra. With this background, can I read any algebraic geometry? ...
2
votes
3answers
85 views

4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
0
votes
0answers
33 views

Are complex subvarieties cycles in the sense of singular homology?

Given a $p$-codimensional complex subvariety $Z\subset M$ of a non singular complex projective variety $M$ of dimension $n$ we can define an element $$\int_\hat{Z}i^*\in ...
0
votes
0answers
10 views

Restriction of an isogeny is still an isogeny?

Given that $E \times F \twoheadrightarrow G$ is an isogeny where $E,F$ are both subgroups of $G$, is its restriction to a subgroup $H$ of $G$, $E \times (F \cap H) \twoheadrightarrow (G \cap H) = H$, ...
0
votes
1answer
43 views

Commutative Algebra “mess”: recover the complete local rings of the normalization

Premise and main idea: I'm not an expert in the field of commutative algebra and when I encounter problems regarding local rings I try to solve them by following a sort of geometric intuition. It was ...
1
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1answer
42 views

Homology of $Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$

I want to compute the homology of $M=Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$. I think I have the answer, but I'm not sure how to make it precise. My approach is to consider the affine cover ...
0
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0answers
22 views

Find a linear sequence of the common terms between 2 other sequences?

Given 2 linear sequences $an_1+b$ and $cn_2+d$ generate a sequence of whole numbers that can expressed as $an_1+b$ and $cn_2 + d$. An example to illustrate this: Given $2n+3$ and $3n+6$ the sequence ...
2
votes
1answer
74 views

If two projective lines both intersect four given projective lines, must the two lines be parallel on an affine open?

L.S., I am trying to solve an exercise of my algebraic geometry course, which is as follows. Given four projective lines in $\mathbb{P}^3$, show that the number of lines intersecting them all is ...
2
votes
1answer
32 views

$A$ abelian variety is the multiplication by n $n_A$ surjective?

According with Munford the answer is yes, but there are some obscure points in the proof. We know that there exists a very ample line bundle on $A$ since every abelian variety is projective. Hence, ...
0
votes
1answer
25 views

Dimension of product of affine varieties

Let $X\subset \mathbb{A}^n_K$ and $Y\subset \mathbb{A}^m_K$ be affine varieties. How can I prove that dimension of the product variety $X\times Y \subset \mathbb{A}^{m+n}_K$ is dim$X$+dim$Y$? Here I ...
1
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0answers
41 views

Generalized Jacobian Conjecture

Is there any known generalization of Jacobian conjecture which gives condition for $k[f_1, \ldots, f_m] = k[g_1, \ldots, g_m]$ where all $f_i$ and $g_i$ are functions over $x_1, \ldots, x_n$. Note ...
2
votes
0answers
32 views

Blowing up a model for a plane curve

Let $R = \mathbb{C}[[t]]$ and let $\mathcal{X} \hookrightarrow \mathbb{P}_R^2$ be the arithmetic surface defined by the equation $$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$ The generic fiber ...
9
votes
0answers
98 views

Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
2
votes
0answers
34 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} ...
3
votes
0answers
38 views

Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
4
votes
1answer
63 views

What would an infinite dimensional projective space look like as a scheme?

In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: ...
2
votes
1answer
60 views

Classical results of Algebraic Geometry using cohomology.

I am looking for classical results of Algebraic Geometry that can be proved using cohomology. For example, Riemann-Roch Theorem and Bezout Theorem admits short proofs (providing that you know enough ...
2
votes
1answer
21 views

real affine varieties are hypersurfaces

In $\mathbb{R}^n$, let X be a Zariski-closed set. then $X=\mathbb{V}(f)$ for some polynomial $f$. Elementary formulation: let $X \subset \mathbb{R}^n$ be the set of common zeroes of some ...