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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8
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2answers
156 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
102 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
282 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
620 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
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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
69 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
60 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
234 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
190 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 ...
1
vote
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, ...
2
votes
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
546 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
213 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
392 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
138 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
74 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
189 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
234 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 $|L|$ ...
1
vote
1answer
180 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
114 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 ...
2
votes
0answers
103 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 ...
1
vote
1answer
180 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
269 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 ...
1
vote
1answer
220 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 $f$. ...
1
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
177 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
255 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
91 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 ...
1
vote
0answers
80 views

Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
6
votes
1answer
127 views

Is $M_g$ NEVER proper? And why does $T_g$ contain products?

Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general ...
1
vote
1answer
38 views

Does pull-push by the quotient map of divisors on the symmetric square of an algebraic curve induce multiplication-by-2?

Let $C$ be a smooth projective algebraic curve over a field $k$ of characteristic different from 2, and let $C^2 = C \times C$ be the square of $C$. Let $C^{(2)} = \operatorname{Sym}^2(C)$ be the ...
1
vote
1answer
128 views

Are smooth relative curves over an arbitrary base normal?

Let $X/S$ be a smooth, projective scheme of relative dimension one over a scheme $S$ (which we may assume is affine Noetherian, but need not be reduced nor irreducible nor even connected). For $s \in ...
2
votes
2answers
68 views

Factorizing rational functions of curves

Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$. Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
6
votes
2answers
164 views

A question on Newton's “theorem about ovals”

This is a question about a result from Newton's Principia. It says, roughly, that the if you intersect lines $ax + by + c$ with a smooth, closed, convex curve, then the area of the curve that the line ...
1
vote
1answer
627 views

Fitting curves to a set of points

Basically, I have a set of up to 100 co-ordinates and their orders, in a 2D plane, along with the desired tangents to the curve at the first and last point. I have looked into various methods of ...
3
votes
1answer
81 views

Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
0
votes
1answer
131 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
7
votes
1answer
155 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
3
votes
1answer
220 views

the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
3
votes
1answer
124 views

How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...