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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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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593 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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1answer
969 views

Simple example of an ample line bundle that is not very ample

I am looking for a very concrete and simple example of a line bundle $L$ (on a curve or a surface) which is ample, but not very ample. I would also like that $L^{\otimes k}$ is very ample for a small ...
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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 ...
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1answer
445 views

Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
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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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1answer
200 views

Why does Mumford want to avoid “reduction to Jacobians”?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
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351 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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228 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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1k views

The Picard Group of the Affine line with double origin

Let $X$ be the affine line with double origin over a field $k$. It is the scheme obtained gluing two copies of the affine line $\mathbb{A}^1_k$ along the open sets $U_1 = U_2 =\mathbb{A}^1_k - (x)$, ...
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871 views

What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across the Albanese map. For a compact Kähler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the ...
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829 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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325 views

Show the set of points $(t^3, t^4, t^5)$ is closed in $\mathbb A^{3}$

Show the set of points $X = \{(t^3, t^4, t^5) \}$ with $t\in k$ is closed in $\mathbb A^{3}$ and find three generators of $\mathcal{I}(X)$. This is a homework question, so please don't provide ...
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517 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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252 views

Spectrum of $\mathbb{Z}^\mathbb{N}$

Is anything known about the spectrum of $\mathbb{Z}^{\mathbb{N}}$? Notice that the fiber of $\mathrm{Spec}(\mathbb{Z}^{\mathbb{N}}) \to \mathrm{Spec}(\mathbb{Z})$ at a non-zero prime ideal $(p)$ is ...
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Toy sheaf cohomology computation

I asked this question a while back on MO : One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$) I am curious, as a sort of followup if ...
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1answer
496 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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288 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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305 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
266 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 ...
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1answer
226 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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269 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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156 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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687 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
149 views

Elliptic curve over algebraically closed field of characteristic $0$ has a non-torsion point

Let $E/k$ be an elliptic curve over an algebraically closed field $k$ of characteristic $0$. Can one prove that the abelian group $E(k)$ is non-torsion? Better yet, can one prove that $E(k) ...
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1answer
560 views

Sard's theorem for algebraic varieties

(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular values ...
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500 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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716 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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251 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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188 views

Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
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372 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 ...
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1answer
280 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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Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
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267 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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656 views

Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ ...
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344 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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458 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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498 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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Zariski Open Sets are Dense?

Is it true than any nonempty open set is dense in the Zariski topology on $\mathbb{A}^n$? I'm pretty sure it is, but I can't think of a proof! Could someone possibly point me in the right direction? ...
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339 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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501 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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Picard group of product of spaces

Suppose $X,Y$ are varieties over an algebraically closed field $k$. Can we compute $\operatorname{Pic}(X \times_k Y) $ in terms of $\operatorname{Pic}(X),\operatorname{Pic}(Y)$? It seems that ...
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Dominant rational maps

The University I go to doesn't have any courses in (classical) Algebraic Geometry so I am trying to learn myself. I am fairly comfortable with the content I have covered so far aside from a so called ...
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Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
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Picture of spec?

I recall seeing a hand-drawn picture of spec of a ring (maybe of $\mathbb Z$?) that had been passed around in the early days of the Zariski topology. Does anyone know where I can find a copy?
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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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674 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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387 views

Torsion Chern class?

Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using ...
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658 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
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Prove that sheaf hom is a sheaf.

Suppose $\mathcal{F}$ and $\mathcal{G}$ are sheaves on $X$. The sheaf hom from $\mathcal{F}$ to $\mathcal{G}$ is defined by $U \mapsto $ Hom($\mathcal{F}|_{U}$,$\mathcal{G}|_{U}$), where the Hom is ...