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**14**

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599 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

487 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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178 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 ...

**10**

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**0**answers

174 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?

**9**

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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 ...

**9**

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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 ...

**9**

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438 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 ...

**8**

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141 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 ...

**7**

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290 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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400 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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143 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**

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323 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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180 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**

votes

**0**answers

491 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

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 $g$ ...

**5**

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126 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

164 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 ...

**4**

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109 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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80 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

167 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

164 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

249 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**

votes

**0**answers

531 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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**0**answers

122 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 ...

**3**

votes

**0**answers

99 views

### Automorphisms of function fields under constant reduction

Let $K=\mathbb{Q}(x,y)$ be a function field of genus at least 2, with defining equation $f(x,y)=0$ (say, absolutely irreducible and with coefficients not divisible by $p$), and let $k$ be the mod-$p$ ...

**3**

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**0**answers

102 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

187 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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212 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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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 ...

**3**

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**0**answers

339 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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334 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

120 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

66 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

80 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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192 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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139 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

121 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

102 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 $C_{red} \hookrightarrow \mathbb{P}^n$ ...

**2**

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144 views

### branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...

**2**

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101 views

### n-canonical embedding

Let $C$ be a nodal curve and let $ \omega $ be its dualizing sheaf. Let $n$ be a integer larger than 2. Does anyone knows how to show that $\omega^{\otimes n}$ separates points and tangent vectors? ...

**2**

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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 ...

**2**

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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 ...

**2**

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**0**answers

204 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 ...

**2**

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**0**answers

117 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 ...

**2**

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218 views

### cubic curve closest to the given set of points

Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bezier spline) with fixed startpoint in such a way, that the ...

**2**

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227 views

### quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...

**1**

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**0**answers

84 views

### Families of curves with “almost-general” moduli

The Brill-Noether theorem says that, if $\rho(d, g, r) := (r + 1)d - rg - r(r + 1) \geq 0$, then there exists a unique component of the Hilbert scheme of curves of degree $d$ and genus $g$ in ...

**1**

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67 views

### The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials

Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...

**1**

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76 views

### Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...

**1**

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**0**answers

118 views

### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...