For questions about surfaces.

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6
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2answers
73 views

Ideal sheaf on a surface

Let $S\subset\mathbb{P}^n$ a smooth complex projective surface. I consider the exact sequence $$0\rightarrow I_S\rightarrow\mathcal{O}_{\mathbb{P}^n}\rightarrow\mathcal{O}_S\rightarrow 0,$$ where ...
2
votes
0answers
42 views

What kind of surface is this?

I'm not a math guru, but just fascinated by it, so sorry if my questions are only curiosity and not high level. In some contemporary art website I have found this image: In the right side there is ...
0
votes
2answers
30 views

Implicit Representation of Surfaces - Basic Quesion

I am reading on implicit representations of surfaces and cant quite come around the following example. Take $F : \mathbb{R^3} \rightarrow \mathbb{R}$, where $F(x,y,z)=x^2+y^2+z^2$. Now we want to ...
2
votes
2answers
49 views

Quadric surface as a $\mathbb{F}_n$ surface

The minimal models for rational projective smooth surfaces are $\mathbb{P}^2$ or the surfaces $\mathbb{F}_n$ for $n\neq 1$, where ...
3
votes
1answer
68 views

Del Pezzo surface of degree $4$

I'd like to show that the del Pezzo surface $S_4\subset\mathbb{P}^4$ (i.e. the complete intersection of two quadrics) is rational. I've got two possibilities: 1- I show that is the blow-up of ...
2
votes
1answer
55 views

Smooth surface that is a complete intersection

I have this definition of a projective complex algebraic surface that is a complete intersection. A surface $S\subset\mathbb{P}^{r+2}$ is said to be a complete intersection if it is a trasversal ...
0
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0answers
65 views

equivariant map $\mathbb{CP}^1 \to \mathbb{CP}^2:$ where does it send the boundary

Start from a map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x = au^2, \quad y = av^2, \quad z = uv,$$ where $a \in \mathbb{R}$ is a parameter. The image of the map is the ...
1
vote
1answer
119 views

Covering of orientable surface (Hatcher)

The following is an exercise from Hatcher, Algebraic Topology, that I'm struggling with (exercise 2.2.23): Show that if the closed orientable surface $M_g$ of genus $g$ is a covering space of $M_h$, ...
1
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2answers
211 views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
1
vote
1answer
25 views

holomorphic disk and crosscap as quotients of $\mathbb{CP}^1$ by antiholomorphic involutions

Consider $\mathbb{CP}^1 \ni (u:v)$ and the maps $$ \sigma_{\pm}: \quad (u:v) \mapsto (\overline{v}:\pm\overline{u})$$ How do we show that the quotient $\mathbb{CP}^1/\sigma_+$ gives a disk, and ...
1
vote
1answer
64 views

Linear systems and hyperplane sections on surfaces

Let $S$ be a smooth projective surface. If $H$ is an hyperplane section on $S$ and $D$ a divisor (that can be not effective) such that $(H.D)<0$, why can we conclude that the linear system $|D|$ is ...
5
votes
1answer
117 views

Book with color pictures of algebraic surfaces

I have a pretty specific question: I'm looking for a book with color pictures of algebraic surfaces. Could anyone point me in the right direction?
1
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1answer
37 views

$\int_{S_{n-1}} \exp(i a\cdot\xi) \,\mathrm{d}S(\xi)$

Let $n\in\mathbb{N}$ and $a\in\mathbb{R}^n$. The question is to find the value $$A_n := \int_{S_{n-1}}\exp(i a\cdot\xi)\,\mathrm{d}S(\xi),$$ where $S_{n-1}$ denotes the $(n-1)$-dimensional sphere in ...
4
votes
1answer
97 views

Hyperplane sections on projective surfaces

I am studying Beauville's book "Complex Algebraic Surfaces". At page 2 he defines the intersection form (.) on the Picard group of a surface. For $L, L^\prime \in Pic(S)$ ...
0
votes
0answers
48 views

Auto-intersection of a line on a smooth cubic surface

Can someone help me with the following idea? I think that i made a mistake: Let $X$ be a smooth surface of degree $d$ in $\mathbb{P}^3$ and $L$ denote the divisor class of a line on $X$. We have ...
4
votes
1answer
138 views

The Definition of the Second Fundamental Form

Let $r:M\rightarrow{\mathbb{R}^{n+1}}$ be an isometric immersion and $M$ is an $n$-dimensional Riemannian Manifold. That is to say, $M$ is the hypersurface in $\mathbb{{R}^{n+1}}$. Then we can ...
0
votes
2answers
103 views

Is a closed compact 2-Manifold that is embedded in euclidean 3-space always orientable?

I am sorry if this is a trivial question but I am a little confused right now so please bear with me. Since non-orientable compact 2-manifolds without boundary cannot be embedded in three-dimensional ...
0
votes
0answers
62 views

The Geometrical Mean of $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$

This is my homework: $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$ Below is what I have think: From the book of M.A.Armstrong, I find $RP^2$#$RP^2$#$RP^2$ is a sphere which is attached by three Mobius ...
20
votes
3answers
242 views

Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
1
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0answers
45 views

Help with Plateau's Laws

Can someone please explain mathematically what is meant by the term 'smooth' in Plateau's First Law: "Soap films are made of entire smooth surfaces" Thank you in advance!
2
votes
2answers
448 views

Surface area of intersection of two cylinders

Let $$R=\{(x,y,z):y^2+z^2\leq 1\,\, \text{and}\,\, x^2+z^2\leq 1\}.$$ Compute the volume of $R$. Compute the area of its boundary $\partial R$. I'm fine with #1. For #2, I have a ...
1
vote
1answer
51 views

What is a rational elliptic surface?

