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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1answer
32 views

Homogeneous ideals and homogeneous elements of them

Let $J \subseteq \mathbb{K}[x_o,\ldots,x_n]$ be a homogeneous ideal. I am struggling to 'prove' (more to understand) this statement in my Algebraic Geometry book: Every homogeneous element of $J$ ...
0
votes
1answer
44 views

Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$)

I am working on Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$) and I would like some assistance. The exercise states: Let $B$ be a ring. If $X$ is a ...
5
votes
2answers
149 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
14
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0answers
111 views

Representability of diagonal of $\mathscr{M}_g$

Let $\mathscr{M}_g$ be the moduli stack of genus $g$ curves ($g \geq 2$). That is, $\mathscr{M}_g$ is the category whose objects are proper smooth morphisms $f: C \to S$ whose geometric fibers are ...
2
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0answers
18 views

definition of cycle theoretic fibre

I am studying the definition of Chow variety on Kollar's Rational Curves on Algebraic Varieties, and I am having some trouble in understanding Definition 3.9. Here we have a proper morphism of ...
1
vote
0answers
32 views

Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
2
votes
0answers
21 views

Étale Fundamental group for $\mathbb{A}^n$ (prime to $p$-part)

I apologize if this is silly - let $k$ be a separably closed field, I wish to calculate $\pi_1(\mathbb{A}^n)$ completed away from the characteristic :($Hom(\hat{\pi_1(\mathbb{A}^n)}, G)$ classifies ...
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0answers
27 views

Compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$.

This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff. Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual ...
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0answers
16 views

Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
4
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0answers
82 views

Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers?

Does anyone know a nice (combinatorial?) proof and/or reference for the following identity? $$\left( \frac{\alpha}{1 - e^{-\alpha}} \right)^{n+1} \equiv \sum_{j=0}^n \frac{(n-j)!}{n!} |s(n+1, ...
5
votes
3answers
278 views

morphism from a local ring of a scheme to the scheme

Let $X$ be a scheme, and $x \in X.$ Let $U=\text{Spec}(A)$ be an open affine subset containing $x,$ then we have the natural morphism $\mathcal{O}_X(U) \to \mathcal{O}_{X,x}$ inducing a morphism $ ...
0
votes
0answers
30 views

Working with affine Varieties

Hi guys I just wanted to hear some input on this, $A=(x^4+y^4-1)$ and $B=(p^2+w^2-1)$ are affine varieties in $C^2$. We want to show that if we apply the map $f(a,b)=(a^2,b^2)$ then $f(A) \subset ...
5
votes
0answers
315 views

Higher dimensional analogue for Riemann Hurwitz formula

There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and ...
5
votes
1answer
123 views

Showing closed immersions are stable under base extension without using that they are affine.

This question is based on question $3.11$ from chapter $2$ of Hartshorne, found on page $92$. Part $a)$ of said question asks to show that closed immersions are stable under base extension. In other ...
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0answers
35 views

Is smoothness of $X\to Y$ for noetherian $X$ a local property on $X$?

Let $X$ be a noetherian scheme. If $X$ is regular, then the scheme $\operatorname{Spec}(\mathcal{O}_{X,x})$ is regular for all points $x\in X$. I wonder if something analog is true for smoothness of a ...
1
vote
1answer
59 views

Analogue of splitting field in several variables

Let $k$ be a field, and $P \in k[X]$. Consider the extensions $k \subset L \subset K$, where $L$ is a splitting field for $P$ over $k$ and $K$ is the algebraic closure of $k$. Then (by definition) all ...
0
votes
1answer
25 views

CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?

In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to ...
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0answers
43 views

Covering of $\mathbb{P}^n$ and the complement of a point

Let $p$ be a closed point in $\mathbb{P}^n$ for some integer $n$ and $\{U_i\}$ be an affine open covering of $\mathbb{P}^n\backslash p$. Does there exists an open set in the covering, say $U_0$ for ...
6
votes
2answers
177 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
2
votes
1answer
53 views

Any rational map can be extended to codimension one.

If I understand correctly: Given a rational map $f$, between two (smooth) varieties $X$ and $Y$, with indeterminacy locus $\Sigma$ of codimension 1 in $X$, then $f$ can be extended to a regular map ...
2
votes
1answer
67 views

Connected vs irreducible Variety

I am asking if there is any particular criterion for a connected component of given variety to be irreducible (you can assume suitable conditions on the variety) thanks
2
votes
2answers
345 views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
1
vote
1answer
51 views

Open embedding and localization.

Let $X, Y$ be algebraic varieties. If we have an open embedding $X \hookrightarrow Y$, then we have a map $\mathbb{C}[Y] \to \mathbb{C}[X]$. Is $\mathbb{C}[X]$ a localization of $\mathbb{C}[Y]$? For ...
2
votes
1answer
137 views

“Equivalent” definitions of the gluing axioms

I tried to convince myself that the two caracterizations of a presheaf that is a sheaf given in wikipedia are equivalent but I couldn't. (F presheaf and notations from wiki) Let's take a simple ...
0
votes
0answers
42 views

Locally free sheaf on Cohen-Macaulay scheme and Serre's criterion

Let $X$ be a projective locally Cohen-Macaulay scheme and $\mathcal{F}$ be a locally free sheaf on $X$. If I understand correctly the definition of Serre's criterion $S_k$, $\mathcal{F}$ satisifies ...
2
votes
2answers
45 views

