The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.
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280 views
Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz?
This is an exercise in a ...
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2answers
977 views
Precise connection between Poincare Duality and Serre Duality
The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from ...
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4answers
272 views
Spectrum of $R[x]$
The spectrum of $\Bbb Z[x]$ is well known : a prime ideal of $\Bbb Z[x]$ is or $(Q, p)$, with $Q \in \Bbb Z[x]$ zero or irreducible modulo $p$, and $p$ prime or zero.
If I'm not mistaken, we have a ...
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2answers
807 views
Intuition for Blow-up.
If I blow up a complex manifold along a submanifold, can you give me a picture to have in mind for the blown-up manifold? Can you also tell me why this is the right picture?
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1answer
839 views
What is the intuition behind the concept of Tate twists?
For any field $K$ we can define the cyclotomic character $\chi: \operatorname{Gal}(K)\rightarrow GL_1(\hat{\mathbb{Z}})$. For any representation $V$ (I will view this as a module over ...
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2answers
583 views
Software for solving geometry questions
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every ...
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2answers
241 views
Where do Chern classes live? $c_1(L)\in \textrm{?}$
If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like ...
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1answer
219 views
How to properly use GAGA correspondence
currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces.
Now i understood that by GAGA, ...
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125 views
Are sets given in parametric form always algebraic?
If a set is given in parametric form by polynomials, is this set always closed (Zariski topology), i.e algebraic?
For example, take $X=\{(t,t^{2},t^{3}): t \in \mathbb{A}^{1}\}$ and ...
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3answers
110 views
Product of two algebraic varieties is affine… are the two varieties affine?
Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then?
If this is not true, could you give a counterexample?
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321 views
Does Hom commute with stalks for locally free sheaves?
This is somewhat related to the question Why doesn't Hom commute with taking stalks?.
My question is this: If $F$ and $G$ are locally free sheaves of $\mathcal{O}_X$ -modules on an arbitrary ...
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2answers
688 views
Theories of $p$-adic integration
What is the compelling need for introducing a theory of $p$-adic integration?
Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a ...
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2answers
122 views
The algebraic de Rham complex
Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
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2answers
223 views
Motivating (iso)morphism of varieties
I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field):
Let $X$ be a quasi-affine or quasi-projective ...
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2answers
619 views
Luroth's Theorem
I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...
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1answer
150 views
algebraic versus analytic line bundles
If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective?
I assume the latter ...
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1answer
144 views
Geometric meaning of completion and localization
Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.
The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
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2answers
150 views
Finite extensions of rational functions
I know that finite extensions of $\mathbb{C}(x)$ correspond to finite branched covers of $\mathbb{P}^1$, and this leads to an abstract characterization of the absolute Galois group of $\mathbb{C}(x)$ ...
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1answer
249 views
Concrete example of calculation of $\ell$-adic cohomology
Let $p$ and $\ell$ be distinct prime numbers.
Consider in the affine plane $\mathbb{A}^2_{\mathbb{F}_p}$ with coordinates $(x,y)$ the union $L$ of the axes $x = 0$ and $y = 0$.
How does one compute ...
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1answer
155 views
principal G-bundles in zariski vs etale topology
Let $G$ be an (affine) algebraic group over say $\mathbb{C}$. A principal $G$-bundle is a scheme $P$ with a $G$ action and a $G$-invariant morphism of schemes $\pi:P \to X$ that is etale locally on ...
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1answer
286 views
Monic (epi) natural transformations
Let $C$ and $D$ be categories and let $F : C \rightarrow D$, $G : C \rightarrow D$ be two functors such that they are either both covariant or both contravariant. Under what most general hypotheses is ...
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1answer
133 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 ...
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2answers
290 views
When do equations represent the same curve?
Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
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1answer
268 views
Inverse Limit of Sheaves
It is well-known that if you have an inverse system of abelian groups $(A_n)$ (this works in several other nice categories) in which all the maps are surjective (or at least satisfy the Mittag-Leffler ...
