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3
votes
0answers
70 views

How order of divisor with support at infinity is changed at reduction?

Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$. To ...
1
vote
0answers
42 views

Understanding the Exp map from a moduli of smooth curves

The setup: Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$. Let $\mathscr{M}$ ...
3
votes
1answer
60 views

Second symmetric product of a hyperelliptic curve

Let $C$ be a hyperelliptic curve of genus $g\geq 3$, let $C^{(2)}$ be the second symmetric product of $C$ with itself, i.e. the quotient of $C\times C$ by the involution $(p,q)\mapsto (q,p)$ and let ...
1
vote
0answers
72 views

On Different Ways of Proving Isoperimetric Inequalities [closed]

Update: Thanks to Douglas Zare's comment, My previous questions in this thread turned out to be equivalent to the Isoperimetric problem. Thus I edited my question to make it a bit different. ...
10
votes
1answer
447 views

A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry. It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ ...
2
votes
1answer
129 views

Parametric smooth curve that vists all integer points of the plane [closed]

Does there exist a parametric smooth curve that visits all integer points $(x,y),\, x,y \in \mathbb{N}$ of the plane? Something similar to this: $$\begin{align} x = &\theta ...
4
votes
1answer
148 views

A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem

I propose a conjecture variant of Cayley-Bacharach's theorem. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
3
votes
2answers
351 views

A conjecture like Cayley–Bacharach theorem

Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...
1
vote
0answers
37 views

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 ...
6
votes
1answer
285 views

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 ...
5
votes
0answers
84 views

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 ...
4
votes
1answer
125 views

“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 $$ ...
2
votes
1answer
96 views

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 ...
0
votes
0answers
125 views

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 ...
19
votes
2answers
3k views

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 ...
1
vote
1answer
131 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, ...
0
votes
1answer
239 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 ...
16
votes
1answer
674 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 ...
0
votes
1answer
129 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 ...
3
votes
1answer
187 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 ...
1
vote
1answer
307 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 ...
2
votes
1answer
186 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 ...
9
votes
1answer
251 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 ...
3
votes
1answer
212 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 ...
3
votes
2answers
287 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 ...
0
votes
0answers
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?
2
votes
2answers
205 views

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 $X$. ...
5
votes
2answers
469 views

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 ...
5
votes
2answers
596 views

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 ...
11
votes
4answers
1k views

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?