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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39 views

sheafification and quasi-coheret sheaf

My problem comes from this post: http://math.stackexchange.com/a/467252/115619 The example he gave is $j_!\mathcal O_U$. But I think $j_!\mathcal O_U$ is the sheafification of the presheaf $V \to ...
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34 views

Elliptic Curve Group and Multiplicative Inverse of an element.

Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, ...
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27 views

induced isomorphism on the blow up

Let $k$ be algebraic closed field and let $\mathbb{P}^2$ the projective space over $k$ of dimension 2. Consider the birational map $$f:\mathbb{P}^2 ---> \mathbb{P}^2, [x_0, x_1, x_2] \mapsto ...
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1answer
74 views

Flatness under reduction

Suppose that $f : X \to Y$ is a flat morphism of schemes. Is $f_\text{red} : X_\text{red} \to Y_\text{red}$ necessarily flat? Are there any hypotheses that would guarantee this?
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1answer
30 views

Working with an Affine Variety and Maps

We have a $\phi :\mathbb C^4 \rightarrow\mathbb C^4$ $$\phi (a_1,a_2:b_1,b_2) = \left(\begin{array}{cc} a_1b_1 & a_1b_2 \\ a_2b_1 & a_2b_2 \end{array}\right) $$ We want to argue there exists ...
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1answer
33 views

morphisms of projective varieties and induced ring homomorphisms

If $\phi: X \rightarrow Y$ is a morphism of affine varieties, then we get an induced homomorphism $\tilde{\phi} : A(Y) \rightarrow A(X)$ of their affine coordinate rings (and vice versa). Question ...
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24 views

homogeneous coordinate rings of isomorphic projective varieties

As i understand, there are two notions of "equivalence" of projective varieties. The first is projective equivalence, under which projective varieties $X,Y$ of $\mathbb{P}^n(k)$ are mapped to each ...
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1answer
50 views

Punctual Hilbert scheme of four points

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know ...
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1answer
34 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let ...
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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 ...
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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 ...
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1answer
32 views

The number of conditions on D that $mult_x(D)=k$

Let $X$ be a smooth projective variety of dimension $n$ and $H$ an ample divisor on $X$. I want to know the number of conditions on $D\in |mH|$ that $x$ be a point of multiplicity$=k$ on $D$. The ...
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21 views

Normalization of curve

How do to normalize the curve $ax^2+y^2=1+bx^2y^2$ (hard exercize)? I tried the substitution $t=xy$, $u=y$. But I get $at^2/u^2+u^2=1+bt^2$ can i multiply it on $u^2$.
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1answer
57 views

How to show that an object is a discrete valuation ring? (Fulton, Exercise 2.14)

I need some help to solve the following problem that appears on page 31 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ V = \mathbb{A}^1 $, $ \Gamma (V) = k[X] $, $ K = ...
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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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30 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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17 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 ...
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85 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, ...
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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 $J$ ...
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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 ...
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34 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 ...
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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 ...
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53 views

Locus of image of point in a line.

I am given the following question: Find the locus of the image of the point $(2,3)$ in the line $$\text{L}:(2x-3y+4)+k(x-2y+3)=0$$ where $k$ is any real number. Attempt at solution. I ...
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67 views

Dimension of the set of forms of degree $d$ in the homogeneous coordinate ring of $V$ (Fulton, Exercise 4.10)

I need help to solve the following problem that appears on page 55 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ R = k[X,Y,Z] $, $F \in R$ an irreducible form of degree ...
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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 ...
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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 ...
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1answer
52 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 ...
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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 ...
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0answers
68 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 ...
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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 ...
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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 ...
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0answers
76 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.
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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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1answer
68 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
3
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1answer
53 views

Sheaf cohomology of ringed space

Why sheaf cohomology on a ringed space $(X, \mathcal O_x)$ are defined as derived functors to $\Gamma: \mathfrak{Ab}(X) \to \mathfrak{Ab}$, not to $\Gamma: \mathfrak{Mod}(X) \to ...
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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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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
18 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 ...
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20 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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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. ...
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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 ...
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29 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 ...
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1answer
50 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[ ...
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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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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 ...
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1answer
50 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 ...
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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 ...
4
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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 ...
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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 ...