Domain of rational map $\mathbb{P}^2 \to \mathbb{P}^2$

Let $\phi:(t_0:t_1:t_2) \mapsto (\frac{1}{t_0}:\frac{1}{t_1}:\frac{1}{t_2})$. I think that we cat extend $\phi$ to rational map $\hat{\phi}$ with domain:$\mathbb{P}^2-\{(1:0:0),(0:1:0),(0:0:1)\}$. How ...
0
votes
0answers
28 views

Rational section of the canonical line bundle of a smooth curve

Let $C$ a complex Riemann surface with genus $g>0$, $L$ a theta characteristic on $C$ i.e $L \in Pic(C)$ such that $L^2 \equiv \omega_C$ where $\omega_C$ is the caninical line bundle on C and ...
2
votes
0answers
66 views

Understanding the Falting's Theorem

I'm an undergraduate student of mathematics, but soon I'll graduate, and as a personal project I want to understand Falting's Theorem, specifically I want to understand Falting's proof; but yet I have ...
3
votes
0answers
50 views

Geometric interpretation of Ideals in a Prime ideal

I have been told this has a geometric meaning, $I_j \in F[x_1,...x_n]$ be ideals such that $\cap_1 ^n I_J= P$ for P been a prime ideal, then we know that $P= I_j$ for some j=1,..n My Understanding I ...
0
votes
1answer
31 views

Finite étale cover of $\mathbb{P}^1$ has finitely many connected components?

I am reading Hartshorne's proof of $\mathbb{P}^1$ being simply connected as a scheme. It seems one ingredient of the proof is that if $X\rightarrow\mathbb{P}^1$ is an étale covering, then X has only ...
4
votes
0answers
105 views

What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow ...
1
vote
2answers
103 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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0answers
75 views

Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
4
votes
1answer
117 views

Fibres of the base change of a scheme

I am trying to gain a better understanding of the notion of fibre products of schemes. Two major applications that I've began to study are: 1) Making an $S$-scheme $X$ into an $S'$-scheme via a ...
0
votes
2answers
38 views

$V=\mathcal{Z}(xy-z) \cong \mathbb{A}^2$.

This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. ...
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0answers
20 views

Finding the “elbow” in a set of numbers

I have point X on a map. I have other points also (call them A, B, C and so), and I know how far away they are from point X. For example: A: 1 unit; B: 2 units; C: 2 units; D: 9 units. I want to ...
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votes
0answers
35 views

In $P^n$(projection of $C^{n+1}$) is a variety isomorphic to $P^1$ irreducible?

In $P^n$(projection of $C^{n+1}$) is a variety isomorphic to $P^1$ irreducible? I think not because is the union of a line and a point at infinity
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0answers
25 views

$\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$

Let $f(x) \in k[x]$. Show that $\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$ if and only if $f$ is the product of distinct linear factors in $k[x]$. Here, $\mathcal{Z}$ is the zero locus and ...
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0answers
17 views

Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
1
vote
1answer
52 views

Definition of a smooth complete integral pointed algebraic curve

Can anybody give me a reference to understand the definition of "a smooth complete integral pointed algebraic curve"? I'm beginning to study the paper "upper bounds for the dimension of moduli spaces ...
6
votes
1answer
62 views

Visual understanding for “the genus” of a plane algebraic curve

I am trying to understand the genus of an algebraic curve in the complex plane $\mathbb{C}P2$. I am looking for a visual or intuitive understanding. The difference between a sphere and a torus as a ...
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0answers
19 views

Dual polyhedral cone

The dual of a polyhedral cone is given by $\sigma^\vee =\{m \in M_\mathbb{R} \mid \langle m,u\rangle =0 \ \forall \ u \in \sigma \}$ Where $M_\mathbb{R}$ the abelian group of all one parameter ...
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0answers
53 views

Is projective space really a moduli space for lines through the origin?

The Wikipedia page for Moduli spaces states that real projective space $\mathbb{RP}^n$ is a moduli space which parametrizes the space of lines in $\mathbb R^{n+1}$ passing through the origin. ...
0
votes
1answer
49 views

a well defined map …

Consider a variety $V$ in $\mathbb{A}^n$, $I=I(V) \subset k[X_1, \cdots, X_n]=R$ ($k$ a field) and $P \in V$. We define the following : \begin{equation*} \mathcal{O}_P(\mathbb{A}^n) = \left\{ \left[ ...
4
votes
0answers
29 views

What are Mumford's 'moduli topologies'?

I've been reading Mumford's Paper 'Picard Groups of Moduli Problems'. Stated in modern language, the most famous result from the paper is that the moduli stack of elliptic curves has Picard group ...
4
votes
0answers
19 views

Nonsigular curve of degree $3$ in $\mathbb P^2$ over a field of characteristic $3$ [duplicate]

I am trying to do problem $1.5.5$ from Algebraic Geometry by Robin Hartshorne. The problem states: For every degree $d>0$, and every $p=0$ or a prime number, give the equation of a nonsingular ...
2
votes
1answer
49 views

Surjective morphism of varieties with finite fibers but not “finite”

Let $X$ and $Y$ be affine varieties, and $f : X \to Y$ a dominant regular map. Following Shafarevich, I will call $f$ finite if the induced map on coordinate rings is integral. One consequence of ...
1
vote
1answer
36 views

How does one show something is not an affine variety

Sorry for the random question, in a class we are talking about affine varieties. I have a problem trying to show a set of points in $R^2$ is not an affine variety. I just wanted to ask what is the ...
4
votes
1answer
93 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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votes
0answers
16 views

Morphism of schemes determined by their induced maps of $Z$ valued points

I am doing an exercise that states: morphism of schemes $X \rightarrow Y$ is determined by their induced maps of $Z$ valued points, as $Z$ varies over all schemes. I am a bit confused with this ...