The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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2
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1answer
46 views

Noetherian local ring, detail in theorem 1.3.16 in Liu

I can't understand a detail in the proof of theorem 1.3.16 in Liu. The theorem is: let $(A,\mathfrak{m})$ a Noetherian local ring, $\hat{A}$ its $\mathfrak{m}$-adic completion, $(B,\mathfrak{n})$ an ...
2
votes
2answers
49 views

Irreducible Curve (Variety)?

I want to construct an irreducible variety of a plane curve. I would like it to be of the curve $f(x,y)=xy^s$, where s is an arbitrary integer. Since the ideal of this would be generated by one ...
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0answers
19 views

The Chow variety of conics in $\mathbb{P}^{3}_{k}$

Consider the family of all conics in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Such curves all have degree $2$ and genus $0$, and they can be uniquely defined to be all curves in ...
7
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1answer
188 views

Total space of line bundle $\mathcal{O}(1)$ same as blow up of plane?

We recall the following facts about total spaces of bundles: Let $X$ be a scheme and $\mathcal{E}$ an invertible sheaf on $X$. The total space of $\mathcal{E}$, $\Bbb{V}(\mathcal{E})$ is defined as ...
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0answers
58 views

Complex line bundle at symplectic manifold

Let's say that there is a symplectic manifold $(M,\omega)$ with condition of $[\omega / 2\pi ]\in H^2(M;\mathbb{Z})$. Then in what condition can I get a complex line bundle $L\twoheadrightarrow M$ in ...
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57 views

$A'_{m'}$ is a finitely generated $A_{m}$-module?

Let $A$ be a finitely generated $k$-algebra that is a domain. Let $A'$ be the integral closure of $A$ in $\operatorname{Frac}(A)$. By finiteness of integral closure $A'$ is a finitely generated ...
2
votes
1answer
26 views

Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
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1answer
63 views

help in an example.

In page 226 of David Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry there is an example which I need help in some parts of it: why $codim I= 1$? why $dim M = dim R = ...
3
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0answers
35 views

Hilbert Scheme and Chow variety in the case of Conics in $\mathbb{P}^{3}$

My question concerns the relationship between chow varieties and hilbert schemes in the case of conics in $\mathbb{P}^{3}_{k}$. More precisely, consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, ...
5
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0answers
42 views

Mittag-Leffler Problem

We have: $X$ a compact Riemann surface defined by $y^{2}=1-x^{6}$ and $P=(0,1) \in X$ a point given in local coordinates $(x,y)$. Furthermore, we have a meromorphic function $f(x,y)=y/x$ such that $f ...
4
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1answer
97 views

What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
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0answers
29 views

Determining how accurate an ellipse fit is

So I have an image of bacteria particles which are often shaped very irregularly with many grooves. Im trying to fit ellipses onto these particles so I can get a better, more smooth analysis of the ...
3
votes
1answer
57 views

Weird definition of Kodaira-Spencer map (What's a relative Kähler differential on a manifold?)

When I was reading "Advances in Moduli Theory" by Shimizu Yuji, I´ve found a weird way of writing the Kodaira-Spencer map $\rho$. For a given analytic family of complex compact manifolds $\pi ...
3
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1answer
105 views

When does a field extension canonically determine a morphism of schemes?

If I have an extension $L/K$ of number fields, then I can take the inclusion $\mathcal{O}_K \hookrightarrow \mathcal{O}_L$ and get a morphism of "curves" $\operatorname{Spec} \mathcal{O}_L \to ...
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0answers
15 views

height to width ratio decimal

For a shape of either rectangle or eclipse. Given the Height to Width ratio in decimal and the area of the shape. How is length and width calculated ?. Example : Area = 20 Height to Width ratio : ...
4
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1answer
35 views

Fixed points of the torus action on $\textrm{Hilb}_n(\mathbb C^2;d)$

On the affine plane $\mathbb C^2$ we have the action of the torus $T=(\mathbb C^\times)^2$ given by rescaling: $$(t_1,t_2)\cdot (a,b)=(t_1a,t_2b)\in\mathbb C^2.$$ This action extends to the Hilbert ...
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0answers
36 views

Chern class of line bundle and vector bundle

Let $L$ is a Line bundle and $E$ a vector bundle of rank $r$ then how can we prove that $$c_1(L\otimes E)=rc_1(L)+c_1(E)$$ where here $c_1$ means first chern class
3
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1answer
67 views

Do rational functions separate points?

