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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2
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0answers
149 views

How to calculate the restriction of a divisor to itself?

I am wondering whether there is a notion of restriction of a divisor to itself? More specifically, let $X=\mathbb{P}^3$ be the projective space and let $x, y$ be two points in $X$, let $l$ be the line ...
4
votes
1answer
464 views

Calculating the distance between a camera and a target using camera output

I have a 640x480 camera that recognizes a rectangle that is 1ftx2ft. Is it possible to calculate the distance between the camera and the rectangle? Edit: The horizontal angle of view is 54°.
3
votes
1answer
146 views

Basic Question on the Ideal-Variety Correspondence and Adjoint Functors

Let $k$ be an algebraically closed field. The ideal-variety correspondence says that the equations \begin{align} \mathbf{I}(X) &= \left\{f\in k[x_1,\dotsc,x_n]:p\in X\Rightarrow f(p)=0\right\} \\ ...
2
votes
1answer
191 views

Metric tensor of complex numbers & Hamiltonian Mechanics

The Euclidean $\mathbb{R}^2$ geometric space can be mapped onto $\mathbb{C}$. In other words I see it like this $$\vec{v} = x\vec{x}+y\vec{y} = x\vec{1}+y\vec{i}= \begin{bmatrix}x \\y\end{bmatrix} ...
3
votes
0answers
120 views

Continuing a Divisor

I need some help, to find the strategy for solving the following problem: Given an analytic surface $S^0$, its compacitfication $S$ and a horizontal Divisor $D^0$ on $S^0$, I have to continue $D^0$ to ...
3
votes
1answer
189 views

Algebraic variety as a union of nonsingular subvarieties

Definitions Let $M$ be algebraic variety and let $I$ be the defining ideal of $M$, that is $$I(M) = \{ f \in K[X_1,...,X_n] \mid \forall x \in M : f(x)=0 \}$$ Let $f_1,...,f_m$ be the generators of ...
3
votes
1answer
137 views

How to compute $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1})$ and $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1}(n))$

Let $\mathcal{O}_{\Bbb P^1}$ be the structure sheaf of the projective line $X=\Bbb P^1_k$ over some field $k$ (algebraically closed of characteristic $0$). What is a good (and, preferably, easy) way ...
4
votes
1answer
374 views

Pullback of very ample sheaf again very ample? And other questions.

Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
3
votes
1answer
132 views

Base-point-free invertible sheaves on smooth projective curves

Let $C$ be a projective curve over $k$ (geometrically integral, nonsingular). I am confused about the following argument in Vakil's notes on algebraic geometry: (1) In (20.6.2), he writes: Fix a ...
5
votes
1answer
357 views

Projective closure: How to determine?

In the exercise 2.9 of the book Algebraic Geometry by Hartshone, the author questions us about the projective closure of an affine variety. Let $Y$ be an affine variety in $\mathbb{A}^n$, ...
1
vote
1answer
70 views

Why $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$

There is an argument saying that because $\Delta(\tau)\in\mathcal{S}_{12}(\text{SL}_2(\mathbb{Z}))$, then we have $\Delta(N\tau)\in\mathcal{S}_{12}(\Gamma_0(N))$. I don't understand the logic here. ...
2
votes
0answers
29 views

Problem about the order of a automorphic function on an elliptic curve.

Let $f:X(\Gamma)\rightarrow\mathbb{C}$ be an automorphic function of weight 0, saying that $f$ is $\Gamma$-invariant. Now at each noncusp point $\tau\in\mathbb{C}$, $f$ has an order $\nu_\tau(f)$, ...
0
votes
1answer
64 views

Irreducible topological spaces covered by infinitely-many closed sets

An irreducible topological space cannot be the union of two proper closed subsets, by definition. Is it possible thai It can be the the union of infinitely-many proper closed subsets? How about the ...
4
votes
1answer
200 views

Does the singular locus of a conical variety (or scheme) determine the singular locus of its projectivization?

