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**14**

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660 views

### New(?) reciprocity law

Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$.
I have found the following equality:
$$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$
Here ...

**13**

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**0**answers

514 views

### Degrees of maps from curves to $\mathbb P^1$

Let $a$ and $b$ be two relatively prime natural numbers. What is the largest number $c$ such that there is a curve with maps to $\mathbb P^1$ of degree $a$ and $b$ but no map to $\mathbb P^1$ of ...

**11**

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205 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...

**11**

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**0**answers

213 views

### Infinitely many curves with isogenous Jacobians

Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?

**10**

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**0**answers

152 views

### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...

**9**

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346 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 ...

**9**

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365 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 ...

**9**

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**0**answers

448 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 ...

**7**

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**0**answers

194 views

### Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...

**7**

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**0**answers

317 views

### The curve $(x+y+z)^3=27xyz$

Can someone point me to literature about the curve defined by $F(x,y,z):=(x+y+z)^3-27xyz$? I'm sure this curve must be well-studied, due to the remarkable property that
$$
F(x^3,y^3,z^3) = ...

**7**

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438 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 ...

**7**

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**0**answers

151 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 ...

**7**

votes

**0**answers

353 views

### Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity

Let $X$ be a hyperelliptic curve of genus $g \ge 2$ over a field $k$ (of characteristic not 2, 3 or 5, if you like, but could be positive in general). Let $J$ be the Jacobian of $X$, thought of as ...

**7**

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**0**answers

181 views

### sheaves on thickened nodal cubics

Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...

**6**

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**0**answers

237 views

### Global sections for a locally free sheaf over curves

Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg ...

**6**

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507 views

### Bezout Theorem in $\mathbb P^3$

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq ...

**6**

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**0**answers

177 views

### Counting plane curves over various fields

Fix two integers $d$ and $g$. The number of genus $g$ and degree $d$ curves passing through $3d+g-1$ generic points on the complex projective plane is finite and doesn't depend on the choice of ...

**6**

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**0**answers

167 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 $g$ ...

**5**

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143 views

### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$.
Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...

**5**

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**0**answers

107 views

### Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind.
Let $X$ be a smooth quasi-projective geometrically ...

**5**

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**0**answers

140 views

### Strengthening of Suslin's rigidity argument?

To fix the situation, let $k$ be an algebraically closed field, and let $C$ be a smooth projective curve over $k$. Suslin's rigidity argument implies in particular that any class in ...

**5**

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**0**answers

182 views

### Is the moduli space of genus three smooth quartics affine?

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm ...

**4**

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**0**answers

169 views

### Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1.
One can show that for a general ...

**4**

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**0**answers

156 views

### Map of the Klein quartic from $CP^2$ to $R^3$

The Klein quartic $\mathcal{X}$ is cut out of $\mathbb{C}P^2$ by the homogeneous equation $$x^3 y + y^3 z + z^3 x = 0.$$ It has 168 orientation preserving automorphisms and includes several copies of ...

**4**

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209 views

### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...

**4**

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**0**answers

90 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

**4**

votes

**0**answers

196 views

### Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...

**4**

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**0**answers

183 views

### Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...

**4**

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**0**answers

280 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 $G$. ...

**4**

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**0**answers

537 views

### Soft proof of multiplicity one for character groups of Shimura curves?

Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the ...

**3**

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206 views

### Symmetric power of an etale map of curves

Let $k$ be an algebraically closed field and $f\colon X \rightarrow Y$ an etale morphism of smooth curves over $k$. Let $f^{[n]}\colon X^{[n]} \rightarrow Y^{[n]}$ be the induced morphism on $n$-th ...

**3**

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323 views

### A Step in the Proof of the Drinfeld-Simpson theorem

I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``$B$ ...

**3**

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**0**answers

110 views

### state of the art for kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

What is the state of the art about the kodaira dimension (and rationality, unirationality, etc.) of the moduli spaces of $n$-pointed curves of genus $g$? When it is known and when not? It would be ...

**3**

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**0**answers

225 views

### Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...

**3**

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**0**answers

251 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).

**3**

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257 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 ...

**3**

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**0**answers

345 views

### 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 ...

**3**

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**0**answers

338 views

### Is the strict transform a finite morphism?

Woe is me! I'm again resorting to this forum to ask a silly question.
Here is the example I had in mind: observe the (complex) curve $y^3=x^2(x-1)$. In attempt to normalize this curve, I've begun by ...

**2**

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**0**answers

214 views

### A question about Weil restriction

Let $\pi:\tilde{C}\rightarrow C$ be a ramified cover between two smooth curves. And consider a group scheme $\mathcal G$ over $\tilde{C}$, I have found two definitions for Weil restriction:
...

**2**

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**0**answers

68 views

### Locus of line bundles with given base points on a curve

Let $C$ be a smooth projective curve over an algebraically closed field. If $D$ is an effective divisor on $C$ (let's say reduced to make things easier) of degree $m$ and $d>m$, is the dimension of ...

**2**

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70 views

### Abel-Prym map for Prym-Tyurin varieties

Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...

**2**

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56 views

### Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, ...

**2**

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**0**answers

73 views

### Criterion for the existence of finite locally free resolution

Let $X$ be a projective variety over an algebraically closed field $k$, $S$ be a $k$-scheme, $E$ be a coherent sheaf on $X \times_k S$, flat over $S$. We know that if $X$ is smooth then $E$ has a ...

**2**

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**0**answers

126 views

### A generalization of the Weil reciprocity law in a case of any two sections of line bundles on a curve

It seems to me that there should exist a generalization of the Weil reciprocity law on curves, where instead of functions one takes arbitrary sections of two line bundles.
More precisely, it might ...

**2**

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**0**answers

72 views

### A basic question on complete intersection liaisons of curves

I am a beginner in the Linkage theory and would like to clarify certain points I am not sure of.
Let $P$ be the Hilbert polynomial of a curve in $\mathbb{P}^3$. Let $L$ be an irreducible component of ...

**2**

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**0**answers

93 views

### Moduli space of sheaves on a ribbon

In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...

**2**

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**0**answers

239 views

### Beautiful curves in Gromov-Witten theory, and in Donaldson-Thomas theory

Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ ...

**2**

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**0**answers

150 views

### When are Hilbert schemes connected by piecewise smooth curves?

Are there examples of Hilbert scheme $H$ of curves in $\mathbb{P}^3$ such that there exists an irreducible component $L$ of $H$ such that for any two points in $L$, there exist smooth projective ...

**2**

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**0**answers

125 views

### On a difference between $i_!$ and $i_*$ over $\mathbb{P}^1$

Let $X$ be a smooth projective surface in $\mathbb{P}^3$ containing a line $l$.
Denote by $C$ the curve corresponding to the divisor $2l$. Let $p \in C$ be a closed point. Denote by $U:=C \backslash ...

**2**

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**0**answers

112 views

### Projective embedding of non-reduced curves

By curve I mean a scheme of pure dimension $1$. Let $(\Gamma(C_{red},\mathcal{L}))$ be a complete linear system on $C_{red}$ which gives a projective embedding $C_{red} \hookrightarrow \mathbb{P}^n$ ...