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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3
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84 views

What applications does abstract algebra and algebraic geometry have in computer science and programming? [closed]

I love math and programming. Abstract Algebra and algebraic geometry seems very pleasant to me. But I also would like to improve my skills as a programmer, but I would love to do so in the fiels that ...
2
votes
1answer
21 views

real affine varieties are hypersurfaces

In $\mathbb{R}^n$, let X be a Zariski-closed set. then $X=\mathbb{V}(f)$ for some polynomial $f$. Elementary formulation: let $X \subset \mathbb{R}^n$ be the set of common zeroes of some ...
3
votes
1answer
43 views

irreducible components of subscheme

Let $f : X \to Y$ be a closed immersion of (noetherian) schemes. Is there any "general" result on $f$ out there ensuring that $X$ has the same number of irreducible components as $Y$ ?
2
votes
1answer
729 views

The cone over a projective variety

I'm trying to prove that $I(C(Y))=I(Y)$, where $C(Y)=\pi^{-1}(Y)\cup \{(0,\ldots,0)\}$ the cone over $Y$ and $\pi:\mathbb A^{n+1}-\{0,\ldots,0\}\to \mathbb P^n$ the projection which sends the point ...
1
vote
2answers
77 views

Algebraic Varieties vs Smooth Manifolds

There are many posts I have read on that subject which seem unclear for me. My main question (it might be silly) is: "Every non-singular algebraic variety over $\mathbb{C}$ is a smooth ...
1
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0answers
23 views

How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups

In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the ...
1
vote
1answer
40 views

smooth complex projective variety with non-surjective cycle class map

I am looking for a smooth complex projective variety such that the rational cycle class map $$CH^k(X)_\mathbb{Q} \rightarrow H^{2k}(X;\mathbb{Q}),$$ which is defined by sending a closed irreducible ...
3
votes
2answers
82 views

Sheaf cohomology with support

Let $X$ be a topological space and $\mathcal F$ is a sheaf of abelian groups on it. Let $Y$ be a closed subspace of $X$. Let $\mathscr{H}^0_Y \mathcal F$ be the subsheaf of $\mathcal F$ with supports ...
3
votes
1answer
63 views

When does a f.g. algebra over a field $F$ make it “look like $F$ is algebraically closed?”

Let $F$ be a field, and let $A$ be a finitely generated algebra over $F$. If $\mathfrak m$ is a maximal ideal of $A$, then $A/\mathfrak m$ is an algebraic extension of $F$, although it is in general ...
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0answers
50 views

Self-extensions of a skyscraper sheaf

Let $V$ be a smooth variety over a field $k$. For a point $x \in V$ we denote the skyscraper sheaf of length 1 by $$ k(x) = \mathcal{O}_x/m_x. $$ Then by taking the Koszul resolution of $k(x)$ one ...
12
votes
3answers
3k views

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? ...
1
vote
1answer
97 views

Line bundle trivial on fibers then isomorphic to the pullback of a line bundle

$\require{AMScd}$ I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let ...
1
vote
0answers
6 views

Closed-form solution for a simple system of concave equations

I am trying to solve what looks like a simple system of equations: $$x_j = A_j\left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, ...
2
votes
1answer
41 views

Blow up of an ideal in $\Bbb C^2$

As described in these notes, I am trying to compute the blow up of $\mathbb C^2=\text{ Maxspec }\mathbb C[x,y]$ along the subvariety corresponding to the ideal $\langle\ x^2,y\ \rangle$ but I am ...
2
votes
1answer
67 views

To prove that an ideal cannot generated by two elements [duplicate]

Let $k$ be an algebraically closed field and let $\ Y\subset \mathbb{A}^n(k)$ be the curve given parametrically by $x=t^3, y=t^4,z=t^5$ I want to show (i) $I(Y)$ is a prime ideal of height 2 (ii) ...
2
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0answers
52 views

On the definition of principal Cartier divisors

In Liu's Algebraic Geometry and Arithmetic Curves, Definition 7.1.17, a few lines after the definition of principal Cartier divisor (as one in the image of $\Gamma (X,K_X^*) \to \Gamma (X,K_X^*/ ...
3
votes
1answer
79 views

Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
4
votes
0answers
61 views

Tropicalization of a line in the projective plane P^2

Lets assume that our field $K$ is the Puiseux series. I have been working with tropicalization from the book "Introduction to tropical geometry" link : ...
3
votes
1answer
32 views

Closed maps in terms of lifting properties (analogousy to formally étale morphisms)?

