The surfaces tag has no wiki summary.
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About fibers of an elliptic fibration.
Consider the pencil of cubics
$\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of $\mathbb P^2$ ...
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4answers
152 views
How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus?
How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
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1answer
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Abelian Elliptic Surfaces
By abelian surface we mean a 2-dimensional algebraic complex torus. Thus
$$ S=\Bbb{C}^2/\Gamma$$
where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
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1answer
28 views
Find position on surface of a lens
If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
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3answers
53 views
Good source to learn about surface singularities?
I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there.
I checked out Miles Reid his lectures on ...
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1answer
30 views
Uniqueness of Seifert graphs
If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph.
If it is not a directed and weighted graph, can we ...
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4answers
111 views
Prove that Gauss map on M is surjective
Let $M$ be a closed, orientable, and bounded surface in $\mathbb{R}^3$.
(a) Prove that the Gauss map on $M$ is surjective.
(b) Let $K_+(p) = \max \{0, K(p)\}$. Show that
$$
\int K_+dA \ge 4\pi.
$$
...
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1answer
42 views
Graphs from Seifert surfaces
Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem,
For ...
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0answers
52 views
Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as
$\Omega_f(x) \triangleq ...
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2answers
49 views
Uniqueness of Seifert surfaces of knots
I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
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1answer
23 views
The continuity of principal coordinate system
$X$ is a $C^k$ hypersurface in $\mathbb R^{n+1}$ and $y$ is a fixed point on $X$. Can we find an orthogonal system $\{e_1(x),e_2(x),\cdots,e_{n+1}(x)\}$ on a neighborhood $U$ of $y$ such that 1. ...
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1answer
32 views
The connectivity of the intersection of hypersurface and ball
$u$ is a function defined on a connected open set $\Omega$ of $\mathbb R^n$ containing $0$ such that $u \in C^2(\Omega)$ and $u(0)=0$. Consider the hypersurface $X=\{(x,u(x))~|~x\in\Omega\}$ and the ...
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2answers
49 views
Algebraic surface as a smooth manifold
Let $S$ be the set of points $x=(x_1,x_2,\ldots,x_9)\in \mathbb{R}^9$ which satisfy the following conditions:
$$x_1^{2}+x_2^{2}+x_3^{2}=x_4^{2}+x_5^{2}+x_6^{2}=x_7^{2}+x_8^{2}+x_9^{2}=1$$
...
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1answer
82 views
Surface Integral directly
Surface integral that has me stumped.
Q: Calculate $\int \int_{S} F \cdot dA$
Where $F(x,y,z)= xi+yj+zk$ S is the boundary of the region $x^{2}+y^{2} \leq z \leq (2-x^{2}-y^{2})^{1/2}$ oriented so ...
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Properties of some hypersurface
$S$ is a compact surface in $\mathbb{R}^n$ of positive definite second fundamental form.
$V= span \left\lbrace e_1, \dots , e_m\right\rbrace .$
$S^\prime \subset S$ is the preimage of $V\cap ...
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1answer
46 views
When is the Gauss Map on a surface bijective?
If $\mathcal{S}$ is a smooth, compact hyper-surface in $\mathbb{R}^n$ with positive definite second fundamental form, can we say that its Gauss map is bijective? If so, why?
Thanks!
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1answer
41 views
Property about revolution surfaces
"Consider $f:[a,b]\rightarrow\mathbb{R}$ of class $C^1$, limited, such that $f(x)\neq 0$ for all $a\leq x\leq b$. After this, consider the revolution surface by turning graphic of $f$ around the $x$ ...
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Calculating the Solid Angle
I've done some steps in this problem, but I got stuck at certain point.
$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization ...
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1answer
69 views
tricky surface integral
I am studying for my final and my prof gave us review questions but with no answers so I am lost with this question. If anyone can help I would really appreciate it.
Question: Find the area of the ...
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1answer
70 views
Is this the right equation for this 3D surface?
Is $\frac{\sin \sqrt{x^2+y^2+z^2}}{\sqrt{x^2+y^2+z^2}}$ the right equation for this surface? I am confused what $z$ is doing in there (unless this is an implicit equation). I get something fairly ...
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3answers
66 views
Interpreting the Surface Integral over a Vector Field
I have seen the fact that in certain instances, the Surface Integral over a Vector Field gives the quantity of fluid flowing through the surface in unit time (as in here, or in any standard Vector ...
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2answers
81 views
Find surface area that lies above a triangle
Determine the area of the part of the surface $z=2 + 7x + 3y^2$ that lies above the triangle with vertices $(0,0)$, $(0,8)$, and $(14,8)$.
I do not know what formula to use to attempt this problem!
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1answer
82 views
Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free
Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
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1answer
34 views
Surface area of a plane inside a cone
Determine the surface area of the part of the plane $z=1+x+2y$ which is inside the cone surface $z=\sqrt{2x^{2}+10y^{2}}$.
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3answers
160 views
What curve is this?
This is my earring (see the image please) and my question is: Does this curve have a name? If it does, which one?
Regards! And thank you.
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1answer
71 views
Stokes theorem problem to find alpha and beta so that I is independent of the choice of S
I have a question that I got half through but can't finish it. If anyone could help I would appreciate it.
Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle ...
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1answer
76 views
How do we define ample vector bundles
Let $X$ be a smooth projective variety over $\mathbf C$. How do we define an ample vector bundle $E$?
Do we just ask its determinant $\det $ to be ample?
Is it the same as saying that $f^\ast E$ is ...
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0answers
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Does the dual of a vector bundle with ample determinant have global sections
Let $E$ be a locally free sheaf on a smooth projective variety $X$ over $\mathbb C$. Suppose that $\det E$ is an ample line bundle on $X$.
