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. ...

learn more… | top users | synonyms (1)

5
votes
1answer
124 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, ...
5
votes
2answers
556 views

Quasiseparated if finitely covered by affines in appropriate way

I've been reading Vakil's notes on algebraic geometry (on my own -- this is not part of a class), and I'm stuck on one problem (number 6.1.H). It goes as follows. Let $X$ be a scheme. Prove that ...
4
votes
1answer
158 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
703 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
2answers
4k views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
3
votes
1answer
742 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
89 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
528 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
66 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
154 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
120 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
2answers
259 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
109 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
48 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
91 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
193 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
90 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
3answers
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 $$(x-g)^2 + (y-f)^2= r^2.$$
1
vote
4answers
241 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
66 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
78 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 ...
-1
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 ...
60
votes
4answers
4k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
28
votes
4answers
2k views

Examples of morphisms of schemes to keep in mind?

What are interesting and important examples of morphisms of schemes (especially varieties) to keep in mind when trying to understand a new concept or looking for a counterexamples? Examples of what ...
33
votes
3answers
3k views

What use is the Yoneda lemma?

Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse ...
14
votes
2answers
1k views

Why is Hodge more difficult than Tate?

There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow: "[...] we ...
14
votes
1answer
1k views

How to compute localizations of quotients of polynomial rings

At the moment I'm trying to understand the concept of localizations of rings / modules. I have done some exercises (using the book of Atiyah / MacDonald) and I will do some more, but a more practical ...
28
votes
2answers
2k views

Original works of great mathematician Évariste Galois

Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
15
votes
3answers
3k views

Best way to learn Algebraic Geometry?

I've been reading the book Commutative Algebra with a view towards Algebraic Geometry. I was wondering is the best way to learn algebraic geometry through commutative algebra? As the book I'm ...
16
votes
3answers
2k views

Hensel's Lemma and Implicit Function Theorem

In the literature and on the web happened to me several times to read confused or simply cryptic assertions regarding the fact that Hensel's Lemma is the algebraic version of Implicit Function ...
23
votes
2answers
792 views

The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
12
votes
2answers
4k views

How does one calculate genus of an algebraic curve?

I've been reading about parametrization of algebraic curves recently and the idea of the "genus of a curve" appears quite often (my impression is that a curve is parametrizable exactly when it has ...
14
votes
2answers
1k views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
22
votes
1answer
709 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
20
votes
3answers
1k views

Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?

I know that there are isomorphic projective varieties which have nonisomorphic coordinate rings, but I'm a little mystified as to "why" this is the case. Why doesn't a usual functoriality proof go ...
19
votes
1answer
2k views

What is an intuitive meaning of genus?

I read from the Finnish version of the book "Fermat's last theorem, Unlocking the Secret of an Ancient Mathematical Problem", written by Amir D. Aczel, that genus describes how many handles there are ...
18
votes
3answers
1k views

Polynomial map is surjective if it is injective

A friend of mine told me the following fact: If $k$ is any algebraically closed field, then a polynomial map $f\colon k^n\to k^n$ of affine space $k^n$ is surjective if it is injective. The ...
14
votes
2answers
632 views

Usefulness of completion in commutative algebra

After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions: (i) What is the usefulness of the concept of completion in ...
11
votes
3answers
514 views

Example I.4.9.1 in Hartshorne (blowing-up)

Let $Y$ be the irreducible curve of $\mathbb{A}^2$ given by $y^2 = x^2(x+1)$. Let $t,u$ be homogeneous coordinates of $\mathbb{P}^1$. Then the total inverse image of $Y$ under the blowing-up $\phi: X ...
5
votes
2answers
518 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
25
votes
1answer
780 views

Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
24
votes
0answers
2k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where ...
23
votes
1answer
318 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
21
votes
3answers
2k views

What is a local parameter in algebraic geometry?

Shafarevich offers the following theorem-definition: "At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
9
votes
3answers
849 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
8
votes
1answer
1k views

What is the definition of surjective morphism of schemes?

Let $f: X \to Y$ be a morphism of schemes, when talking about the surjectivity of $f$, there are at least several possibilities. (1) $f$ is surjective at the level of sets, that is $\forall \ y \in ...
25
votes
0answers
460 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...