The algebraic-curves tag has no wiki summary.

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### Schottky locus in genus 2

Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized ...

**2**

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**2**answers

413 views

### Quartic Space Curves

It is an exercise in Hartshorne to classify nonsingular quartic curves in projective 3-space. I am interested in what happens when we allow singularities. In particular, I am looking for an ...

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**2**answers

409 views

### What is the advantage of the approach of valuations to the Riemann-Roch Theorem for curves (a la Chevalley)? AKA theory of algebraic functions in one variable

Hello,
Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any algebraic geometry ...

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382 views

### Max Noether's residual intersection theorem (Fundamentalsatz): importance and applications

I would like to ask a couple of naive question about the following theorem of Max Noether:
http://en.wikipedia.org/wiki/AF%2BBG_theorem
In the book of Fulton, page 60
...

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**0**answers

315 views

### Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...

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355 views

### Is there an algorithm which determines if a curve has good reduction outside a given set of primes

Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.
Is there an algorithm which tells you if $X$ has good reduction outside ...

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**2**answers

1k views

### Nagata's conjecture, Seshadri constant

What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric ...

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**1**answer

278 views

### Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...

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vote

**1**answer

233 views

### Manin-Drinfeld and constructing a finite morphism with two given ramification points

Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian.
Can we always find a ...

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votes

**1**answer

234 views

### Does each finite morphism of curves have a model whose minimal resolution is semi-stable

Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$.
Question. Does there exist a finite field extension $L/K$ and a regular model ...

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**1**answer

180 views

### Is there an easier argument to prove that almost all of these curves have no semi-stable reduction

Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X_d$ over $K$ by the equation $dy^2 = F(x)$. Note that the curves $X_d$ ...

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**1**answer

275 views

### What is the mod l monodromy of a generic trigonal curve?

For a hyperelliptic curve H, the mod 2 monodromy is smaller than $GSp_{2g}(F_2)$ -- since the two torsion of the Jacobian H is generated by differences of Weierstrass point, the monodromy of a generic ...

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560 views

### Automorphisms of Riemann surface and mapping class

For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?

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431 views

### Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...

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**1**answer

191 views

### Is there an upper bound and a lower bound on the contribution to the genus, for a singularity of codimension k?

To make my question precise, suppose you have a complex curve locally given by
$$f(x,y) =0 $$
and $f$ has singularity of type $\chi_k$ at the origin. The codimension of
this singularity is $k$. Let ...

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759 views

### Reducibility (or not) of algebraic curves [closed]

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...

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471 views

### Can we always find a curve which doesn't have semi-stable reduction

Let $K$ be a number field and let $g$ be a positive integer. Does there exist a smooth projective geometrically connected curve $X/K$ of genus $g$ such that $X$ does not have semi-stable reduction ...

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196 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...

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**1**answer

354 views

### Are Weierstrass points algebraic

Let $X$ be a compact connected Riemann surface of genus $g>0$.
Suppose that $X$ can be defined over a number field (as an algebraic curve). Then, is it clear that each Weierstrass point of $X$ is ...

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218 views

### Comparing heights of rational points on curves through covers

Let $a$ be a closed point in $\mathbf{P}^1_{\overline{\mathbf{Q}}}$.
Let $Y \cong \mathbf{P}^1_{\overline{\mathbf{Q}}} $ and let $\pi:Y\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ be a finite morphism ...

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201 views

### Is there an algebraic analogue of the degeneration of riemann surfaces in M_g

Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite alot. The idea is to ...

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267 views

### The locus of cyclic covers in the moduli space of curves

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. Let $Z$ be the closure in $\mathcal{M}_g$ of the set of smooth curves of genus $g$ which are a cyclic cover of the projective ...

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551 views

### $Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]

I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...

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421 views

### Weil bound for characters sums. (reference-request )

Do you know on any good reference on Weil bound for charcter sums over algebraic curves.
I prefer reference which assume few previous knowlage.