Is $$H^0(X,E^\vee) =0?$$
In fact, if $E$ of rank $1$, it is ...
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1answer
72 views
2-cell embeddings of graphs in surfaces and Euler formula
I have a few questions regarding 2-cell embeddings of graphs in surfaces. Suppose $G$ is a 2-cell embedded graph in an orientable surface $S$,
a) Is any connected subgraph of $G$ 2-cell embedded in ...
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0answers
53 views
Find area of a curvilinear triangle that includes hyperbolic functions
We were given this question in class and I tried to compute it and it looks to e pretty crazy. Can anyone take a look and let me know if I did it correctly... I would really appreciate it.
...
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2answers
92 views
Find area of a simple, smooth, closed curve lying in a plane
I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
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2answers
37 views
How can we parametrize the following surface?
How to parametrize the following surface in $\mathbb{R}^3$: the intersection of $S=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2\leqslant 1\}$ and $D=\{(x,y,z)\in\mathbb{R}^3:x+y=1\}$.
Any hints are ...
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0answers
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Can one characterize which surfaces are capable of being described by a closed-form parameterization?
Speaking intuitively, I can visualize a lot of surfaces in my mind; but it seems that some of the ones I can imagine are not capable of being described by the 'usual suspects', i.e., elementary ...
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2answers
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Why are the fibers of the Albanese map of a nonrational ruled surface copies of $\mathbb{P}^1$?
I'm currently reading "Rational surfaces with many nodes" by Dolgachev et al., avaliable here:
http://www.math.lsa.umich.edu/~idolga/lisbon.pdf
A "surface" is always smooth and projective and let us ...
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1answer
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Is the universal covering surface orientable?
Let $M$ be a smooth, say also closed (compact and without boundary) surface. Is it true that its universal covering surface is orientable?
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1answer
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Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf
Let $B$ a smooth projective connected variety over $\mathbf C$.
Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero.
Does the converse hold? That is, suppose that $B$ ...
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2answers
107 views
Are endomorphisms of degree one always automorphisms
Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one.
Do I understand correctly that $\sigma$ is an automorphism?
I believe this ...
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1answer
50 views
Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf
Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
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1answer
51 views
Compute the surface area of an oblate paraboloid
Consider the surface S: $z=4-4x^2-y^2, z\geq0$. Compute its surface area.
I've tried the following: $Area(S)=\int\int_D \sqrt{(8x)^2+(2y)^2+1}dxdy$ with D being the interior of the ellipse ...
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1answer
95 views
The Gaussian and Mean Curvatures of a Parallel Surface
This is a homework problem from do Carmo. Given a regular parametrized surface $X(u,v)$ we define the parallel surface $Y(u,v)$ by $$Y(u,v)=X(u,v) + aN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ ...
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1answer
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Show that there is a fixed $p \in \mathbb{R}^n$ such that for all $s \in I, \gamma(s)=\beta(s)+p$.
Suppose that $\beta,\gamma : I \to \mathbb R^3$ are two unit speed smooth curves. Suppose that the curvatures and tortions are everywhere positive, and that $B_\beta(s)= B_\gamma(s)$ for all $s\in I$. ...
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Difference between tangent space and tangent plane
I’ve avoided doing any manifold (regretting it somewhat) courses, however do have some understanding. Let $p$ be a point on a surface $S:U\to \Bbb{R}^3$, we define:
The tangent space to $S$ at $p$, ...
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2answers
46 views
Implicit form of a parametric surface
Let $\Sigma$ be the surface in $\mathbb{R}^3$ parametrized by
$$ (u,v) \mapsto \Big(\;p_X(u,v),\; p_Y(u,v),\; p_Z(u,v)\;\Big), $$
where $p_X, p_Y, p_Z$ are polynomials. Is there a standard way to ...
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1answer
111 views
Must a surface fibered over a curve with constant fiber have a local trivialization?
Let us work over an algebraically closed field $k$ and suppose $\pi:S\rightarrow C$ where $S$ is a surface, $C$ is a smooth curve, and the fibers over closed points are all isomorphic to a fixed ...
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1answer
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Representation of (co)homology classes of $3$-manifolds by embedded surfaces
Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify
$$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$
where (co)homology is meant with integer ...
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1answer
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Cutting a sphere along a curve
i have a question. I think that is not difficult, but i can't find a solution.7
I want to show the following:
Cutting a sphere along a curve always results in two discs.
Therefore i want to use the ...
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1answer
25 views
Math question related to three dimensional surfaces?
So I have to determine and draw the surfaces
$$z-2x^2-4y^2 ≥0,\qquad \mbox{and}\qquad 4y^2-x^2+4z^2-1 ≥0$$
so the first one in my opinion should be transformed like this
$$z ≥2x^2+4y^2$$ then we ...
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What is an equation (not a parametric equation) for the Fresnel Elasticity Surface?
There is little information on the Wolfram website. I know the surface is a quartic. I've tried some equations that I've found by web searches, but I do not get a surface that looks like the one at ...
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Do K3 surfaces with an Enriques involution have a polarization of bounded degree
Does there exists a real number $C$ with the following property.
For any Enriques surface $E$ over a number field $K$ with K3 cover $X\to E$, there exists an ample divisor $H$ on $X$ such that $H^2 ...
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0answers
78 views
Area of a surface sphere between two parallel planes
I am given the following question:
Consider the surface of a sphere (that is the boundary of the sphere) of radius $R>0$ in $\mathbb{R}^3$ and two parallel planes which are $R$ units away from ...
