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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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
330 views

Existence of divisors of degree one on a curve over a finite field

Let $C$ be a smooth, geometrically irreducible projective curve defined over a finite field $\mathbb{F}_q$. Given a (scheme-theoretic) point $x \in C$, define the degree of $x$ to be the degree of ...
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1answer
377 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
13
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1answer
229 views

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 := \{ ...
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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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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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456 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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347 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 $\mathbb{C}^{n}$. ...
13
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453 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 ...
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561 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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376 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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634 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 ...
12
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2answers
516 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 ...
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5answers
883 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 ...
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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 ...
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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 ...
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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 ...
12
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2answers
531 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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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 ...
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3answers
562 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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Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
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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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977 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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301 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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313 views

Hartshorne exercise III.6.2 (b) - $\mathfrak{Qco}(X)$ need not have enough projectives

Let $X=\mathbb P^1_k$, with $k$ an infinite field. Show that there does not exist a projective object $\mathcal P$ either in $\mathfrak{Qco}(X)$ or $\mathfrak{Coh}(X)$ together with a surjection ...
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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
528 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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1answer
982 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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793 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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473 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 ...
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1answer
882 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 ...
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1answer
475 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 ...
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2answers
298 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 ...
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1answer
502 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
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1answer
238 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 $y=0$ ...
12
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2answers
182 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 ...
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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 ...
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2answers
652 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
373 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
650 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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229 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, ...
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1answer
637 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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277 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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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
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1answer
261 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 ...
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704 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 ...