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1answer
155 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 $Y'$ ...
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0answers
170 views
Application of Hilbert's basis theorem in representation theory
In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand:
Two orders are defined on the set ...
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6answers
947 views
Reference for Algebraic Geometry
I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is ...
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3answers
256 views
Do the pictures in Hartshorne Ex. 1.5.1 make sense?
I have done exercise 1 of section 1.5 of Hartshorne and am able to determine that the curves (a),(b),(c) and (d) are respectively those with a tacnode, node, cusp and triple point. Now when I did this ...
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3answers
190 views
Is every algebraic curve birational to a planar curve
Let $X$ be an algebraic curve over an algebraically closed field $k$.
Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$?
I think I can prove this ...
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1answer
2k views
Divisor — line bundle correspondence in algebraic geometry
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of Cartier ...
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178 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 ...
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283 views
Variety vs. Manifold
In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
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2answers
118 views
Is a divisor in the hyperplane class necessarily a hyperplane divisor?
Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$.
...
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2answers
387 views
Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.
In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that ...
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263 views
Good books/expository papers in moduli theory
I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves.
I began ...
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6answers
478 views
Visualizations of some of the abstractions of algebraic geometry
Where, or do there exist, good visualizations of sheaves, stalks, stacks, and/or schemes? I'm a better visual thinker than I am a symbolic thinker, and it would be easier for me to follow some of the ...
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2answers
128 views
Can an algebraic variety be described as a category, in the same way as a group?
Can an algebraic variety be described as a category, in the same way as a group? A group can be considered a category with one object, with elements of the group the morphisms on the object.
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1answer
422 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 ...
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2answers
201 views
Good problems in Algebraic Geometry
I am now using Fulton's book Algebraic Curves to learn algebraic geometry from and have just finished chapter 2. However I feel that the problems are not very inspiring (at the moment at least) and ...
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1answer
329 views
Geometric meaning of primary decomposition
In the book "Commutative Algebra with a view toward Algebraic Geometry of David Eisenbud, he wrote about the Geometric interpretation of primary decomposition.
I summary as follows :
Let ...
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2answers
128 views
Elliptic curves over Spec Z
I want to show that there are only finitely many elliptic curves over Spec $\mathbf Z$ without appealing to Siegel's theorem or Shafarevich' theorem.
Firstly, I think (but I am not sure) that such an ...
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1answer
178 views
intrinsic proof that the grassmannian is a manifold
I was trying to prove that the grassmannian is a manifold without picking bases, is that possible?
Here's what I've got, let's start from projective space.
Take $V$ a vector space of dimension n, and ...
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2answers
133 views
What does $Tor_{R}^n(M,N)$ represent?
Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
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1answer
424 views
My first course in algebraic geometry: two simple questions
I'm attending my first course in algebraic geometry, and my professor has chosen an approach which is a middle-way between the basic algebraic geometry done in $\mathbb A^n_k$ and the approach with ...
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1answer
216 views
Why study schemes?
Why study schemes instead of only affine/projective varieties, given by zeros of polynomials in the affine/projective space? I mean, what is gained by introducing the concept of schemes?
Thank you!
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1answer
105 views
Curious about Hilbert-Zariski theorem involving homogeneous variety and set of zeroes.
I got myself in a confusing situation the other week while trying to read a bit of algebraic geometry. I'm hoping someone can pull me out.
Suppose $k$ is a field, and $V$ a homogeneous variety with ...
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1answer
80 views
Ext between two coherent sheaves
Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$.
How to show, that all $Ext^i ...
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2answers
220 views
Can there be a point on a Riemann surface such that every rational function is ramified at this point?
Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset.
Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$?
I'm ...
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1answer
311 views
Stacks in arithmetic geometry [closed]
Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
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1answer
288 views
vector bundles on affine schemes
Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that ...