I know what is an elliptic surface, and I also know what is a Rational surface. However I can't find the definition of Rational elliptic surface. Can any one help me? I read this on the Kulikov ...
1
vote
1answer
65 views

Showing that a map between surfaces is a local isometry

Currently studying Differential Geometry of Curves and Surfaces. We have: $$\sigma:(0,\infty) × \mathbb{ R} \to \mathbb{R}^3, \quad (u,v) \to \frac{1}{\sqrt{2}} (u \cos v,u\sin v ,u)$$ We need to ...
2
votes
0answers
30 views

Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
2
votes
1answer
133 views

A lemma on the growth of the number of certain edge paths for a given train track

How to prove the following lemma from the book "Closed curves on surfaces" written by Francis Bonahon? Lemma: For any fattened train track $\Phi$, the number of edge paths of $\Phi$ of length $r$ ...
1
vote
1answer
55 views

Parametrize $|x|+|y|+|z|=1$

How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize. I think we may need to divide it into 8 pieces and consider them respectively.
1
vote
1answer
55 views

Determining the surface with given polynomial borders

Let's say we'd like to guess the shape (I'm not sure the word 'approximate' is appropriate here) of some surface when we are given its borders via third order polynomials (i.e. we are given their ...
0
votes
0answers
32 views

Integration by substitution on surface

Suppose I have an integral on the boundary of a Lipschitz domain $\Omega \subset \mathbb{R}^n$ $$\int_{\partial\Omega} f(x-y)dS$$. where $dS$ is the surface element $dS = g(x)dx$. Can I do a ...
1
vote
1answer
55 views

Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
3
votes
2answers
54 views

Are there odd-sheeted coverings of non-orientable surfaces by orientable surfaces?

For any non-orientable surface (compact,connected) $X$ with genus $h$, we have a $2n$-sheeted cover of $X$ by an orientable surface $Y$ first by covering $X$ by $\Sigma_{h-1}$ (a double cover) and ...
1
vote
1answer
80 views

bounds for surface integral of a plane

I need to calculate the next surface integral, but i'm having troubles with the bounds; $$\iint -2 \, dS,$$ where $S$ is the part of the plane $x+2y+z=2$ that is cut off in the first octant. My ...
1
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0answers
106 views

Question about integral on hypersurface

Let $S$ be a (n-1) dimensional hypersurface in $\mathbb{R}^n$. I see something like this written: $$\int_S f(s)d\sigma(s)$$ where $d\sigma$ means the surface measure. Now if $\Phi\colon T \to ...
2
votes
2answers
74 views

Height at 2D coordinate on a 3D rectangular surface

The Problem: How can I obtain every 3D coordinate on a rectangular surface given x and z? For those who are visual, picture looking down on the surface, and finding the height at where the x and z ...
0
votes
0answers
21 views

Relationship between the set of all canonical knots and the set of all knot genuses

What is the relation between the set of all canonical knots and the set of all knot genuses? I understand that there is at least not bijection between them.
2
votes
1answer
79 views

Mathematically explaining a trapped surface?

I'm currently writing my thesis, and the general area is that of minimal surfaces. I have a deep interest in cosmology so have directed it towards space-time, ie applying minimal surfaces in space, ...
2
votes
0answers
24 views

Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$ S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}. $$ If we denote by ...
1
vote
1answer
104 views

Surface area of a sphere bounded by a pyramid

I'm trying to calculate the surface area of a sphere that is bounded by the walls of a triangular pyramid. $o$ is the origin and centre of a sphere of radius $R$, $a$ is a point at the tip of the ...
1
vote
1answer
85 views

Is there a common name for the surface z = xy?

I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?
1
vote
2answers
29 views

Does $g(x,y,z)$ (the equation of the surface) need positive $z$ or negative $z$ when doing a surface integral?

$\quad$If a smooth surface $S$ is defined by $g(x,y,z)=0$, then recall that a unit normal is $$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$ where $\nabla g=\dfrac{\partial g}{\partial ...
2
votes
0answers
64 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
1
vote
2answers
55 views

Surface Integral of a Vector Field Over a Torus

Let $S$ be the surface obtained after rotating $(x-2)^2+z^2=1$ around the $z$-axis. What is the value of $$\int_{S}\mathbf{F\cdot n } dA$$ where $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y))$$
2
votes
0answers
92 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
0
votes
1answer
36 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
1
vote
1answer
45 views

Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis

So here is my problem: Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis Just thinking about the problem I feel ...
2
votes
1answer
64 views

Calculate the surface of revolution (area) for $x^2/4+y^2/2=1$

$$x^2/4+y^2/2=1$$ The curve is rotating around the x-axis and I'm suppose to calculate the area. My attempt: $$y=\sqrt{2-x^2/2}$$ The integral goes from -2 to 2. I tried to simplify the integral ...
0
votes
1answer
43 views

Paremetric surface revolved around y-axis

if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation $2\pi x\sqrt{(x')^2+(y')^2}$ and I'm given $$x=(2/3)t^{3/2}$$ $$y=2\sqrt{2}$$ and I got ...
1
vote
2answers
47 views

Is there a Covering Map $\Sigma_3^1\to \Sigma_2^1$

Let $S_{g,n}^b$ be a genus $g$ surface with $b$ boundary components and $n$ punctures. I'm having some trouble with these past qualifying exam questions: Is there a covering map $p\colon ...
8
votes
1answer
165 views

Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous ...
1
vote
0answers
108 views

Surface Integral

The glass dome of a futuristic greenhouse is shaped like the surface $z = 8 - 2x^{2} - 2y^{2}$. The greenhouse has a flat dirt floor at $z = 0$ Suppose that the temperature T, at points in and around ...
17
votes
2answers
355 views

How to determine that a surface is symmetric

Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane. The special case where $f$ is a polynomial is of some interest. ...