In continuation to this MSE question, where closed maps are characterized by "fiber thickenings", I trying to formulate this fiber thickening condition as some lifting property of $f$ against some ...
2
votes
0answers
34 views

Generators of $ Z[x_1, x_2,\ldots , x_n] $

Is there any characterization of n-element generators of $ Z[x_1, x_2,\ldots , x_n] $? They obviously need to be algebraically independent.
2
votes
1answer
54 views

Intuition behind fibers of a morphism of schemes

Let $X,Y$ be schemes over a field $k$ and $f:X\rightarrow Y$ a morphism. Let us suppose that the fiber $f^{-1}(y)$ of $f$ at a point $y\in Y$ has two connected components $Z_{1},Z_{2}$. I have read ...
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0answers
27 views

Are there curves of genus 2 and higher over number fields with everywhere good reduction?

a theorem of Fontaine states that there are no curves of genus $\geq 1$ over $\mathbb Q$ with everywhere good reduction. For curves of genus one over number fields, this is not true. There are number ...
2
votes
1answer
132 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
2
votes
2answers
21 views

If a closed set in projective space is stabilized by an automorphism, so are its components

Let $X$ be a closed set in $\mathbb{P}^n$, and let $\phi$ be an automorphism of varieties $\mathbb{P}^n \rightarrow \mathbb{P}^n$. If $\phi$ stabilizes $X$, does it follow that $\phi$ stabilizes the ...
1
vote
1answer
33 views

Does every open covering of a scheme also contain an affine covering?

Let $U_i$ denote a family of open subsets of a scheme $X$, which is not affine, so that $X = \bigcup U_i$. Can this covering be somehow transformed into an affine covering of $X$?
1
vote
1answer
24 views

smooth affine algebraic curves and their subschemes

I am reading a lot about curves at the moment and I am a little confused: Let $X= Spec K[X]$ denote a smooth affine algebraic curve. Then, according to some sources, the ring $K[X]$ is a Dedekind ...
3
votes
0answers
41 views

Are closed points of a scheme $\frac{X}{k}$ the same $\overline{k}$-points, modulo Galois group action

Let $k$ be a field, and $X$ a scheme locally of finite type over $k$. Let $\overline{k}$ be the algebraic closure of $k$. Is it true that the set of closed points of $X$ is in bijection with ...
-1
votes
1answer
35 views

how to distance circles drawn on another circles

I need to do some calculations in order to do this drawing (sorry for the quick sketch): I need to define a set of variables and do simple calculations as much as possible in order to come up with ...
2
votes
2answers
46 views

Example of a dominating map

Unfortunately the book that i am reading (Algebraic curves by Fulton) has no examples, so i am trying to find an example of a dominating map that would be helpful for the understanding of the ...
2
votes
1answer
24 views

How to prove that $H^q(X_{et},F) \cong \prod_{i=1}^n H^q(X_{i_{et}}, F|_{X_i})$ for $X = X_1 \sqcup … \sqcup X_n$

Let $X = X_1 \sqcup ... \sqcup X_n$ be a disjoint union of schemes, and let $F \in \text{Ab}(X_{et})$ be an abelian sheaf on an etale site of $X$. I need to show that $H^q(X_{et},F) \cong ...
2
votes
2answers
62 views

Confusion regarding definition of regular function

This is quoted from Basic algebraic geometry Shafervich. Let X be a closed set in the affine space $\mathbb{A}^n$ over the ground field $k$. A function f defined on X with values in $k$ is regular ...
3
votes
1answer
51 views

A version of Bezout's Theorem

I have read the following version of Bezout's Theorem, but I don't get to understand how it implies the classical version. Let $F,G\in K[X_{0},X_{1},X_{2}]$ be non-constant homogeneous polynomials ...
10
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0answers
127 views

When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that ...
3
votes
1answer
40 views

Prove that a general monomial curve is smooth

Let $k$ be a field, $n_1<n_2<\cdots<n_r$ positive integers, and $C:=\{(t^{n_1},...,t^{n_r})\mid t\in k\}\subset \mathbb{A}^r$. Show that $C$ is a smooth curve iff $n_1=1$. This is what ...
2
votes
0answers
25 views

Degree of an algebraic variety given by a polynomial parametrization

I'm trying to prove that the degree of an algebraic variety given by a polynomial parametrization of the form $$x_1=P_1(u_1,\dotsc,u_{m-g+1}), \dotsc, x_m=P_m(u_1,\dotsc, u_{m-g+1}),$$ where all ...
4
votes
0answers
55 views

Elegant formalism for gluing spaces over open subsets (Vakil 17.2.B.)

