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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77 views

differentials on formal schemes

Let $A$ be a topological ring, we say that it is pseudo-compact if : there is family of ideals $\Lambda_A$ which gives a basis of neighborhoods of $0$ and such that $A/\mathfrak{a}$ is artinian for ...
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votes
1answer
90 views

I want to learn MATHEMATICS [closed]

I want to learn MATHEMATICS complete from Basic level to master level in a detailed way. What should I do. You may help me. I am searching my answer on net but I didn't find my answer. Please help me ...
2
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0answers
10 views

The process of alternation on an n-polytope

I am currently working on a problem involving algebraic geometry and as a part of the research it would be helpful for me to understand the process of alternation, also called partial truncation, ...
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0answers
39 views

Dimension of embeddings of Segre variety (product of projective spaces)

The Segre map gives an embedding of the Segre variety $\Sigma_{n,m}$ (i.e. of the categorical product of two projective spaces of dimension $n$ and $m$) into a projective space of dimension $nm+n+m$. ...
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1answer
47 views

Question about solvable cocompact subgroups in linear algebraic group over a finite extension of the p-adic numbers

Let $Q_p$ be the p-adic numbers, where p is any prime number. Then $Q_p$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological field. Let $GL(n,Q_p)$ be the general linear ...
2
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1answer
26 views

Proving that a map is a birational equivalence

I am trying to prove that the map $\phi:P^1\to X = Z(x^2y^3-z^5)$, given by $[r:s]\mapsto [u^5:v^5:u^2v^3]$ is a birational equivalence, i.e. that there exists some map $\psi:X\to P^1$ such that ...
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1answer
31 views

Relation of “irreducible” polynomials and vareities

In $\mathbb{C} [x,y]$, is a variety irreducible iff the corresponding polynomial is irreducible?
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37 views

Is there a general way to have a polynomial in two variables over C (a plane curve) be irreducible?

Is there a general way to have a plane curve be irreducible? If the curve $C \in \mathbb{C} [x,y]$, would it be sufficient for it to factor into linear terms? What about if I have an equation of the ...
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0answers
19 views

Multiplicities of tangent line

What does it mean for a curve to have two tangent lines at a point such that the tangent lines intersect the curve with different multiplicities? Could someone give an example?
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1answer
43 views

If $S$ is the integral closure of $R$ in it's field of fractions and $S\subset R_{m}$ is $R_{m}$ integrally closed?

Let $R$ be a domain and $K$ be it's field of fractions. Let $S$ be the integral closure of $R$ in $K$. Let $M$ be a maximal ideal of $R$. If $S\subset R_{M}$ is $R_{M}$ integrally closed in $K$? My ...
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0answers
21 views

intersection theory on projective space

Let $\mathcal{L} := O(1) \in Pic(\mathbb{P}^d)$ be considered as an element of $CH^1(\mathbb{P}^d)$. What is its $d$-fold power in $CH^d(\mathbb{P}^d) = \mathbb{Z}$?
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54 views

Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
4
votes
1answer
37 views

Normalization bijective on smooth points?

Suppose we take an algebraic variety $X$ over $\mathbb{C}$ (I assume reduced). Is the normalization $$\pi:\tilde{X} \to X$$ always bijective on the smooth points of $X$?
4
votes
1answer
42 views

Is the fiber product of the connected component of a group scheme connected?

Let $G$ be a group scheme over a field $k$. Let $G^0$ be the connected component containing the identity. Is it true that $G^0\times_k G^0$ is connected? I know that this is true if $G^0$ is ...
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0answers
36 views

Check a non-projective morphism

The following is from the wiki: http://en.wikipedia.org/wiki/Proper_morphism Projective morphisms are proper, but not all proper morphisms are projective. For example, it can be shown that the ...
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votes
3answers
58 views

How to tell if algebraic set is a variety?

I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite ...
2
votes
1answer
51 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
2
votes
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
47 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 ...
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1answer
169 views
+100

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
57 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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0answers
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 ...
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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 $ ...
2
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1answer
62 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
34 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 $\mathsf{hilb}^{2t+1}_{3}$, ...
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0answers
40 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
votes
1answer
96 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
55 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
votes
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
13 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
votes
1answer
34 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
33 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
votes
1answer
66 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
50 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
votes
1answer
23 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
28 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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0answers
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 ...
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0answers
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 ...
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1answer
50 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
39 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 ...
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1answer
39 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
39 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 $\mathbb{C}$?
3
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0answers
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
votes
1answer
55 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
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0answers
38 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 $M(r',d')^n\to M(r,d)$ ...