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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3
votes
1answer
292 views

There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.

I have been working on the following problem from Ulrich-Görtz today and I can't seem to find a nice solution. Let $f:X \rightarrow Y$ be a morphism of schemes and let Y be irreducible and $\eta$ ...
2
votes
1answer
96 views

A chain ring with Krull dimension greater than one

Recall that a commutative ring $R$ with identity is a chain ring if the set of ideals of $R$ is linearly ordered under inclusion. I want to know if there a chain ring with Krull dimension greater ...
2
votes
0answers
110 views

First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.

Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a ...
2
votes
1answer
131 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
2
votes
1answer
121 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
2
votes
1answer
59 views

Vector bundle on a quadric $Q$

The problem: Consider the smooth quadric $Q=V(X_{0}X_{1}+X_{2}X_{3}+X_{4}^{2})\subset\mathbb{P}^{4}$ and the line $L=V(X_{0},X_{2},X_{4})$ contained in it. Prove that there exists a vector bundle $F$ ...
2
votes
1answer
226 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...
2
votes
1answer
186 views

Blowing up at a subvariety

Let $Y\subseteq\mathbb{A}^n$ be an affine variety with $\mathbb{I}(Y)=(f_{1},\ldots,f_{s}) \subseteq k[x_{1},\ldots,x_{n}]$. Define $\psi:\mathbb{A}^n \to \mathbb{P}^{s-1}$ by ...
2
votes
1answer
307 views

Separated scheme

How to show, that the affine line with a split point is not a separated scheme? Hartshorne writes something about this point in product, but it is not product in topological spaces category! Give the ...
1
vote
1answer
57 views

Are the $C$-points of a simply connected algbraic group simply connected?

Let $G$ be a simply connected algebraic group defined over $C$. Note : I see the definition of simply connected as "Every isogeny to $G$ is an isomorphism" as given in Hochschild's "Basic Theory of ...
1
vote
2answers
73 views

A non-trivial closed polynomial function must not be surjective?

Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is a nonconstant polynomial function. If $f$ is a closed mapping, then $f$ must not be a surjection? Thanks in advance.
1
vote
4answers
205 views

Proving algebraic sets

i) Let $Z$ be an algebraic set in $\mathbb{A}^n$. Fix $c\in \mathbb{C}$. Show that $$Y=\{b=(b_1,\dots,b_{n-1})\in \mathbb{A}^{n-1}|(b_1,\dots,b_{n-1},c)\in Z\}$$ is an algebraic set in ...
1
vote
1answer
382 views

The image of a proper scheme is closed

Let $X$ be a scheme, $Y$ a proper $X$-scheme and $Z$ a separated $X$-scheme. Let $f : Y \to Z$ be a morphism of $X$-schemes. I would like to prove that the image of $f$ is closed in $Z$. Here is what ...
0
votes
1answer
78 views

Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
-2
votes
2answers
437 views

Hartshorne Exercise II. 3.19 (b)

How do we prove the following exercise of Hartshorne? Let $A$ be a subring of an integral domain $B$. Suppose $B$ is a finitely generated $A$-algebra. Let $b$ be a non-zero element of $B$. Then there ...
-3
votes
1answer
236 views

Classic Circle and Adjacent Arrangement Problem

Given: A circle with nine distinct positive integers. n is a fixed positive integer. So these positive integers are arranged in the circle in such a way that the product of any selected two ...
-3
votes
2answers
913 views

Hartshorne Exercise II. 3.19 (c)

Hartshorne Exercise II. 3.19 (c) is as follows. Prove the following theorem of Chevalley by using Exercise II. 3.19 (a) and (b) and noetherian induction on $Y$. How do we prove this? Theorem of ...
9
votes
1answer
673 views

Determine if a conic is degenerate with the determinant.

There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices: $$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc} a&b&d\\ ...
7
votes
1answer
600 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
6
votes
1answer
184 views

Is the global section ring of a Noetherian Scheme Noetherian as well?

As the title suggests, I am asked to prove that, given a Noetherian scheme $(X,\ \mathcal{O}_{X})$ and any open subset $U\subseteq X$, $\Gamma(U,\ \mathcal{O}_{X}):=\mathcal{O}_{X}(U)$ is a Noetherian ...
6
votes
1answer
663 views

Set that is not algebraic

I'd like some hints for the problem: Show that the following set is not algebraic: $$ \{ (\cos(t),\sin(t),t) \in \mathbb{A}^3 : t \in \mathbb{R} \} $$ Thanks.
5
votes
1answer
438 views

Why degree of a reducible projective variety is the sum of the degree of its irreducible components

Could anyone show me how to prove that The degree of a reducible projective variety is the sum of the degree of its irreducible components? The definition of the degree I know is quite vague, ...
5
votes
1answer
139 views

Proving that 4 specified sets are not algebraic

I have four sets that I've come up with, which I think fail to be algebraic. However, I don't know how to prove this. They are: The graph of $\mathbb{C} \rightarrow \mathbb{C} : z \mapsto e^z$ {$(z, ...
4
votes
1answer
184 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...
4
votes
2answers
184 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
4
votes
1answer
165 views

For any ideal $ \mathfrak{a}\subseteq A$, $I(Z(\mathfrak{a})) = \sqrt{\mathfrak{a}}$, the radical of $\mathfrak{a}$

I'm trying to prove that $I(Z(\mathfrak{a})) = \sqrt{\mathfrak{a}}$, where $\mathfrak{a}$ is an ideal of $A = K[x_1, ... , x_n]$ and $K$ is an algebraically closed field. In case this notation is ...
4
votes
1answer
754 views

Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root

I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my ...
3
votes
1answer
146 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
3
votes
1answer
132 views

Closed morphism between schemes of finite type over a field induces a closed map between varieties?

