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### Characterizations of cycloid

There are several constructions of a cycloid. I have some examples below. Are there any others?
Trace of a fixed point on a rolling circle
Evolute of another cycloid (the locus of all its centers of ...

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### Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...

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### Cusp point and straightness of a smooth curve.

I have a smooth curve of length $L$ with a single cusp point $P$ occuring at length $s = L_P$. Let the curve in arc length parametrization be $\alpha_t(s) \equiv (X_t(s),Y_t(s)) $. They are actually a ...

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### “Inverse problem” for the zeta function [duplicate]

Let $C$ be a smooth, projective, geometrically irreducible curve, of genus $g$, over a finite field $\mathbb{F}_q$. By the Weil conjectures, the zeta function has the shape
$$
...

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### Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research.
Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...

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### support of embedded points in a curve

Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded ...

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### Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...

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125 views

### endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...

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199 views

### Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...

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625 views

### Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...

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126 views

### meaning of $k(C)/1+\mathfrak{m}_x$ [closed]

Let $C$ be a smooth projective curve over some field $k$ and $x$ a closed point of $C$. I've seen some constructions in which people use
$k(C)^\times / 1+\mathfrak{m}_x$.
What's the meaning of ...

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184 views

### Etale covers of products of curves

Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...

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279 views

### Kodaira dimension of the moduli space of curves

It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...

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172 views

### Are Isom-schemes geometrically connected

This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...

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243 views

### Shimura surfaces that do not contain a Shimura curve

Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...

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196 views

### Structure of fundamental groups arising from smooth projective morphisms

Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...

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283 views

### Lipschitz parametrization of a symmetric convex curve

Assume that $\gamma$ is a convex curve which is symmetric with respect to two orthogonal lines (the ellipse is such a curve).
I want to know if there exists a $(l,L)$-bi-Lipschitz mapping of the ...

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71 views

### Dimension of the Representation of the Suzuki and Ree Groups?

What are the dimension of the group representation of $^2B_2$ and $^2G_2$? All what I know is that the first is 4 and the second group has two representation of dimension 7 and 13. Are there any?

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### Relating the toric rank of a semistable curve and the first Betti number of its reduction graph

Introduction
Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$.
Let $J$ be the Jacobian of $X$. ...

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### radius of tubular neighborhood

Hi there,
Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?
For example, given a simple smooth curve on R^2, what's the radius ...

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### Is it possible to check two curves on birational equivalence by some computer algebra system?

I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them ...

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### Moduli space of genus 2 curves

Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?