The algebraic-curves tag has no wiki summary.

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**1**answer

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### Genus two pencil in K3 surface

It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...

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**3**answers

406 views

### Smooth Models of Hyperelliptic Curves (+ a concrete question)

My general question is : given a hyperelliptic curve $y^2 = f(x)$ with disc$f(x) \ne 0$, is there a general formula for finding a smooth complete model of the curve?
Specifically, I want a smooth, ...

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**3**answers

822 views

### Quadratic reciprocity and Weil reciprocity theorem

I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...

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**2**answers

329 views

### Recognize this plane curve?

An aspect of my work led to a plane curve with implicit equation
$$
x^2+y^2 = 3 (y/2)^{2/3} + 1
$$
Actually, I started with the parametrization below and derived from it the
equation above:
...

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votes

**1**answer

2k views

### Does there exist a curve of degree 11 having 15 triple points?

Does there exist an irreducible curve of degree 11 in the projective plane which would have 15 triple points?
For information, such a curve would be rational, if it exists, and would be smooth at ...

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**2**answers

261 views

### Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$

Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or ...

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votes

**1**answer

160 views

### Affine model of a hyperelliptic curve

We know that any hyperelliptic curve over a field with characteristic not equal to $2$ has an affine model given by $y^2 = f(x)$, with $deg(f) = 2g+1$ or $2g+2$.
Can we always find a model such that ...

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votes

**1**answer

432 views

### Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...

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votes

**1**answer

391 views

### Correspondences on curves and their induced maps on differentials?

How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.
More specifically, I was reading a ...

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**3**answers

724 views

### Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...

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**0**answers

211 views

### Reference for Jacobians in characteristic $p$

I am looking for a basic reference for Jacobians of algebraic curves in characteristic $p>0$. I just want basic facts about the relation between the curve and its Jacobian.
I dont want to assume ...

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**1**answer

335 views

### Is Castelnuovo bound tight?

Castelnuovo bound says that if we have a function field(algebraic curve) $F$ and a divisor on it $D$ then:
$g\leq c\frac{\deg(D)^2}{\ell(D)}$(where $c$ is some global constant say 2 and $g$ is a ...

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**1**answer

358 views

### A question about the Tannakian etale fundamental group of a curve

Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let ...

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**4**answers

672 views

### A question on deformations of Theta divisor in the Jacobian of a complex curve

Suppose $C_g$ is a smooth compact complex curve (of genus $g$), and let $J$ be its Jacobian. Recall that the Jacobian $J$ of a curve $C_g$ is a complex torus that can by obtained by contractions of ...

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**0**answers

128 views

### genus four curve with $|3p|= \mathfrak{g}^1_3$

Let $M_4$ be the moduli space of genus four curves. Let $\Sigma \subset M_4$ be the locus such that for $X \in \Sigma$, there is a point $p$ on $X$, with $\dim H^0(X, \mathcal{O}(3p)) =2$. What is ...

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**1**answer

366 views

### Hyperelliptic curves over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...

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**1**answer

326 views

### Families of three dimensional algebraic curves

Let's consider spatial algebraic curve $C\subset \mathbb P^3$.
How could I describe a family of such curves, for example the set of all curves genus $g$ passing through $k$ points?
I'd like to some ...

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votes

**1**answer

260 views

### Defining ideals for rational curves in space

A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...

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votes

**1**answer

224 views

### Calculating the local index of intersection of two algebraic curves.

Let $F_1,F_2$ be two polynomials of two variables $(x,y)$ (say complex variables).
Suppose that $F_1$ and $F_2$ have no common factors and $F_1(P)=F_2(P)=0$.
What is in practice the quickest way to ...

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votes

**5**answers

826 views

### The use of embedding a curve into its Jacobian

I'm looking for as many examples/applications as possible of the use of embedding a smooth projective geometrically connected curve $X$ over a number field $k$ with $X(k)\neq \emptyset$ into its ...

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**1**answer

493 views

### A curve with bad reduction for which the jacobian has good reduction

Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the ...

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**1**answer

1k views

### Trace of Frobenius over $F_q$

Let $q_0$ be a prime and $q$ = $q_0^n$.
Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is ...

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**1**answer

494 views

### Does the Mordell conjecture imply the Shafarevich conjecture

The base field is a number field.
It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin).
Is the converse also true?
Note that both conjectures are now ...

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**0**answers

228 views

### Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.
A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...

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**0**answers

367 views

### determinant of cohomology

Hi,
I have probably a silly question. Take $X,Z$ proper and Gorenstein schemes $p:X\times Z \rightarrow Z$, $q: X\times Z\rightarrow X$ the projection maps, $L$ a torsion free sheaf of rank 1 on ...

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**1**answer

278 views

### How big is the locus of Galois covers in the moduli space of curves

Let's consider the moduli space $M_g$ of curves of genus $g$ over $\mathbf{C}$.
Not every curve of genus $g$ is a Galois cover (of the projective line) if $g\geq 3$.
How big is the locus of Galois ...

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**1**answer

932 views

### Bound for the number of rational points on the modular curve

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ...

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**2**answers

350 views

### Zograf's bound on the index of a modular curve for Shimura curves

I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...

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**0**answers

148 views

### Relations among Hodge classes?

Let $\pi : C_g \to M_g$ be the universal curve over the moduli stack of genus $g$ curves. Let $\omega_\pi$ be the relative canonical bundle. Then $\mathbb{H} := \pi_\ast \omega_\pi$ is a rank $g$ ...

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**1**answer

192 views

### $\psi$ class in $\overline{M}_{0,n}$

Basic question, but I found no reference.
Is the $\psi$ class the only one which is not a boundary class in the PIcard group of the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$? Or can it ...

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**1**answer

346 views

### fano moduli varieties of vector bundles

Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is ...

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**1**answer

180 views

### Maximal sets of algebraic curves, closed under rotation, dilation, and translation, that pairwise intersect at most twice

Consider a set of nontrivial algebraic curves on the plane groovy if that set is closed under rotation, dilation, and translation, and has the property that no two members of the set intersect more ...

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**5**answers

647 views

### 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 ...

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**2**answers

427 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

425 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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**2**answers

414 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

324 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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357 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

281 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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**1**answer

239 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

246 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

181 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

279 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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576 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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### 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

195 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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votes

**1**answer

902 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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**1**answer

487 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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202 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 ...