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7
votes
1answer
282 views

Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$. Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$. ...
22
votes
1answer
954 views

Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$? By a Néron model, I mean ...
1
vote
1answer
172 views

Constructing a curve with good reduction over a function field

Let $K$ be the function field of a smooth projective connected curve $B$ over $\mathbf{C}$. Let $g\geq 0$ be an integer. Does there exist an nonsingular integral $\mathbf{C}$-scheme $X$ with a ...
2
votes
1answer
311 views

Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer. Does there exist a curve $Y$ over $K$ with a finite etale ...
1
vote
1answer
251 views

What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$. Let $S$ be non-empty finite set of finite places of $K$ and suppose ...
5
votes
1answer
299 views

kapranov's realization of $\overline{M}_{0,n}$ over other fields

Kapranov gave a very nice desciption, over $\mathbb{C}$ of the moduli space of stable pointed rational curves $\overline{M}_{0,n}$ as a series of blow-ups of $P^{n-3}$. Does this, or a similar result, ...
5
votes
1answer
308 views

Genus 2 curves vs Abelian surfaces

In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$ $lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & ...
3
votes
1answer
145 views

genus two curve with special automorphisms

Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its ...
1
vote
1answer
128 views

action of automorphisms on the Galois cohomology of the function field of a variety

Let $C$ be a (quasi-projective) variety over an algebraically closed field $k$ and let $k(C)$ be its field of rational functions. Then for any birational map $\sigma: C \dashrightarrow C$ there is an ...
7
votes
1answer
457 views

Algebraic curve mapping to elliptic curve - how to check whether this is possible?

Question: Let $C$ be an algebraic curve over some field (like the rationals) given by a plane projective model (possibly with singularities). Is there an easy way to see if this curve has a ...
2
votes
2answers
452 views

Moduli spaces of vector bundles and stability conditions

Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is $$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$ This induces the ...
1
vote
0answers
167 views

dimension of spaces of rational curves in a variety!

Do you know calculate the dimension of the space of rational curves of degree m, through d given points, contained in some projective variety?
1
vote
1answer
179 views

invariants of plane quartics

Does anybody know a good reference where the invariants for plane quartic curves are developed?
3
votes
2answers
271 views

Secant variety of an irreducible non-degenerate projective curve

I would like to know why every non-degenerate irreducible projective curve has a three-dimensional secant variety. It is clear to me that the dimension can't be larger. Thanks for your help!
9
votes
1answer
554 views

Rank two vector bundles on a curve of genus two

I recently learned of an interesting result of Narasimhan and Ramanan from 1969, which says that moduli space of rank two vector bundles with trivial determinant on a curve $X$ of genus two is ...
3
votes
2answers
664 views

Cartier divisors on singular curves

(1) Let $X$ be a projective (integral) curve over $\mathbb{C}$ and let $P$ be a singular point of $X$. Is there always a Cartier divisor whose support is exactly $P$ (set-theoretically)? The ...
7
votes
0answers
410 views

Elementary proof of the Hurwitz formula

I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective ...
8
votes
1answer
392 views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
1
vote
1answer
162 views

Is the number of twists of a curve with a section in a given field finite

Let $X$ be a smooth projective geometrically connected curve over a number field $k$ of genus $g\geq 2$. Is the number of twists of $X$ always infinite? (The answer is no, because there aren't any ...
6
votes
1answer
199 views

Is the class of $k$-gonal curves dominant

Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero. Let $\mathcal C$ be a class of ...
7
votes
1answer
586 views

Picard group of a singular projective curve

Let $X$ be a singular irreducible projective curve over an algebraically closed field and $\pi : \widetilde{X} \to X$ the normalization morphism. In the book on Neron models by Bosch et al. (I have ...
7
votes
0answers
146 views

Unicritical rational functions on curves in characteristic $p$

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$. How precisely can one describe ...
1
vote
2answers
125 views

Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$

I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in Gauss' ...
4
votes
0answers
264 views

On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields). Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...
1
vote
1answer
153 views

pull back of hodge bundle via glueing map

Hi, I need a precise reference for the following fact, which is certainly well known, but I do not find any. I consider the natural glueing map of pointed curves $\overline{M}_{g_1,n}\times ...
1
vote
1answer
237 views

Riemann-Roch and dim of deformation space.

Let's consider curve $C\subset \mathbb P^n$ of degree $d$ and genus $g$. We want to calculate dimension of deformation space of $C$, i.e. $h^0(C,L)$ where $L$ is the normal bundle. We can decompose ...
2
votes
1answer
257 views

Maps of algebraic curves (and their Jacobian)

When people consider a map $\varphi: C \rightarrow C$ between algebraic curves and they mention the "associated map" on the Jacobian of $C$. Which map do they mean? Do they mean $\varphi^{\*}$ or ...
3
votes
0answers
225 views

Igusa-Clebsch invariants

I am looking for a good reference for definitions and basic properties of the Igusa-Clebsch invariants for curves of genus $2$ (especially, but not only, in positive characteristic).
2
votes
1answer
367 views

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 - ...
3
votes
3answers
432 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, ...
7
votes
3answers
856 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 $ord_xf$ ...
2
votes
2answers
341 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: ...
18
votes
1answer
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 ...
1
vote
2answers
265 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 ...
2
votes
1answer
166 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 ...
2
votes
1answer
437 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 ...
2
votes
1answer
427 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 ...
9
votes
3answers
848 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 ...
2
votes
0answers
214 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 ...
1
vote
1answer
349 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 ...
3
votes
1answer
366 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 ...
7
votes
4answers
717 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 ...
2
votes
0answers
129 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 ...
1
vote
1answer
375 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 ...
2
votes
1answer
332 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 ...
3
votes
1answer
263 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 ...
2
votes
1answer
227 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 ...
5
votes
5answers
878 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 ...
11
votes
1answer
519 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 ...
3
votes
1answer
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 ...