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### Unbounded convex domains in 2D

Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior ...
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### A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems: A chain of six circles ...
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### A problem of four curves

This is a generalization of my previous question, a problem of a cubic and six conics. Let a curve $(K)$ of degree $m$ and three curves $(C_i)$ of degree $n$, for $i=1,2,3$. Let $(C_1)$ meets $(K)$ ...
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### Automorphisms of cartesian products of curves

Let $C$ be a smooth projective curve. Is it true that $$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$ and in case, what would be a reference for this? Thanks.
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### Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid. Could you ...
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### Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...
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### 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 ...
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### 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 $\mathscr{M}$ be ...
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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 $... 1answer 494 views ### A tricky tractrix question about vertical tangents This is raised by a recent question occurring in combinatorial geometry. It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius$r>\frac12$(... 1answer 134 views ### Parametric smooth curve that vists all integer points of the plane [closed] Does there exist a parametric smooth curve that visits all integer points$(x,y),\, x,y \in \mathbb{N}of the plane? Something similar to this: \begin{align} x = &\theta \cos(2\sin(\theta\pi)... 1answer 166 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 ... 1answer 398 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 ... 0answers 40 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 ... 1answer 304 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 ... 0answers 103 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 ... 1answer 132 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 Z_C(t)=\frac{P(t)}{(1-t)(... 1answer 98 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$-smooth (just to ... 0answers 149 views ### support of embedded points in a curve Let$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}$does not have embedded ... 2answers 3k views ### Is this statement which relates the Fourier transform of a function to its singularities correct? I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ... 2answers 222 views ### endomorphisms of the Jacobian of a curve Let$C$be a smooth, projective curve of genus >1 over the complex numbers and let$J(C)$be its Jacobian. The Torelli theorem relates the automorphisms of$C$to the automorphisms of$J(C)$. ... 1answer 260 views ### Lipschitz boundary vs rectifiable curve boundary I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ... 1answer 704 views ### Number of curves over a finite field Let$K$be a finite field. Is there a formula for the number of isomorphism classes of genus$g$smooth curves over$K$? In other words does there exists a formula for the number of rational points ... 1answer 129 views ### meaning of$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$. What's the meaning of that?... 1answer 188 views ### Etale covers of products of curves Is a finite etale cover of a product of curves again a product of curves? The answer is no in general. Here's one way to construct an example. Take the product$A$of two elliptic curves and an ... 1answer 329 views ### Kodaira dimension of the moduli space of curves It is known that the moduli space$\overline{M}_{g}$of genus$g$curves is of general type for$g\geq 24$. By Theorem 2.4 of Logan, Adam The Kodaira dimension of moduli spaces of curves with ... 1answer 200 views ### Are Isom-schemes geometrically connected This question is about properties of Isom-schemes that are well-known over algebraically closed fields. Let$K$be a field of characteristic zero, let$C$be a smooth projective geometrically ... 1answer 259 views ### Shimura surfaces that do not contain a Shimura curve Let$S$be a Shimura surface i.e. a Shimura variety with$dimS=2$. Does$S$necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ... 1answer 223 views ### Structure of fundamental groups arising from smooth projective morphisms Let$f:X\to B$be a smooth projective morphism of complex algebraic varieties. If$f$is of relative dimension zero, i.e.,$f$is a finite etale cover, then the image of the topological fundamental ... 2answers 290 views ### Lipschitz parametrization of a symmetric convex curve Assume that$\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)$-bi-Lipschitz mapping of the ... 0answers 72 views ### Dimension of the Representation of the Suzuki and Ree Groups? What are the dimension of the group representation of$^2B_2$and$^2G_2$? All what I know is that the first is 4 and the second group has two representation of dimension 7 and 13. Are there any? 2answers 208 views ### Relating the toric rank of a semistable curve and the first Betti number of its reduction graph Introduction Let$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\$. ...
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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?