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**1**answer

579 views

### Genus of algebraic curves with unknown degree

I am not sure if this is a valid question but posting any way:
Say I am over $\mathbb{F}_{p}$ for a prime $p$.
I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence ...

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**1**answer

169 views

### Does the Newton polytope characterize the equisingular i.e topological type?

Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some ...

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473 views

### the blowing up of a plane curve playing me tricks.

Sorry for the easy question but this is driving me crazy. Consider the blowing up of the curve $(y^2-x^3)^2+y^5$ at the origin.
On the first blowing up, on the chart that intersects the exceptional ...

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608 views

### Closed but not rational points of a real cubic

In Mumford's Red Book of Varieties and Schemes, page 102, he gave the example of the closed but not rational points (that is to say points having residue field the complex field and not the real ...

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### Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?

Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
Is there an implementation of this in the MAGMA program?

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416 views

### Arithmetic and moduli spaces of higher genus curves

Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I ...

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517 views

### Which curves have stable Faltings height greater or equal to 1

Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.
Question 1. Can one classify or describe the ...

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368 views

### Picard group of $\mathcal{M}_{0,n}$

Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?

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488 views

### Automorphism groups of Elliptic curves as Galois module

Let $E/k$ be an elliptic curve over a field of characteristic $\neq$ 2, 3. Then we have an isomorphism $ [ \ \ ] :\mu_n \rightarrow\mathrm{Aut}_{\overline{k}}(E)$, $[ \zeta ] : (x,y) \rightarrow ...

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225 views

### About maps induced for a divisor D in P^1

Let $D$ be an effective $ \mathbb{Z}$-divisor on $ \mathbb{P}^1$. Is there a form to associate a curve $C$, and morphism $C \to P^1$ to the divisor $D$ ? For example, let $Y$ be a singular plane ...

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### Curves which are not covers of P^1 with four branch points

The following interesting question came up in a discussion I was having with Alex Wright.
Suppose given a branched cover C -> P^1 with four branch points. It's not hard to see that the field of ...

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414 views

### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...

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**1**answer

239 views

### Global functions on generalized Jacobians

Let $X/\mathbb{C}$ be a smooth projective curve of genus $g>0$ (here $\mathbb{C}$ is any algebraically closed field, say of characteristic $0$). Let $S$ be a finite set of closed points of X and ...

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### Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...

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527 views

### Maximal number of connected components of complement to an affine plane real algebraic curve

Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?

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113 views

### Tangent cones to Severi strata

Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the ...

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### Existence of certain identities involving characteristic 2 “thetas”

Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...

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848 views

### Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
The modular curves $X(n)$. They are ...

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440 views

### Why does the parameterization (F:F':1) happen?

1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).
2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we ...

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### Image of projective 1-space contained in projective 1-space over a smaller field?

This is inspired by
Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field? .
Say $K/F$ is a finite separable extension of fields. ...

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548 views

### resolution of singular points on curve

After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of ...

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### Does there exist a fibration of genus two over P^1 with only 3 singular fibres but two are semi-stable fibers for algebraic surfaces?

We will work over the complex numbers C.
there exist a fibration of genus two over P^1 with only 3 singular fibres but one is semi-stable fiber.
there not exist a fibration of genus two over P^1 ...

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### Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...

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### How do the number of plane curves over a finite field of a fixed genus increase with the degree?

Let $k$ be a finite field. We define $N(d,g)$ to be the number of plane curves $f(x,y)$ defined over $k$ of degree $d$ with (geometric) genus $g$. If $D(d) := (d-1)(d-2)/2$ (the maximum possible ...

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### What is the minimal degree of a smooth projective embedding of a hyperelliptic curve?

(partly motivated by this question, but different: Degree of a Variety)
For a hyperelliptic curve $C$ of genus $g$ (over an algebraically closed field of characteristic not two) what is the smallest ...

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906 views

### What do poles of differentials on a curve mean?

Let C be a (smooth proper) curve. I think of an element f of the (algebraic) function field of C as, literally, a map from C to P1. Poles of f literally mean the fiber over infinty, and div f makes ...