Let $X$ be an irreducible, normal variety over an algebraically closed field of characteristic zero. Let $x,y\in X$ be two points such that $f(x)=f(y)$ for every $f\in K(X)$ which is defined at $x$ ...
2
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1answer
54 views

Existence of $g_{d}^{r}$ implies existence of which $g_{d'}^{r'}$'s?

Suppose I have a Riemann surface with a $g_{d}^{r}$. I am wondering what $g_{d'}^{r'}$'s exist for sure. For instance, since $h^{0}\left(L\left(-p \right)\right)$ is either $h^{0}\left(L\right)$ in ...
2
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1answer
25 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
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2answers
29 views

How do we define touching lines?

If two curves are touching at one point and intersect one another, how do we define it? If two lines are touching at a point then $L\cap K=\{q\}$ for two lines L and K and q is the touching point. ...
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25 views

Find the third platonic solid

So here is one math problem I cannot seem to get a grip on: We have one hexahedron (No. 1) and one dodecahedron (No. 2) plus one third solid that has either 4/6/8/12 or 20 sides. Each side has a ...
2
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23 views

moduli spaces of vector bundles

Let $X$ an elliptic curve and $M(r,d)$ the moduli space of S-equivalence of semistable bundles over $X$. I'd like to prove that if $(r,d)= \eta > 1$ then $M(r,d) \simeq Div^{\eta}(X)$, where ...
3
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1answer
42 views

Fermat Curve example and questions from coding theory.

I've been studying the basics of Algebraic Geometry for coding theory using the Pless-Huffman book. However since this is mostly self study, and without good resources I still feel a little shaky on ...
4
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1answer
52 views

Working out an example of a Chern class

I'm trying to understand page 161 of Fulton's "Young tableaux" in an explicit example. I'm looking at flags in $\mathbb{C}^4$, which I think of as flags in $\mathbb{CP}^3$ (and I'm really just able ...
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0answers
40 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
2
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1answer
40 views

Intersection Multiplicity and Multiplicity of Zeros in Polynomial

I study coding theory and we use the textbook Fundamentals of Error-Correcting Codes . In the section related to Algebraic Geometry Code, we need to compute Intersection Multiplicity of two curve in ...
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1answer
42 views

Regular immersion and canonical exact sequences between conormal sheaves

Let $f:X \rightarrow Y$, $g:Y \rightarrow Z$ be regular immersion of locally noetherian . One can then show that $g \circ f$ is a regular immersion. In the book I am reading it is stated that we have ...
3
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1answer
41 views

How to determine the local ring

In general, how does one determine a local ring. And in particular, how would one do it for $O_{A}(A $ \ $ \{(0)\})$, where A is 1-dim affine space in $\mathbb{C}$?
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23 views

Associated projective bundle on a projective scheme

Suppose that $X$ is a quasi-projective $k$-scheme with a right $B$-action (where $B$ is a linear algebraic group or Lie group) and that the quotient $X/B$ exists. Let the canonical projection $X \to ...
3
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1answer
56 views

Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ ...
3
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1answer
35 views

Image of Regular Map

Determine the image of the regular map $f: A^2 \to A^2$, $f(x,y)=(x,xy)$ and describe it from the point of view of topology. Would the image of f be $A^2$, because every point of $A^2$ is still in the ...
4
votes
0answers
39 views

Semistable vector bundles elliptic curve

Let $n=(r,d)$, r=r'n, d=d'n and $M(r,d)$ the moduli space of $S-$equivalence classes of semistable bundles of rank $r$ and degree $d$. How can I construct a finite morphism $M(r',d')^n\to M(r,d)$ ...
2
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0answers
26 views

Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
3
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0answers
63 views

Spectral sequence differentials

Let $\mathcal{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$ be complexes of coherent sheaves on a variety $X$. There is a spectral sequence ...
4
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1answer
36 views

Confusion with computing kernel of an isogeny between two elliptic curves

Consider the two elliptic curves $$E_3: y^2+y=x^3+x^2+x \enspace [Cremona:19A3]$$ and $$E_1: y^2+y=x^3+x^2−9x−15 \enspace [Cremona:19A1]$$ Let $\varphi$ be the $3$-isogeny from $E_3$ to $E_1$. I want ...
2
votes
1answer
47 views

Prime Spectrum of A Ring

I was given the definition that the spectrum of a ring R, denoted Spec R, is the set of the prime ideals of R. Then for an arbitrary subset $S \subseteq R$, then $V(S) = \{P \in SpecR | S \subseteq R ...
5
votes
1answer
74 views

Vector bundles on elliptic curves

Let $F$ be a stable vector bundle of degree $d$ and rank $r$, with $(r,d)$ coprime and $X$ an elliptic curve. I know that I can construct an extension $$0 \to H^0(F) \otimes O_X \to G \to F \to 0 $$ ...
3
votes
1answer
71 views

Weierstrass Point of a Riemann surface

I have that $X$ is a compact Riemann surface defined by the curve $y^{2}=1-x^{6}$ and a point $P=(0,1) \in X$ in the usual coordinates $(x,y)$. Ultimately, I want to solve a Mittag-Leffler problem on ...
3
votes
2answers
84 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
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0answers
36 views

Resolution of an affine variety

Did exists a method to find a resolution of singularities of an affine algebraic variety over $\mathbb{C}$ such that its resolution is affine too?
4
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1answer
68 views

Complex manifold with no divisors

I read in Griffith Harris P132 that a complex manifold of dimension greater than one can have no divisors on it at all. I want to find examples. Is there an example? Does the Hopf manifolds $S^1\times ...
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3answers
190 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
4
votes
1answer
41 views

Profinite completion of the fundamental group

Let $X$ be a complex algebraic variety. Is the functor of the algebraic fundamental group $X\mapsto \pi_1^{alg} (X)$ the composition of the functor of the classical fundamental group ...
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0answers
53 views

Hartshorne Lemma ( I 6.4 )

I have difficulty to understand the proof of this lemma : Lemma 6.4 Hartshorne Let $Y$ be a qausi-projectiue variety, let $P,Q\in Y,$ and suppose that $\mathcal{O}_P\subset\mathcal{O}_Q$ as subrings ...
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0answers
20 views

Dual Translation plane

An affine plane $\mathcal{A}$ is called a translation plane if the translation group of $\mathcal{A}$ operates transitively on the point set of $\mathcal{A}$. So how do we define the dual translation ...
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13 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
3
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4answers
49 views

Morphism between two $K$-schemes restricted to an affine subscheme

Suppose that $f:X\longrightarrow Y$ is a morphism between two $K$-schemes. If $U\subseteq X$ is an affine open set, then can we conclude that $f(U)$ is contained is some affine open subset of $Y$? ...
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votes
2answers
61 views

How to determine that a set of equations has a solution or infinite many solution or no solution?

I have a set of polynomials (the variables are $a,b,c,d,e,f,g,h,i,j,a_1,b_1,c_1,d_1,e_1,f_1,g_1,h_1,i_1,j_1$, the polynomials are $L_i$, $i=1,\ldots,20$): $$ aa_1=0 ; [L_1]\\ e a_1 + b e_1=0; [L_2]\\ ...
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0answers
24 views

Algorithm for finding nearest distance from a point to a curved surface in space

I need to write an algorithm which can find the nearest distance from a point in space to a 3D curved surface which is straight in vertical direction but its projection is an arc of a circle (Similar ...