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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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
votes
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
votes
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 ...
0
votes
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
votes
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
40 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
votes
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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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
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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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
votes
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
votes
1answer
35 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
45 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
68 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 ...
3
votes
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
67 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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votes
3answers
189 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
40 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
50 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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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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0answers
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
votes
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
60 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 ...
3
votes
1answer
48 views

Hom functor of quasi-coherent sheaf maybe not quasi-coherent

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give ...
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votes
1answer
28 views

meaning of definition of intersection transversally (in Hartshorne book)

Let $X$ be a surface, $C,D$ be curves on $X$ and ley $p \in C\cap D$. In Hartshorne book, $C$ and $D$ intersect transversally at $p$ if the local equations $f,g$ on $C,D$ at $p$ generate the maximal ...
3
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0answers
32 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
2
votes
1answer
25 views

Equivalence of weak forms of Hilbert's Nullstellensatz

The version of the Nullstellensatz with which I am familiar states that if $K$ is an algebraically closed field, and $f_1,\dots,f_n\in K[X_1,\dots,X_m]$, then the family $\{f_i\}$ has a common zero ...
3
votes
1answer
44 views

Projective points of a Fermat Curve

This is a problem from my coding theory book which I am trying to wrap my head around. Consider the curve $f_3F(q)$ given by $x^3+y^3+z^3=0$ A) Find the three projective points (x:y:z) of $P^2(F_2)$ ...
6
votes
0answers
45 views

Rank two vector bundle on $\mathbb P^1$, with trivial canonical bundle

I would like to show that the total space $X$ of the rank $2$ vector bundle $E=\mathscr O_{\mathbb P^1}(-1)\oplus\mathscr O_{\mathbb P^1}(-1)$ on $\mathbb P^1$ has trivial canonical bundle $\omega_X$. ...
3
votes
1answer
93 views

Exactness of the pullback of the Euler sequence

Let $(L,V)$ be a base point free $g^r_d$ on a curve $C$. Then we have an exact sequence $0\rightarrow M_{L,V}\rightarrow\mathcal{O}_C^{r+1}\xrightarrow{\theta} L\rightarrow 0$, where we just let ...
0
votes
2answers
150 views

Non-cohomological proof that a quasi-coherent sheaf over an affine scheme is quasi-flasque

Let $\mathcal F$ be a quasi-coherent sheaf over an affine scheme $X$. Let $0 \rightarrow \mathcal F \rightarrow \mathcal G \rightarrow \mathcal H \rightarrow 0$ be an exact sequence of sheaves on $X$, ...
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votes
1answer
27 views

Möbius transforms with a common fixed point

Let $f,g$ be two Möbius transformations with a common fixed point $z_0$. Show that the Möbius transformation $f \circ g \circ f^{-1} \circ g^{-1} $ is either parabolic or the identity. Möbius ...
2
votes
1answer
33 views

Line Meeting a Plane Curve at One Point

Given a curve (smooth, projective, irreducible) $X$ in $\mathbb{CP}^2$, this curve meets all other curves in the same space. Generically, it will meet a line (a copy of $\mathbb{CP}^1$ in ...
7
votes
2answers
443 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
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vote
2answers
42 views

space of sections of homogenious spaces

Let $G/H$ be a homogeneous space and then for homogeneous line bundle $L$ of $G/H$ the space of sections can be written as functions related to character of $H$. what about $\Gamma (G/H, L^2)$. then ...
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1answer
49 views

Definite integral of product of two bessel functions of different order and different argument

What is the solution of the integral: $\int_0^a J_m(k_2\rho)J_{m+1}(k_1\rho)d\rho$ where the integer $m\geq0$
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0answers
24 views

Why is the restricted nullcone a variety?

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted ...
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vote
2answers
75 views

Product topology of Affine Varieties

I want to prove that $X \times Y \subseteq \Bbb A^{2n}$ is an affine variety, given that $X,Y \subseteq \Bbb A^n$ are affine varieties. Is this proof correct? Since both $X$ and $Y$ are affine ...
3
votes
1answer
38 views

Showing the set of diagonalizable matrices is constructible

Identifying $M_n(k)$ with $k^{n^2}$ with $k$ algebraically closed, I am asked to show that the subset of diagonalizable matrices, $D_n$ is constructible. Constructible is defined as being the finite ...
8
votes
4answers
106 views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
4
votes
1answer
46 views

Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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0answers
35 views

Irreducible Linear Subspace

Let k be an infinite field. Prove that any linear subspace of $A_k^n$ is irreducible. My first question is, what would a linear subspace be? Is is a variety that is generated by linear equations? ...
0
votes
2answers
41 views

Singular points of a variety

I am trying to understand the proof of this result as formulated in chapter 1 of Hartshorne's Book : Let $Y$ be a variety. Then the set $\mathsf{Sing}(Y)$ of singular points of $Y$ is a proper ...
1
vote
1answer
48 views

Intersection of all maximal ideals containing a given ideal

Let $I$ be a proper ideal in $k[x_1,....,x_n]$, where $k$ is an algebraically closed field. Show that $\sqrt{I}= \cap M$, where $M$ runs through all maximal ideals containing $I$. I am confused ...
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vote
1answer
25 views

Irreducible Variety of Irreducible Polynomial

Prove that if $f \in k[x_1,...x_n]$ is an irreducible polynomial, then the variety $V(f) \subseteq A^n_k$ is an irreducible variety. Basically, I think that I want to prove that the ideal which ...