For questions about surfaces.

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0
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1answer
75 views

Find the area of the indicated surface

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = a^2$ inside the circular cylinder $x^2 + y^2 = ay$ ($r = a\sin(\theta)$ in polar coordinates), with $a > 0$. First time posting ...
3
votes
1answer
44 views

Why is the partial derivative of a surface a curve?

I'm trying to understand the proof for Green's Theorem and I've stumbled upon a few problems. In my notes, it says that: If $E$ is a simple (flat?) surface in $\mathbb{R}^2$ (I've been trying to ...
2
votes
1answer
28 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
3
votes
1answer
28 views

Proof of Castalnuovo's rationality criterion

Let $S$ be a complex projective smooth surface. If $D$ is a divisor on $S$, let's write $h^i(D)$ for $dim H^i(S,\mathcal{O}_S(D))$, where $\mathcal{O}_S(D)$ is the invertible sheaf associated to $D$. ...
1
vote
2answers
68 views

Total Curvature of 4 pi

What does it mean for a surface to have a total curvature of $4\pi $? I have seen that both the catenoid and Enneper surface are the only minimal surfaces that have this total curvature, but I don't ...
0
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1answer
47 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
2
votes
1answer
55 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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2answers
37 views

Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?
4
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1answer
85 views

Any interesting properties of Fermat's Last Theorem Surfaces?

I wonder if there are any interesting geometric (as opposed to number-theoretic) properties of what might be called Fermat's Last Theorem surfaces, i.e., $x^d + y^d = z^d$. Below are the surfaces for ...
2
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0answers
23 views

diagrams of twist spun torus knots

Kindly can you explain to me how to obtain the double twist spun of torus knots from tangle diagram of the given torus knot. I found the method here ...
0
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2answers
91 views

2D data fitting

I have some numbers as a function of 2 variables: $(x, y) \mapsto z$. I would like to know which function $z=z(x,y)$ best fits my data. Unfortunately, I don't have any hint, I mean, there's no ...
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0answers
12 views

For any diagram of a surface-knot, what are the possible connections of edges?

The singular set of a surface-knot diagram is a disjoint union of i) a graph, which has 1- and 6-valent vertices corresponding to branch points and triple points respectively. ii) circles which has no ...
0
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1answer
46 views

Is every quadratic surface in $\mathbb{P}^3$ ruled?

Let $\mathbb{P}$ be the projective line over an algebraically closed field $k$. Is it true that every quadratic surface in $\mathbb{P}^3$ is ruled? How can one see that this is the case? This ...
1
vote
1answer
42 views

Linear systems and rational maps

I'm following Beauville's book on Complex Algebraic Surfaces. If $D$ is a divisor on a surface $S$, we write $|D|$ for the set of all effective divisors linear equivalent to $D$ and we call it a ...
0
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0answers
32 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
2
votes
1answer
85 views

Flux and Gauss theorem

I have a problem; There seems to be something wrong with my understanding of gauss theorem. Let's say $F = [y ; x^2y; y^2z]$. I want to calculate the flux of $F$ going out of $$D = \{1 \le z \le 2 - ...
2
votes
1answer
58 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...
2
votes
1answer
45 views

Cohomology of the moving part of a linear system

Let $X$ be a smooth projective complex surface, $L$ a line bundle decomposed in its fixed and moving part as $|L|=F+|M|$. Intuitively, the inclusion of $|M|$ into $|L|$ yields an isomorphism ...
1
vote
1answer
26 views

Fixed components of linear systems on K3 surfaces

On a K3 surface, let $D$ be an effective divisor with $D^2\geq0$. Let $$D\sim D'+\Delta$$ be its decomposition in moving part and fixed part, respectively. Let $\Gamma$ be a prime component of ...
2
votes
1answer
36 views

Degree of blow up of a smooth projective surface

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$ and $\{x_i\}_{i \in I}$ be a finite set of closed points in $X$. Let $X'$ be the blow up of $X$ at these points. Then, $1)$ Is there a ...
2
votes
2answers
98 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
3
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1answer
45 views

Irreducible hypersurfaces vs irreducible polynomials

I know there exists a bijective correspondence between affine irreducible hypersurfaces and irreducible polynomials. This correspondence associates to each irreducible hypersurface ...
0
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1answer
44 views

Find the mass of a Ring

A ring $(x^2 +y^2=a^2)$ is made of a thin wire weighting $|x|+ |y|$ at all $(x,y)$. What is the mass of the ring?
0
votes
1answer
202 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
0
votes
1answer
26 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
0
votes
2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
0
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0answers
20 views

Catmull - Clark subdivision surfaces with a given border

Is there a method to decide the best mesh topology for a Catmull - Clark subdivision surface with a given border. I ask this because, when modeling with "blueprints", often we have only the ...
2
votes
1answer
40 views

Intersections with elliptic curves on a K3 surface

This is a fairly simple question. Suppose $E$ is an elliptic curve on a K3 surface $X$. Can we say that $E$ must intersect any curve $D\subset X$ of genus $g(D)\geq3$ ?
9
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1answer
68 views

K3 surfaces as complete intersections

I'm following Beauville's book "Complex Algebraic Surfaces". If $S$ is a K3 surface and $C$ is a smooth not hyperelliptic curve of genus g, then we have a birational morphism $\phi : ...
3
votes
1answer
37 views

Cubic surface and birational equivalence

Following Shafarevich "Algebraic Geometry II", I found this example. Let $X_3\subset\mathbb{P}^3$ a smooth cubic surface. To prove that $X_3$ is rational he claims that there is a birational map ...
0
votes
1answer
60 views

Show that space of mxm matrices form a surface

How can we show that the space of $m ~ \mathbb{x}~ m$ matrices with determinant equal to one form a $C^1$ surface of dimension $m^2-1$ in $\mathbb{R^{m^2}}$? I know there is probably a catch to it, ...
5
votes
2answers
63 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
29 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
37 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
60 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
41 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
63 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
79 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 $M_h$, ...
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vote
2answers
81 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
21 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
50 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
110 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
36 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
76 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
45 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
128 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
71 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
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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
232 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? ...