For questions about surfaces.

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3
votes
2answers
138 views

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 ...
2
votes
2answers
137 views

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?
6
votes
1answer
211 views

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 $B$ ...
7
votes
2answers
179 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 ...
8
votes
1answer
208 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 ...
1
vote
1answer
159 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 ...
3
votes
1answer
874 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 $X$ ...
0
votes
1answer
45 views

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 $s\in I$. ...
2
votes
2answers
352 views

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 $p$, ...
2
votes
2answers
136 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 ...
3
votes
1answer
174 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 ...
4
votes
1answer
173 views

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 ...
3
votes
1answer
62 views

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 ...
1
vote
1answer
31 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 ...
7
votes
1answer
110 views

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 ...
0
votes
0answers
277 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 ...
3
votes
1answer
49 views

Does this diagram of Chern classes and push forwards commute

Let $p:Y\to X$ be a birational proper surjective morphism of regular surfaces, and let $D$ be a divisor on $Y$ such that $p(D)$ is a point. Then $p_\ast D =0$ by definition. Is there an easy way to ...
3
votes
2answers
250 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
3
votes
1answer
239 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
2
votes
0answers
33 views

Seifert surface and crossing number

i am sitting here with the problem of Seifert Surfaces. I know from a theorem that every knot does have a Seifert surface. We can also make a so called disc-and-band surface $F$ by gluing $v$ discs ...
1
vote
0answers
37 views

Why does the surface area integral need the arc length differential but the volume doesn't? [duplicate]

When calculating the surface area of a revolution you need to use the arc length differential $$\sqrt{1 + y'^2}$$ but you don't need to use that when calculating the volume. Why is that? Thanks!
1
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0answers
58 views

Curvature of surface

So lets say I have a mesh and for each face I have the position of its $3$ vertices and the area of the face. So let's say I have a point $p$ on this face and a vector $v$ that goes from $p$ to the ...
0
votes
1answer
97 views

Tetrahedralize Mesh

Say I have a triangle mesh which forms the shell of an object which may not be convex. For every triangle, I have the vertices and a normal. I want to turn this mesh into a solid. I want to break up ...
2
votes
2answers
85 views

Circles and surfaces

We say a subset $\overline{D}$ of $\Bbb R^2$ is $x\textbf{-normal}$ is there exist $a,b\in \Bbb R$ with $a<b$ and continuous functions $\psi_1,\psi_2:[a,b]\to \Bbb R$ such that $\forall x\in ...
0
votes
1answer
100 views

Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
1
vote
0answers
1k views

Shortest distance between a point and a surface

What I actually have to do is find the point on the surface which is closest to the point $P$, both of which are given below. Function $z(x,y)=x^2+2y^2$ Point $P = (1,-1,1)$ This is what I have ...
0
votes
1answer
40 views

Approximated distance between two points on a surface

I'm reading this paper and I don't understand this line because I haven't the book (I can't look up Theorem 7.4.2) . How come that the distance between $P_i$ and $P_{i-1}$ is calculated as follows?
1
vote
1answer
60 views

Area of a 3D surface

I need to compute the area of a $3D$ sphere centered on $0;0;0$ and the book I'm following says: "If a curve $y = f (x)$ from $y = a$ to $y = b$ is revolved around the $x$ axis, the surface area of ...
1
vote
0answers
112 views

Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
4
votes
1answer
149 views

Deriving equations for the “Bianchi-Pinkall torus”

I am trying to work out explicit parametric equations for the "Bianchi-Pinkall flat torus" as depicted in this note, but I seem to have gotten stuck in understanding the descriptions given in that ...
2
votes
1answer
135 views

How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
0
votes
1answer
93 views

Bounded vector field on a closed surface

Let $S\subseteq\mathbb{R}^3$ a closed surface and let $X\in\mathfrak{X} (S)$ a vector field on $S$ such that $\mid\mid X_p\mid\mid \le M$ $\forall p\in S$ for some constant $M>0$. Prove that $X$ ...
6
votes
1answer
155 views

