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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2
votes
1answer
76 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 ...
17
votes
3answers
812 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
332 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 ...
8
votes
1answer
567 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
votes
2answers
167 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 ...
2
votes
1answer
109 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
57 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
1answer
62 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 ...
7
votes
1answer
727 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
vote
0answers
109 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
votes
1answer
131 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 ...
1
vote
0answers
37 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 ...
0
votes
0answers
116 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
votes
1answer
95 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
4answers
211 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
85 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 ...
7
votes
0answers
102 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
370 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
276 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
276 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
194 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 ...
8
votes
1answer
270 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 ...
1
vote
0answers
47 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 ...
12
votes
2answers
1k 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 ...
11
votes
2answers
196 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 ...
6
votes
1answer
114 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
94 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
95 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
332 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
votes
1answer
205 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 ...
3
votes
1answer
252 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
562 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
53 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
98 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
1answer
232 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
88 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) ...
6
votes
1answer
326 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
164 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
votes
0answers
59 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: ...
0
votes
0answers
65 views

$C^{\infty}$ 1-form on a Riemann surface is unique.

Let $X$ be a Riemann surface and $\mathcal{A}$ be a complex atlas on $X$. Suppose that $C^{\infty}$ 1-forms are given for each chart of $\mathcal{A}$, which transform to each other on their common ...
5
votes
1answer
220 views

hyperelliptic curve on abelian variety

There is serious flaw in the following argument, but I have yet to see what it is: Let $C$ be a genus $g$ hyperelliptic curve in an abelian variety $A$ (over an algebraically closed field with ...
4
votes
2answers
212 views

Every curve in $\mathbb{A}^3$ is the zero-locus of $3$ polynomials

I have the following problem: Let $X \subset \mathbb{A}^3$ which doesn't contain a vertical line and let $g \in K[x,y]$ such that $g$ hasn't any double factor and the clousure of the projection of ...
8
votes
2answers
159 views

Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
3
votes
1answer
104 views

Factorization over a relatively minimal surface

Let $A$ be a DVR with an algebraically closed residue field $k$. Consider a morphism $f: X \to Y$ of arithmetic surfaces (regular integer projective and flat schemes over $A$ of dimension 2), such ...
7
votes
2answers
303 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
6
votes
1answer
681 views

Basis for the Riemann-Roch space $L(kP)$ on a curve

Let $C$ be a smooth algebraic plane complex curve of degree $d$ defined as the zero locus of a homogeneous polynomial $F$. Let $P\in C$. For a positive integer $k$ consider the divisor $$D=kP$$ ...
1
vote
0answers
53 views

Why these two field extension have the same index?

Let $h:C\rightarrow C'$ be a nonconstant morphism between two algebraic curves $C,C'$over $k$, and let $h^*:\bar{k}(C')\rightarrow \bar{k}(C)$ be the pullback of $h$ given by $f\mapsto f\circ h$. My ...
3
votes
0answers
72 views

Is the number of automorphisms of a hyperelliptic curve bounded

Certainly, if we fix the genus $g$ of a curve $X$, we have $\# $Aut$(X) \leq 84(g-1)$. Let $X$ be a hyperelliptic curve. Is there a bound on $\#$Aut$(X)$? (Note that I do not want to fix the genus!) ...
3
votes
0answers
63 views

Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
6
votes
2answers
243 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...