An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...
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Definition of simple spectrum
From the book "Spinning Tops" by Audin, given Lax equation $[A_{\lambda},B_{\lambda}]$ where $\lambda$ is a parameter (so called spectral parameter), he claims that we have spectral curve ...
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Puiseux series and Resolution of Singularities
I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities.
So the Newton's method gives us a Puiseux series parametrizing the branches of ...
4
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3answers
50 views
Good source to learn about surface singularities?
I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there.
I checked out Miles Reid his lectures on ...
6
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1answer
46 views
Existence of a variety with prescribed properties
In these notes that give a proof of the Weil conjectures for curves, the author writes on page 17 that given a smooth projective curve $X$ over a finite field $k = \mathbb{F}_q$ for a fixed prime $q$, ...
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When is an intersection of varieties finite
Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
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32 views
deg functions and maps
For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman.
For algebraic functions on elliptic ...
9
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2answers
119 views
Is a divisor in the hyperplane class necessarily a hyperplane divisor?
Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$.
...
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0answers
12 views
How to find the dimension of linear system of curves of degree $d$
Consider two curves $C_1$ and $C_2$ in $\mathbb P^2 (\mathbb C)$ . How can i find the expected and real dimension of the linear system of cuves of degree $d$ passing through points lying on the both ...
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0answers
26 views
Solving the algebraic equations .
I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ .
Basically after taking the partial derivatives and doing some manipulations it reduces to
$$y^2 + (2-k)xz ...
2
votes
1answer
38 views
Extension of prime ideal in $k[V]$ to $\mathcal{O}_P(V)$ is prime?
Let $k$ be an algebraically closed field, $I\subset k[X_1,\cdots, X_n]$ be a prime ideal, $V=V(I) \subset \mathbb{A}^n$ a variety and $P=(a_1,\cdots, a_n)\in V.$ Recall that $\mathcal{O}_P(V)$ is the ...
3
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0answers
32 views
Computing a rational function at a point in terms of a uniformising parameter
I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
2
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1answer
37 views
Number of intersection multiplicity points .
I need help for the following problem :
Consider $C_1 = V(F_1)$ and $C_2=V(F_2)$ be algebraic curves in $\mathbb P (\bar K )$ (where $K$ is a field,) without a common component and $F_1, F_2 \in ...
10
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2answers
290 views
When do equations represent the same curve?
Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
3
votes
2answers
81 views
Intersection of smooth projective plane curves
I need to calculate the number of intersections of the smooth projective plane curves defined by the zero locus of the homogeneous polynomials
$$
F(x,y,z)=xy^3+yz^3+zx^3\text{ (its zero locus is ...
5
votes
1answer
61 views
The genus of the Fermat curve $x^d+y^d+z^d=0$
I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation.
Such curve is defined as the zero locus
...
2
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0answers
13 views
Large Intersection Multiplicity
A cubic curve, say, $x^3+y^3=1$ and some quadratic curve $f(x,y)=0$ generally have six intersection points in $\mathbb{CP}^2$.
Question: If all the intersection points coincide, what will be the ...
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0answers
23 views
What's the sense in a Hyperelliptic Riemann Surface?
Can someone explain me, possibly using some very intuitive ideas, of what kind of object a hyperelliptic Riemann Surface is? What's the goal of constructing it (my lecture on is was based in Miranda's ...
2
votes
1answer
32 views
Creating and using calibration factors
Perhaps simple question, but I (the simple) need some guidance. The following applies to a project ongoing and is a challenge in that I am not a math whiz!
As example, I wish to measure temperature ...
1
vote
0answers
40 views
Degree of morphism of quotient of upper half-plane
Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
5
votes
1answer
110 views
Circumference of a superellipse?
Could someone help me formulate the circumference of a superellipse?
$$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$
If it makes things easier, I'm considering only the cases $n>2$, and ...
1
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0answers
52 views
A funny condition for ampleness on a curve
Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
8
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1answer
81 views
Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free
Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
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0answers
28 views
Homogeneity of translated polynomial
I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9):
The multiplicity of the complex ...
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30 views
Some basic questions about Jacobians of curves
Let $C$ be a curve defined over $\mathbb{Q}$, of positive genus. Let $J$ denote its Jacobian. I would like to ask a couple of basic (I presume) questions:
0) Why is $J$ an algebraic variety?
1) For ...
4
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1answer
43 views
plane cubic with a singularity must have non-constant morphism from $\mathbb{P}^1$?
If $C$ is a plane projective curve which is defined by an irreducible homogeneous cubic polynomial and has a singularity, why must there be a nonconstant morphism $\mathbb{P}^1\rightarrow C$?
(I'm ...
5
votes
5answers
94 views
Formula with 2 points of inflection
$x^3$ has a point of inflection at $x=0$. How will you modify the formula to add a 2nd point of inflection at $x=1$?
Plot of $x^3$
Plot of $x^3(x-1)^3$
Update
The plot I am aiming to achieve ...
4
votes
1answer
52 views
n-canonical embedding
Let $C$ be a stable curve of genus $g>1$ and let $ \omega $ be its dualizing sheaf. Let $n$ be a integer larger than 2. Does anyone knows how to show that $\omega^{\otimes n}$ separates points and ...
