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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Real points of a complex curve

Since the "real points" of a complex curve can mean a couple of different things, bear with me while I'm annoyingly formal here. Consider first a cubic curve $y^2 = x^3 + a x + b$. Write $$S := \{ ...
13
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1answer
310 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
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3answers
2k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
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1answer
449 views

Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
13
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1answer
338 views

What is the homotopy type of the affine space in the Zariski topology..?

I'm asking this question out of curiosity, as I was unable to come to a conclusion. Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
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2answers
452 views

The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...
13
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1answer
550 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 ...
13
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2answers
373 views

Variety of pairs of product-zero matrices

Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components ...
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632 views

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies? x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n ...
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5answers
855 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
12
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3answers
1k views

Why is there no polynomial parametrization for the circle?

How does one show that the unit circle admits no polynomial parametrization? What is needed for this, are there general criteria? Thanks
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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 ...
12
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3answers
2k views

Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...
12
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3answers
1k views

Question on sheafification of a presheaf

In chapter 2 of GTM 52 by Robin Hartshone there are definition of presheaf and the associated sheaf of a given presheaf. I found that the definition of the sheafification is rather less natural and ...
12
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2answers
1k views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
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2answers
1k 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 ...
12
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3answers
557 views

Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff

In the book of Atiyah and MacDonald, I was doing exercise 3.11. One has to show that for a ring $A$, the following are equivalent: $A/\mathfrak{N}$ is absolute flat, where $\mathfrak{N}$ is the ...
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2answers
3k views

One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I'm beginning to learn some grothendieck's algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one ...
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2answers
965 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 ...
12
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1answer
297 views

What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
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2answers
221 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 ...
12
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2answers
521 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 ...
12
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1answer
971 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 ...
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2answers
788 views

Algebraic versus topological line bundles

Let $X$ be a CW complex. The (isomorphism classes of) complex line bundles on $X$ are classified by the homotopy classes of maps $X \to \mathbb{CP}^\infty$, that is by the elements of $H^2(X, ...
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465 views

Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Serre's criterion for affineness (Hartshorne III.3.7) states that: Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine. ...
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1answer
1k views

Undergraduate roadmap for Langlands program and its geometric counterpart

What are the topics which an undergraduate with knowledge of algebra, galois theory and analysis learn to understand Langlands program and its goemetric counterpart? I would also like to know what are ...
12
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1answer
841 views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
12
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1answer
472 views

Regular local ring and a prime ideal generated by a regular sequence up to radical

Let $R$ be a regular local ring of dimension $n$ and let $P$ be a height $i$ prime ideal of $R$, where $1< i\leq n-1$. Can we find elements $x_1,\dots,x_i$ such that $P$ is the only minimal prime ...
12
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2answers
297 views

Preimaging units to units

I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an ...
12
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1answer
231 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
12
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1answer
218 views

An inverse limit

Let $k$ be a field. Consider the inverse limit $\varprojlim k[x,y]/(y\cdot x^n)$. I wonder if there is a nice description of this ring? Geometrically, we look at the union of the line $y=0$ ...
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2answers
172 views

The coarse moduli space of a Deligne-Mumford stack

This question is just a "definition request". Question. What is the "the coarse moduli space" of a Deligne-Mumford stack? I know (the basics of) DM stacks, but how are their moduli spaces ...
12
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2answers
154 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
12
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2answers
651 views

Problem in Rick Miranda: finding genus of a projective curve

I have just started learning Riemann surfaces and I am using the book by Rick Miranda: Algebraic curves and Riemann Surfaces. #F in section 1.3 asks to determine the genus of the curve in ...
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1answer
370 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 ...
12
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1answer
642 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 ...
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2answers
225 views

Zariski dense implies classically dense?

I was surprised that I wasn't able to find this question already posted; if it has been posted and I just didn't find the right search terms, let me know. Let $X$ be any complex variety. A priori, ...
12
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1answer
629 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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2answers
269 views

Are there applications of noncommutative geometry to number theory?

The marriage of algebraic geometry and number theory was celebrated in the twentieth century by the school of Grothendieck. As a consequence, number theory has been completely transformed. On the ...
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votes
1answer
370 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
12
votes
1answer
250 views

Does the Zariski closure of a maximal subgroup remain maximal?

Let $k$ be an algebraically closed field and let $G\leq\rm{GL}_n(k)$. Assume that $M<G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski closure of $G$ in ...
12
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1answer
696 views

Irreducible components of topological space

Let $X$ be a topological space. Let $\Sigma$ be the set of irreducible components of $X$. Let $X=\cup_{i\in I} X_i=\cup_{j\in J} Y_j$, $X_i,Y_j\in \Sigma $ for some index set $I,J$. $X_i$'s are ...
12
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0answers
168 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...
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0answers
501 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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1answer
326 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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3answers
434 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
380 views

Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb ...
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4answers
302 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
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3answers
471 views

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number field and let $\pi:X\to \mathbf{P}^1_K$ be a finite morphism, where $X$ is a smooth projective geometrically connected curve. Is $\pi$ a Galois cover if and only if the base ...
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2answers
474 views

Is tautological bundle $\mathcal{O}(1)$ or $\mathcal{O}(-1)$?

I always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ...