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710 views

### base points of multiplicity $>1$

Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important).
Consider a ...

**11**

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**0**answers

718 views

### Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...

**9**

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124 views

### Area of the minimal surface of a non-planar quadrilateral in 3d

Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...

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639 views

### What is the relation between hypocycloids and ideals in polynomial rings as alluded to in Arnold's text on teaching mathematics?

While browsing through this site, I came upon the text of Arnold: "On teaching mathematics".
http://pauli.uni-muenster.de/~munsteg/arnold.html
containing the phrase
... it can be said that a ...

**7**

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**0**answers

89 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

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**0**answers

102 views

### Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...

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517 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 ...

**4**

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201 views

### Are there Zoll pancakes?

How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be?
I don't want to impose a ...

**3**

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**0**answers

217 views

### Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...

**3**

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**0**answers

166 views

### Mapping One Curve to another using Dehn Twists

Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...

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113 views

### The central fiber of this family of surfaces?

I have a question on a description of a central fiber of the following family of surfaces.
Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...

**2**

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65 views

### Topological/numerical constraints for the existence of more than one pencil

A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective ...

**2**

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304 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...

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157 views

### An isoperimetric type maximization problem with a barrier.

I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:
Let $(r(\theta), \theta)$ be a ...

**2**

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**0**answers

267 views

### What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...

**1**

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122 views

### Was this particular case of the tube formula known before Weyl and Hotelling?

The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...

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71 views

### General reparameterization of a b-spline

Say I have a bspline function (or curve) of order $k_1$, defined over some knot vector
$\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$
Do you know of a process of finding ...

**1**

vote

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142 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 ...

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41 views

### Connectivity and contarctibility of complexes associated to curves and arcs

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to ...

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197 views

### Embedding a projective curve in a smooth surface (using a Bertini theorem)

Let $C$ be a reduced (not irreducible) projective curve of degree $d$ such that $C$ contains at most double points. By a result due to Kleiman and Altman, we know that there exists a smooth surface ...

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190 views

### Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...

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149 views

### Bonnesen's inequality for non-simple curves

Given a closed curve in the plane $\mathbb{R}^2$, it is well known that $L^2 \geq 4\pi A$ where $L$ is the length of the curve and $A$ is the area of the interior of the curve.
For a simple closed ...

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49 views

### Does the action of a 2-torsion line bundle on $Pic^d(C)$ fix the number of sections?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...

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64 views

### Embedding curves in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface.
For instance, a helix is not contained (or non-embeddable) in ...

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39 views

### Angle at self-intersection points of a curve in hyperbolic surface

Let $F$ be a hyperbolic surface of finite type. Let $\alpha$ be a closed oriented geodesic with more than one self intersection. Suppose all the self-intersections are double points. Let $\angle_p$ ...

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39 views

### Unitary derivative and countable set

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...

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91 views

### Positive curvature of the boundary away from a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...

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43 views

### Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...

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79 views

### Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...

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223 views

### “Spreading out” locally free sheaves

Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...

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221 views

### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

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52 views

### A question on the curvature of smooth and embedded curves in a plane

I am from China. This is my first question in this website.
Given a family of embedded, smooth and closed curves $\{X_i(s)\}_{i=1}^\infty$ in a plane, let us denote by $L_i$ the perimeter and ...

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76 views

### invariance of the dimension of severi varieties of surfaces

Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...

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82 views

### Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...

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328 views

### Approximate a piecewise linear path by a path with bounded curvature

I have a piecewise linear path in two dimensions (with finitely many pieces). I would like to approximate it by a curve whose radius of curvature is bounded away from 0 (i.e. I specify the bound a ...

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298 views

### Boxing the Rational Box

You surely all know the Perfect Cuboid problem. Here is a bit of pondering
on the Euler box (space diagonal doesn't have to be rational).
We start with the generators p,q,r and the surface ...