Let $X$(resp. $Y$) be a scheme of finite type over a field $k$. Let $f\colon X \rightarrow Y$ be a closed morphism. Let $X_0$(resp. $Y_0$) be the set of closed points of $X$(resp. $Y$). Then $f$ ...
3
votes
1answer
768 views

Finding the Transformation to a Canonical form for a Quadric Surface

I am attempting to calculate the intersection of quadric surfaces defined by $0 = X^T A X$, $0 = X^T B X$, $0 = X^T C X$ with X = [x, y, z, 1]. Matrices A, B and C are real and symmetric. There are ...
3
votes
0answers
90 views

Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface

This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ...
3
votes
1answer
540 views

coordinate ring questions

I need your help to solve this question A subset $X \subset k^n$ ($k$ a field) is called algebraic if there exist polynomials $f_1, \dots, f_m \in k[t_1,\dots,t_n]$ such that $$X = \{x \in K^n | ...
2
votes
1answer
83 views

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$ Honestly I don't know where to begin... It's the same as proving that ...
2
votes
1answer
217 views

Blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point

How to show that blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point? I think it is a standard fact, but Google can not help me with it. Update: I found an answer ...
2
votes
1answer
129 views

3 questions about Algebraic Geometry and Zariski topology

I've read this exercise on my book, but I don't know how to prove it... Can someone help me? Let X $\subseteq \mathbb{A}^n$ be an algebraic set, and $f, g:X\rightarrow k$ two regular functions. a) ...
2
votes
4answers
1k views

Intersection of ellipse with circle

I would like know whether a circle is intersecting an ellipse. Here ellipse equation is $$Ax^2 + Bxy + Cy^2 + dx+ey + 1 = 0,$$ and the circle equation is $$(x-g)^2 + (y-f)^2= r^2.$$
2
votes
1answer
456 views

Hartshorne's proof of Proposition 2.5, Chapter II of his book Algebraic Geometry [duplicate]

Let $S = \sum_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $> 0$. Let $D_+(f) = \{\mathfrak{p} \in\operatorname{Proj} S\mid f \notin ...
2
votes
2answers
277 views

Principal ideals having embedded components

Does there exist a noetherian domain $A$ and a principal ideal $I = (x)$ in it having an embedded component?
2
votes
1answer
1k views

Parametrization of a conic and rational solutions

How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$? I'm thinking let $x = \sqrt{5}\cos t$, and $y ...
1
vote
0answers
118 views

A question regarding Grothendieck , topos and (adelic??) points

I am having a look at this conference by Bertrand Toen about Grothendieck's work. At 1:14:30 and after, Toen presents the new objects emerging from topos theory in algebraic geometry. He takes the ...
1
vote
1answer
51 views

Is taking projective closure a functor?

For an affine variety $X\subset \mathbb{A}^n$, we can associate it with $\overline{X}$, which is the closure of $X$ in $\mathbb{P}^n$. Does $\overline{X}$ depend on the choice of embedding ...
1
vote
1answer
105 views

Terminology: Why is it called sheaf COhomology?

I just learned the definition of sheaf cohomology as the derived functors of the global sections functor. I have a question about terminology - why is it called sheaf COhomology and not just homology ...
1
vote
2answers
195 views

1-form on Riemann Surface

Good evening, I can not prove the following result: Let $\omega $ be a meromorphic 1-form on $ \mathbb {C} _ {\infty} = \mathbb {C} \cup \infty $ such that $ \omega_{|\mathbb{C}} = f (z) dz $. Show ...
1
vote
1answer
111 views

when is rational function regular?

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle $V(x^2+y^2-1) \subseteq A^2$is the rational function ...
1
vote
4answers
260 views

Defining/constructing an ellipse

Years ago I was confronted with a (self imposed) problem, which unexpectedly resurfaced just recently... I don't know whether it makes sense to explain the background or not, so I'll be brief. If I ...
0
votes
1answer
82 views

Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.

Let $K$ be a field, and let $I=(XY,(X-Y)Z) \subset K[X,Y,Z]$. Prove that $\sqrt{I}=(XY,XZ,YZ)$. I have no idea how to start with this question, can anybody give me some hint? Thanks a lot.
0
votes
1answer
81 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
0
votes
1answer
77 views

explicitly constructing a certain flat family

Is it possible to construct a flat family $$ \phi:\mathbb{A}_{\mathbb{C}}^8=\operatorname{Spec} \mathbb{C}[x,y,z,w,a,b,c,d]\longrightarrow \operatorname{Spec} \mathbb{C}[t_1, t_2, t_3] ...
0
votes
1answer
81 views

Exterior product of Modules, problem wih tensor product

Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$. Then the exterior product $M \boxtimes N $ is defined ...
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votes
0answers
60 views

Classes of rings C[x,y]/(x²+cy²+ey+f) [duplicate]

I have a question. I would like to describe the classes of rings that appear in $\mathbb{C}[x,y]/I$ up to isomorphism, where $I=(Q)$, $Q=x²+cy²+ey+f$, $c,e,f\in\mathbb{C}$. $Q$ comes from ...