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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249 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
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1answer
557 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 ...
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1answer
52 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
95 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
222 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 ...
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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
315 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
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1answer
162 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, $K$. ...
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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: ...
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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 ...
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votes
1answer
216 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
211 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
157 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
295 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
653 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 $$D=kP$$ ...
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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
71 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
61 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 ...
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votes
2answers
239 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 ...
2
votes
2answers
191 views

Given an algebraic curve $F(x,y)=0$, why do the partial derivatives of $F(x,y)$ being zero at a point imply the plane curve has a singularity?

I'm looking at algebraic plane curves of the form $F(x,y)=0$ and trying to figure out why for points on the curve such that $\frac{\partial F}{\partial x} = \frac{\partial F}{\partial y}=0$, the plane ...
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0answers
41 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
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0answers
33 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
3
votes
1answer
559 views

Normalisation of an algebraic curve.

I need to compute explicitly the normalisation of a singular algebraic curve $C$ which is given by an explicit equation in $\mathbb{A}^2$. This task is mostly reduced to finding the integral closure ...
2
votes
1answer
216 views

proof that sum of ramification degrees is degree of morphism between curves?

If $X$, $Y$ are irreducible smooth projective curves over an algebraically closed field and $\alpha:X\rightarrow Y$ is a morphism, how do we prove that ...
2
votes
1answer
409 views

what is genus of complete intersection for: $F_1 = x_0 x_3 - x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete ...
2
votes
1answer
97 views

Hyperelliptic curve, injectivity of pullback homomorphism

Let $X$ be a hyperelliptic curve of genus $g$ (nonsingular, etc.) with a hyperelliptic cover $X \to \mathbb{P}^1$ corresponding to an invertible sheaf $\mathscr{L}$. The composition with the $(g-1)$ ...
3
votes
1answer
139 views

Base-point-free invertible sheaves on smooth projective curves

Let $C$ be a projective curve over $k$ (geometrically integral, nonsingular). I am confused about the following argument in Vakil's notes on algebraic geometry: (1) In (20.6.2), he writes: Fix a ...
2
votes
1answer
75 views

Pulling-back a divisor and reducing it

Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$. Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, ...
3
votes
0answers
33 views

Methods to prove that points do not lie on an algebraic plane curve

I have an infinite sequence of points in an affine plane and I want to show that these points do not lie on any algebraic plane curve. Are there any standard methods for doing this?
2
votes
1answer
191 views

Is the complement of an ample divisor always affine

Let $X$ be a projective variety and let $D$ be an ample divisor. Is the complement of the support of $D$ in $X$ affine? We can suppose $D$ is very ample. (Just replace it with a multiple.) I'm trying ...
4
votes
1answer
83 views

Why is the rank of $f_\ast L$ the degree of $f$

Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$? Here is my ...
4
votes
0answers
143 views

If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$

Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be ...
2
votes
1answer
91 views

Relating the genus of a curve to its degree, via $n$-canonical embedding.

Let $n\geq 3$ be an integer. If we embed a connected curve $C$ (e.g. a stable curve) of genus $g$ in $\mathbb P^N$ by an $n$-canonical embedding, i.e. using the very ample linear system $|nK_C|$, we ...
1
vote
1answer
235 views

extension of algebraic function field

Let $K$ be a field, $t$ a transcendetal element over $K$ and $F'|K(t)$ an infinite Galois extension. Hence I have a tower of extension $K\subset K(t) \subset F' $. Does exist a subextension $F$ of ...
0
votes
1answer
53 views

Finding the volume when a region is rotated about the $y$-axis?

My basic question is: When we think of the area under the graph and extending it in the 3 dimensions, we actually get a cylinder with height = $f(x)$, thickness = $dx$ and inner radius = $x$. Then the ...
3
votes
1answer
96 views

Curves in a linear system on a surface

I'm looking for references on a very classical question: Let $X$ be a compact surface and let $L \to X$ be an ample line bundle. We assume that $L$ has nonzero sections. Then the linear system $|L|$ ...
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vote
1answer
182 views

Approximating an algebraic curve using cubic bezier splines

Suppose I have an algebraic curve in its implicit form, i.e. described as the set of points $(x,y)$ where some polynomial $P(x,y)$ becomes zero. All of this is in the real Euclidean (or with minor ...
1
vote
1answer
115 views

dimension of moduli space of curves via Hodge structure

It is well-known that the moduli space of genus $g\ge 2$ curves $\mathcal{M}_g$ has dimension $3g-3$. This can be computed for example as $\dim_{C} H^1(C,T_C)$. Is is also known that the structure of ...
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0answers
105 views

Are there infinitely many rational functions of bounded degree and given ramification

It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism). Edit. The branch locus $B$ of $f:X\to ...
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vote
1answer
186 views

Equivalent conditions defining stable curves

I'm learning the basics about stable curves. Suppose we have a connected complex projective curve $C$, at worst nodal and of genus $g\geq 2$. Then I want to prove that the following are equivalent: ...
4
votes
1answer
281 views

Holomorphic Euler characteristics and topological Euler characteristics of curves.

I noticed that the holomorphic Euler characteristic $\chi(C,\mathcal{O}_C)=1-g$ of a smooth complex curve $C$ of genus $g$ is just a half of the topological Euler characteristic $\chi_{top}(C)=2-2g$. ...
2
votes
2answers
68 views

Polar plane curves algebraic

Let $h\in \mathbb{R}[x,y]$ be a nonzero polynomial and define a plane curve in polar coordinates as $r(\theta) = h(\cos\theta,\sin\theta)$. For all the examples I've looked at, it seems like we can ...
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vote
1answer
224 views

Intersection multiplicity as the dimension of a vector space

I'm trying to solve the following problem in Dino Lorenzini's book on arithmetic geometry: Let $f,g\in k[x,y]$ be coprime and assume that $P=(0,0)\in V(f)\cap V(g)$ is a nonsingular point of $f$. ...
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vote
2answers
2k views

Find X given Y in a cubic function.

Having asked this question on the math overflow boards one of the contributors suggested this may be a more appropriate forum. I have a cubic function in the form: $$y = ax^3 + bx^2 + cx + d$$ ...
4
votes
1answer
179 views

Curves not embeddable in the projective plane - examples?

In the chapter on curves in Hartshorne it is proved that every curve can be embedded in $\mathbb{P}_k^3$ and is birationally equivalent to a planar curve with at most nodes as singularities ...
4
votes
1answer
265 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
4
votes
1answer
55 views

Very ampleness of $\omega_{C}^n$

Let $C$ be a genus $g$ curve over complex numbers. How can I prove that $\omega_{C}^n$ is very ample for $n\ge2$ if $g=2$ and $n\ge 3$ if $g\ge 3$? Also, I wonder if this still true for other fields ...
2
votes
1answer
92 views

the divisor of a rational function

Is the section $df$ associated to a rational function $f$ on a curve $X$ a global section of the canonical sheaf $\omega_X$? I know its zeroes are the ramification points, but does it have poles?
2
votes
1answer
86 views

Why do number rings have no endomorphisms

This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question. Consider the projective line over a field. This has many ...