Geodesic of a Surface in $\mathbb{R}^3$

I'm not familiar with geodesics. How can I show that a curve $c$ given by $c(t)=(t,f(t)\cos{\alpha},f(t)\sin{\alpha})$ for $\alpha$ constant is a geodesic on $M$ where $M=\left\{(x,y,z) \in \Bbb{R}^3 ...
1
vote
4answers
343 views

Orientation of Surfaces

I'm having a little trouble understanding how to orient a surface in $\mathbb{R}^3$ For example, how would I orient the ellipsoid given by: $$x^2+y^2+z^2+xy+xz+yz=\frac12$$ for $(x,y,z) \in ...
2
votes
1answer
49 views

Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
1
vote
1answer
147 views

Divisor class group on blowup of nodal surface

All varieties will be over $\mathbb{C}$ and projective unless stated otherwise. In Beauville - complex algebraic surfaces, the following is described: Let $S$ be a smooth surface and $p \in S$ a ...
12
votes
1answer
512 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
1
vote
3answers
127 views

Calculate Area of Surface

I am trying to calculate the area of the surface $z = x^2 + y^2$, with $x^2 + y^2 \le 1$. By trying to do the surface integral in Cartesian coordinates, I arrive at the following: $\int_{-1}^{1}dx ...
4
votes
0answers
134 views

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose ...
3
votes
1answer
80 views

Projective lines which can be viewed in some sense as surfaces

The complex projective line can be viewed as the $2$-sphere. I'd appreciate some examples of other projective lines (over any field or even ring) that can be viewed in some sense as a surface. I am ...
3
votes
0answers
49 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
6
votes
1answer
136 views

Compute the differential of a smooth map

Let $S\subseteq \mathbb{R}^3$ be an oriented regular surface and let $N$ be a field of normal unitary vector on $S$. We consider the map $F:S\times \mathbb{R}\rightarrow \mathbb{R}^3$ defined by ...
0
votes
1answer
163 views

Surface Parameterizations

I've been reading Manfredo Do Carmo's Differential Geometry of Curves and Surfaces and was wondering what are the conditions that need to hold for a surface parameterization as this is not defined in ...
2
votes
0answers
299 views

Surfaces are homeomorphic iff are diffeomorphic.

I have read this statement in several places: "Two surfaces are homeomorphic iff are diffeomorphic". I think the nontrivial implication follows in this manner: First, we triangulate the surface and ...
0
votes
1answer
111 views

Why the solution of this brainteaser a linear function?

I have been asked the following brainteaser: Imagine that you have a grid of dots in 2D placed at regular interval, you draw a convex shape by joining dots. Let us call M the number of dots ...
6
votes
2answers
213 views

Embedded surface in $\mathbb{R}^3$

Let $U \subseteq \mathbb{R}^2$ be an open set and let $\sigma : U \rightarrow \mathbb{R}^3$ be a parametrization of an oriented surface $S$ embedded in $\mathbb{R}^3$ whose unit normal in $\sigma ...
3
votes
2answers
455 views

is this set a regular surface?

I'm reading "Differential Geometry of Curves and Surfaces of Manfredo Docarmo" I'm doing the exercises of the chapter 2. Here is the definition of regular surface that we are following: I have ...
0
votes
1answer
110 views

Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in ...
3
votes
2answers
138 views

A vanishing theorem for differential forms.

I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
4
votes
1answer
66 views

Holomorphic 1-forms in $y^2-(z-a_1)\ldots(z-a_n)$

I know that the surface $y^2-(z-a_1)\ldots(z-a_n)$ is a Riemann Surface (that is the Riemann surface of $\sqrt{P(z)}$ with $P(z)=(z-a_1)\ldots(z-a_n)$) of genus $g$ and that ...