Lets say $X$ is a conical affine algebraic variety (conical meaning $X$ is the zero set of homogeneous polynomials of positive degree, equivalenty $X \subsetneq k^n$ and $x \in X \Rightarrow ax \in X$ ...
3
votes
1answer
175 views

Three kinds of spectra

In commutative algebra while proving the Nullstellensatz one introduces for a while the Rabinowitsch sprectrum as: $$\operatorname{Spec}_{\rm Rab}(R)=\{R\cap \mathfrak m: \mathfrak m \text{ is a ...
13
votes
2answers
370 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 ...
2
votes
0answers
53 views

spectral sequence computing invariants

let $G$ be an abelian group (can also assume $G=\mathbb{Z}^n$ for some positive integer $n$). Let $X\stackrel{g}{\rightarrow} Y$ be a $G$-covering where $X,Y$ are schemes, or topological spaces or ...
3
votes
2answers
70 views

Why function $j(\tau)$ has degree 1?

We have $$ j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_nq^n, a_n\in\mathbb{Z},q=e^{2\pi i\tau} $$ Then it is said that because $j$'s only pole is simple, $j$ has degree 1 as a map ...
3
votes
0answers
88 views

Euler characteristic of the complement of a normal crossing divisor

Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Put $n=\dim X$. It follows from the Hirzebruch-Riemann-Roch theorem that the degree of the top Chern class of ...
2
votes
1answer
74 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
5
votes
1answer
110 views

Geometric interpretation of the fundamental theorem for coalgebras?

Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the ...
2
votes
1answer
153 views

Identity intersection number with degree of invertible sheaf on a surface?

This question is about an argument on page 358 of Hartshorne, proof of Lemma 1.3. Consider curves $C$ and $D$ on a surface $X$ meeting transversely. There is an exact sequence $0\to ...
4
votes
1answer
82 views

What is Weil paring computing really?

I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing ...
3
votes
1answer
152 views

In one version of the Modularity Theorem, what does “arise from modular forms” mean?

One version of Modularity Theorem says that The elliptic curves with rational $j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ...
2
votes
1answer
172 views

Quasi-Coherent modules over sheaves of rings (proof question)

Let $X$ be a scheme with structure sheaf $\mathcal O_X$ and let $\mathcal B$ be a quasi-coherent $\mathcal O_X$-algebra. Let $\mathcal F$ be any sheaf of $\mathcal B$-modules. In EGA I, § 9, Prop. ...
1
vote
2answers
185 views

Segre embedding of $\mathbb{P}^n\times \mathbb{P}^m\times\mathbb{P}^q$

How could I define the segre embedding of three projective spaces? Usually in references only the Segre embedding of two projective spaces are defined: $\mathbb{P}^n \times \mathbb{P}^m \rightarrow ...
7
votes
3answers
254 views

When does locally irreducible imply irreducible?

The situation is this: I have a homogeneous ideal with many generators and variables, too many to simply ask isPrime I in Macaulay2. However, the ideal simplifies ...
1
vote
1answer
70 views

Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
5
votes
1answer
95 views

About twistor space of a K3 surface

I know that for $X=(M,I)$ , where $I$ is the complex structure, a K3 surfaces and $\alpha \in H^2(X,\mathbb{R})$ a Kähler class, there exist a Kähler metric g and J,K complex structures such that 1) ...
2
votes
2answers
359 views

What does degree of an isogeny mean?

The book I'm reading doesn't provide the definition of degree of an isogeny and I failed to google it. Can anyone tell me?
5
votes
1answer
71 views

A question about complex tori.

Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$ m\Lambda\subset\Lambda' $$ The, the book I'm reading says that by the theory of finite Abelian groups ...
1
vote
1answer
101 views

Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
1
vote
0answers
237 views

Great Circle Center of Circle + radius

I need to draw a great circle arc between two latitude and longitude points. For sake of example we will use the coordinates for LAX and JFK. JFK is 40.64°N / 73.78°W LAX is 33.94°N / 118.41°W My ...
1
vote
2answers
49 views

Find $A^{-1}$(W) of linear manifold W

Given linear map $A:\mathbb{R}^2\to \mathbb{R}^4$ defined as $$A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 2 \\ 3 & 1 \end{pmatrix}$$ and linear manifold $ W \subset ...
4
votes
1answer
164 views

Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism? I believe the answer should be yes, ...
3
votes
0answers
85 views

How do one think about $\mathcal{O}_X(D)$?

The notation $\mathcal{O}_X(D)$ appears a lot in algebraic geometry, for which I get confused sometimes. More specifically my question is the following: When is $\mathcal{O}_X(D)$ defined, and is ...
10
votes
0answers
370 views

Self-Intersection Number $-2$

I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some ...
1
vote
2answers
103 views

Involutions of a torus $T^n$.

Let $T^n$ be a complex torus of dimension $n$ and $x \in T^n$. We have a canonical involution $-id_{(T^n,x)}$ on the torus $T^n$. I want to know for which $y \in T^n$, we have ...
3
votes
2answers
130 views

an irreducible quadric $ X \subset \Bbb A^n$ d is birational to some $\Bbb A^m$

I want to prove that an irreducible quadric $ X \subset \Bbb A^n$ defined by a quadratic equation $ F(T_1,\ldots,T_n)=0$ is rational (i.e birational to some affine space $\Bbb A^m$ ). I'm not sure ...
1
vote
1answer
116 views

a ideal of the ring of regular functions of an affine variety.

We assume that $\Bbb k$ is an algebraically closed field. Let $X \subset \Bbb A^n$ be an affine $\Bbb k$-variety , let's consider $ \mathfrak A_X \subset \Bbb k[t_1,...t_n]$ as the ideal of ...
9
votes
0answers
772 views

Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting ...
4
votes
1answer
139 views

$f(X)$ dense in $Spec A$

I have some problems solving the following exercise from Liu's book Algebraic Geometry and Arithmetic Curves, exercise 3.15 from chapter 2. Let $X$ be a quasi-compact scheme, $A=O_X(X)$. Let us ...
5
votes
3answers
386 views

(geometric/intuitive) interpretation of ext

In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext. But I don't have a feeling about ...
2
votes
0answers
65 views

One question on map algebras

Kindly asking for any hints about the following questions: 1- Suppose, $X$ be an scheme and $g$ an finite-dimensional Lie algebra over an algebraically closed field $k$. We denote by $M(X,g)$, the ...
2
votes
0answers
61 views

Finite-dimensional Lie algebra as a scheme

Kindly asking for any hints about the following questions: Suppose $k$ is an algebraically closed field of characteristic zero and $g$ is a finite-dimensional Lie algebra over $k$. Then $g$ is ...
9
votes
1answer
255 views

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces ...
7
votes
2answers
294 views

Infinite coproduct of affine schemes

Let $(X_i)_{i\in I}$ be a family of affine schemes, where $I$ is an infinite set and $X_i = Spec(A_i)$ for each $i \in I$. Let $X$ be a coproduct of $(X_i)_{i\in I}$ in the category of schemes. Let ...
3
votes
2answers
295 views

Is Klein bottle an algebraic variety?

Is Klein bottle an algebraic variety? I guess no, but how to prove. How about other unorientable mainfolds? If we change to Zariski topology, which mainfold can be an algebraic variety?
2
votes
1answer
290 views

Fundamental group of multiplicative group in Zariski topology

What is the fundamental group of the multiplicative group of the complex numbers $\mathbb{G}_m(\mathbb{C})$ with respect to the Zariski topology. More precisely, what are the homotopy classes of ...
30
votes
5answers
4k views

Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...