The curves-and-surfaces tag has no wiki summary.

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### Quermassintegrals as mean curvature integrals

It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral
$$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} ...

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### Generalization of an inequality due to Gage for curve shortening

There is a well known inequality due to Gage which asserts the following.
Let $\Omega$ be a smooth, convex set in $\mathbb{R}^2$ and let $p = \langle X, \nu \rangle$ be the support function of ...

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251 views

### Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$

Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or ...

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### Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...

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163 views

### Bezier Curves question

Hi everyone
I have a fairly simple question about bezier curves: can you represent n bezier curves that have been continuously joined together by a single bezier curve of degree 3n?
My instinct is ...

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### 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 ...

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272 views

### Minimal surface as varities

Minimal surface equation is the following:
$$(1+ \phi_t^2) \phi_{xx} - 2 \phi_x \phi_t \phi_{xt} + (1 + \phi_x^2) \phi_{tt} =0$$
Solution of this equation $\phi(x,t)$ is minimal surface(non ...

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255 views

### Defining ideals for rational curves in space

A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...

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### 'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code ...

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386 views

### Flips of triangulations on non-orientable surfaces

Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures.
I'm looking at ideal triangulations of the surface, namely ...

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525 views

### Example of cone of numerically effective curves which is not polyhedral

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read ...

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235 views

### Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...

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### Nagata's conjecture, Seshadri constant

What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric ...

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234 views

### Does each finite morphism of curves have a model whose minimal resolution is semi-stable

Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$.
Question. Does there exist a finite field extension $L/K$ and a regular model ...

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269 views

### Log resolutions on surfaces and 3-folds in characteristic p

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...

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394 views

### Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...

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433 views

### Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?)

The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ...

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492 views

### Does this surface contain all perfect cuboids?

Probably this is wildly known, but the closest I found was a surface with additional restrictive conditions.
A perfect cuboid is a cuboid having integer side lengths, integer face diagonals and an ...

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494 views

### Directional Distortion of a Surface

Hi everyone,
I am facing a math road block.
I have two surfaces (3D) described by two functions $f_1$ and $f_2$ (known). I would like to create some sort of directional distortion along the loading ...

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609 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 LovÃ¡sz conjecture for Cayley graphs, I started noticing ...

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350 views

### The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.)
Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...

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### Why is it a non-basepoint?

In the proof of the Castelnuovo theorem for curves in $\mathbb{P}^3$ (Hartshorne IV, 6.4.) the following is done: One considers a smooth, complete curve $C$ in the projective space $\mathbb{P}^3$ over ...

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275 views

### Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...

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545 views

### Hirzebruch surfaces

I am sorry for too naive and stupid question,
How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Can F_{2} be realizable as the total space of a bundle over $\mathbb{R}_{+}$ ...

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270 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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271 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 ...

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### Parametrization of the intersection of an ellipsoid with a sphere

Hi,
Fisrt I would like to say that geometry is far away from my domain.
I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy ...

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1k views

### Geometric meaning of derivatives of the curvature

Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ ...

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### The number of singular fibres of a semi-stable arithmetic surface over \Z

This is an arithmetic follow-up to my previous question Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres
Let $k$ be an algebraically closed field and let ...

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### Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres

This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986.
We will work over the complex numbers $\mathbf{C}$.
Let ...

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### Surfaces containing curves of arbitrarily negative self-intersection

Olivier Wittenberg and I are curious about the following :
Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can ...

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373 views

### Meeting point of the vertices of a square cloth on x-y plane

Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...

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### Minimizing geodesic on a convex surface

Let $\Sigma$ be a smooth convex surface in Euclidean 3-space
and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$.
Assume that for some $a < b < c$, we have
...

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### 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 ...

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1k views

### Formulas for equidistant curves

Hello,
I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to ...

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372 views

### Can curves differentiate vector bundles on P^2?

Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question:
If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a ...

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### Gaussian curvature radius

In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where ...

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### What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?

Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely ...

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763 views

### Help with Griffiths & Harris, Surfaces

I believe to have found a typo in Griffiths & Harris.
In the chapter on surfaces, section Rational Surfaces 1, I am trying to read the result that a holomorphic vector bundle over ...

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### Walking around Santa Cruz, track around the soccer field

I was recently walking around the track at UCSC, and I noticed that the track didn't always curve inward. Sometimes it curved the other way. Compare this (A convex track): ...