6
votes
0answers
64 views
A Regular Map Has Finitely Many Ramification Points
Let $C,D$ be nonsingular projective curves, $f \colon C \to D$ nonconstant, $K = k(C), L = k(D)$, $d = \text{deg }f = [K:L]$, and of course $k$ algebraically closed. Furthermore let's suppose that ...
1
vote
2answers
47 views
Next Point in a Curve?
If I have a series of data points that can be plotted as a curve, but I don't know the underlying function responsible for this, how can I calculate the next data point in the curve?
The data points ...
2
votes
1answer
58 views
Elliptic curve as an intersection of quadrics
Let $E$ be an elliptic curve. If one starts with embedding associated with invertible sheaf $\mathcal{O}(3x)$ where $x$ is some point on $E$ then one gets cubic in $\mathbb{P}^2$ and this embedding is ...
6
votes
1answer
80 views
Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf
Let $B$ a smooth projective connected variety over $\mathbf C$.
Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero.
Does the converse hold? That is, suppose that $B$ ...
7
votes
2answers
106 views
Are endomorphisms of degree one always automorphisms
Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one.
Do I understand correctly that $\sigma$ is an automorphism?
I believe this ...
5
votes
1answer
49 views
Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf
Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
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0answers
34 views
$\frac{dy}{dx}$ of a parametric curve
Given $x = sin^2(t)$, $y = cos^2(t)$, I need to find $\frac{dy}{dx}$ in every non-singular point of the curve.
So $\frac{dy}{dt} = -2sin(t)cos(t)$ and $\frac{dx}{dt} = sin(2t)$.
To find the ...
6
votes
2answers
112 views
Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$
I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
6
votes
0answers
50 views
Families of curves over number fields
Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
5
votes
1answer
42 views
Why are finite unions of algebraic sets algebraic?
Suppose $F,G$ are polynomials in $k[x_1,\cdots, x_n]$ (k is a field). Let $$V(F) = \{ (a_1,\cdots,a_n)\in k^n : F(a_1,\cdots,a_n)=0 \}.$$
Then $V(F)\cup V(G) = V(FG),$ essentially because integral ...
2
votes
1answer
85 views
Reference request: Construction of $M_{1,0}$
Does anyone know a reference for the construction of the (Artin) stack $M_{1,0}$ and a result about the corresponding coarse moduli space? In Deligne-Mumford they construct $M_{g,0}$ when $g\geq 2$ ...
2
votes
0answers
57 views
Computing wedge product of two 1-forms.
Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic ...
3
votes
1answer
44 views
Birational Maps of Nonsingular Projective Curves
I'm trying to solve exercise I.6.7 in Hartshorne, stated (in part) here:
Let $P_1,\ldots,P_r, Q_1,\ldots,Q_s$ be distinct points of $\mathbb{A}^1$. If $\mathbb{A}^1-\{P_1,\ldots,P_r\}$ is ...
2
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1answer
55 views
Computing the intersection divisors of a smooth projective cubic curve.
Let $X$ be the smooth projective plane cubic curve defined by $y^2z=x^3-xz^2.$ Compute the intersection divisors of the lines defined by $x=0,y=0,$ and $z=0$ with $X$.
Here is an idea:
Any point ...
2
votes
1answer
89 views
Compute the principal divisors of a hyperlelliptic surface.
Let $X$ be the hyperelliptic surface defined by $y^2 = x^5-x.$ Note that
$x$ and $y$ are meromorphic functions on $X.$ Compute the principal divisors
div($x$) and div($y$).
We have the ...
4
votes
1answer
122 views
Computing the divisors of a meromorphic function defined by a hyperelliptic curve.
Let $X$ be a hyperelliptic curve defined by $y^2=h(x).$ Let $\pi : X\to \mathbb{P}^1$ be the double covering map sending $(x,y)$ to $x$. Let $\omega=\pi^{*}(dx/h(x)).$ Compute div$(\omega)$.
I ...
3
votes
1answer
41 views
Does this diagram of Chern classes and push forwards commute
Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
3
votes
1answer
39 views
Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$
Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...
7
votes
2answers
111 views
Singularities of Curves in Positive Characteristic
Given a collection of polynomials $\mathscr{F}\subset\mathbb{Z}[x_1,\ldots,x_n]$, we can associate to each prime ideal of $\mathbb{Z}$ an affine variety as follows:
$$
(p)\longmapsto ...
0
votes
1answer
75 views
Degree of sum algebraic functions
This question I have asked on mathoverflow already: http://mathoverflow.net/questions/123921/degree-of-sum-algebraic-functions
Let $C$ - curve, $f_1, f_2 \in K(C)$. How to prove that deg$(f_1 + f_2) ...
5
votes
1answer
77 views
How much do I need to learn before I can read about Toric varieties?
I have a copy of the book "Introduction to Toric varieties" by William Fulton, and over the next few months I'd like to make some progress on it.
As a first goal, I'd like to be able to read just ...
6
votes
1answer
84 views
Integral Closures and Affine Curves
Let $C$ be an irreducible affine curve with singular points, and let $A$ be its ring of regular functions. Since $C$ has singular points, $A$ is not integrally closed in its field of fractions, $K$. ...
3
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0answers
35 views
Spectrum theorem for p-adic matrix analysis
Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\
2: ...