This question is about exercise 17.2.B. from Vakil's algebraic geometry notes. Let $X$ be a scheme. The following data is given: For each affine open set $U \subset X$ a scheme $\pi_U :Z_U \to U$. ...
4
votes
0answers
64 views

Algebraic variety determined by closed points

I am in the process of understanding the importance of the closed points of algebraic varieties (taking the scheme point of view). I ask myself the following question: if varieties $X$ and $Y$ over a ...
4
votes
1answer
119 views

Defining the set $\{(t^3,t^4,t^5) : t \in \mathbb{C}\}\subset \mathbb{C}^3$ by two polynomial equations

What are two polynomials $f,g \in \mathbb{C}[x,y,z]$ such that $$\{(x,y,z): f(x,y,z)=g(x,y,z)=0\}\;=\;\{(t^3,t^4,t^5): t \in \mathbb{C}\}$$ holds as an equality of subset of $\mathbb{C}^2$? This ...
2
votes
0answers
48 views

What properties single out $ \operatorname{Spec}(\mathbb{k}) $-schemes that are quasi-projective varieties over $ \mathbb{k} $?

I have a question in algebraic geometry that I would like to ask: Let $ \mathbb{k} $ be an algebraically closed field. Is there a property $ P $, phrased in the language of schemes, such that ...
1
vote
1answer
40 views

Algebraic families of vector spaces that are pairwise dependent

Let $T \subset Gr(n,2n)$ be an algebraic family of complex $n$-dimensional vector subspaces in $\mathbb{C}^{2n}$, $n > 1$, and denote the vector space corresponding to a point $t$ of $Gr(n,2n)$ by ...
8
votes
1answer
354 views

In which cases does pullback commute with the Hom-sheaf?

Assume $f: (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a morphism of locally ringed spaces and E and F are two locally free $\mathcal{O}_Y$-moduels of finite rank. I was wondering if we have ...
2
votes
1answer
37 views

Suppose X is a closed subscheme of Y, with Y locally Noetherian. Is there a locally free resolution of $i_* O_X$?

Let $I : X \to Y$ be a closed subscheme of a locally Noetherian scheme. I am secretly trying to show that sheaf exts of $i_* O_X$ to coherent sheaves on Y are coherent (in order to find a dualizing ...
2
votes
1answer
100 views

Zariski topology questions from Atiyah and Macdonald's Introduction to Commutative Algebra

Exercise 17 in Chapter 1 in Atiyah and Macdonald's Introduction to Commutative Algebra introduces the Zariski topology. There are 7 subquestions, of which 4 I've solved on my own, but the last couple ...
1
vote
1answer
52 views

Zariski topology on Affine spaces, Name of Functor

I've been studying the Zariski topology in my free time. So I found this functor between Polynomial Algebras and Affine Spaces. First, we have this $T$ such that for any affine space ...
8
votes
3answers
1k views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...
3
votes
0answers
45 views

Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm very intrigued by ...
4
votes
2answers
90 views

Fiber of morphism homeomorphic to $f^{-1}(y)$

I want to solve exercise 3.10 (a) of Hartshorne's book, chapter II, which asks to prove the following: Let $f\colon X\to Y$ be a morphism of schemes and let $y\in Y$, then $X_y=X\times_Y ...
3
votes
2answers
180 views

Rank of an elliptic curve

How could we compute the rank of an elliptic curve? I looked for a methodology in my book, but I didn't find anything. Could you give me a hint? I want to find the rank of the curve $Y^2=X^3+p^2X$ ...
5
votes
1answer
57 views

Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and ...
4
votes
1answer
109 views

Automorphisms of an elliptic curve fixing the invariant differential?

If we consider an elliptic curve $E/k$ given in Weierstrass form $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$, then I know that the translation maps $\tau_{P}$ with $P\in{E}$ fix the invariant ...