The curves-and-surfaces tag has no wiki summary.
9
votes
1answer
200 views
How useful/pervasive are differential forms in surface theory?
Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
3
votes
2answers
127 views
Curve of 3-secant lines
Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...
0
votes
0answers
30 views
minimal surface from a variational problem [migrated]
Given the Lagrangian
$$ J(u)= \int_{V} \sqrt{1+|\operatorname{grad}(u)|^{2})} $$
with the constraint $ \int_{V}udx =1 $
(1) Why is the volume constraint there ?
(2) For the case of ...
0
votes
1answer
82 views
Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'
Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...
1
vote
1answer
166 views
On the dualizing sheaf of a curve
Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is ...
1
vote
1answer
235 views
base points of multiplicity $>1$
Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important).
Consider a ...
1
vote
0answers
158 views
Embedding a projective curve in a smooth surface (using a Bertini theorem)
Let $C$ be a reduced (not irreducible) projective curve of degree $d$ such that $C$ contains at most double points. By a result due to Kleiman and Altman, we know that there exists a smooth surface ...
1
vote
0answers
44 views
Number of fixed order elements in the mapping class group of a closed surface [migrated]
Let $S_g$ be a closed surface of genus $g\geq 2$. Given $r \in \mathbb{N}$, what is the number of elements of order $r$ in the mapping class group? Is it finite or infinite? If it is infinite is ...
1
vote
0answers
120 views
Pull-back of globally generated sheaves
Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
12
votes
4answers
703 views
What is the analog of the “Fundamental Theorem of Space Curves,” for surfaces, and beyond?
The "Fundamental Theorem of Space Curves"
(Wikipedia link; MathWorld link)
states that there is a unique (up to congruence)
curve in space that simultaneously realizes
given continuous curvature ...
3
votes
2answers
207 views
Realizing homology classes on surfaces with boundary by simple closed curves
Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...
1
vote
1answer
92 views
Bitangent locus of torus knots
Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...
3
votes
2answers
146 views
Points of a linear system on a cubic surface
Let $S$ be a generic cubic surface and let $C$ be its intersection with a generic quadric surface.
In the linear system of hyperplane sections of $S$, how many points represent the planes $H$ tangent ...
4
votes
3answers
204 views
On discrete version of curve shortening flow
One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...
8
votes
1answer
417 views
Geodesics on the twisted pseudosphere (Dini's surface)
I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...
1
vote
0answers
67 views
invariance of the dimension of severi varieties of surfaces
Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...
1
vote
1answer
69 views
Centralizer of a pseudo-Anosov element
What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?
0
votes
0answers
21 views
B-spline re-parametrization
I have a question regarding the re-parameterisation of a B-spline.
Some info:
The B-spline is of order 4 (degree 5), hence $C^3$ continuity
There is no knot multiplicity
The end conditions are not ...
3
votes
0answers
117 views
Mapping One Curve to another using Dehn Twists
Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...
10
votes
4answers
306 views
How to detect a simple closed curve from the element in the fundamental group?
(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be ...
5
votes
1answer
120 views
Curvature flows for PL closed curves in the plane?
I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane.
There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth ...
5
votes
1answer
122 views
Injective simplicial maps between Arc complexes
Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
6
votes
0answers
468 views
Bezout Theorem in $\mathbb P^3$
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq ...
2
votes
1answer
150 views
winding number for outer-pointing normal
While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...
2
votes
1answer
149 views
Characterization of convex space curve
In general, a smooth curve $C$ in $R^3$ that lies entirely on the boundary of its convex hull, $\mathcal{H}(C)$, is defined to be convex.
Does any one know of a characterization of a curve in space ...
0
votes
0answers
65 views
Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...
8
votes
2answers
301 views
Surface in 3D that realizes all pairs of principal curvatures
This is a question that Willie Wong raised in comments after he answered my question,
Surface analog of clothoid: curvatures covering $\mathbb{R}^3$. Willie's question is
more interesting (and ...
1
vote
3answers
233 views
Surface analog of clothoid: curvatures covering $\mathbb{R}$
The clothoid $C$, a.k.a. the Euler spiral,
is one among many curves with
the property that its curvatures cover $\mathbb{R}$
in the sense that, for every $x \in \mathbb{R}$,
there is a point $p \in C$ ...
5
votes
1answer
209 views
Representation of surface group
Is there a faithful representation of a surface group of genus $>2$ into $GL(n,\mathbb{C})$ for some $n$ for which, for each conjugacy class of each embedded loop in the fundamental group, the ...
7
votes
1answer
99 views
A procedure to determine if an automorphism of a closed 2-manifold extends to an automorphism of a handlebody
In a paper of Casson and Gordon's "A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983), no. 1, 119–137" they give a necessary criterion for a fibred knot to be a ribbon ...
3
votes
0answers
94 views
The central fiber of this family of surfaces?
I have a question on a description of a central fiber of the following family of surfaces.
Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...
2
votes
2answers
324 views
Existence of smooth surfaces containing a curve
Let $C$ be a curve in $\mathbb{P}^3$, possibly non-reduced. Assume, there exists a smooth surface in $\mathbb{P}^3$ containing $C$. Is it true that for $d \gg 0$, a generic element of $I_d(C)$ defines ...
11
votes
2answers
445 views
On closed simple curve with curvature at most 1
I am looking for the reference to the following theorem.
I have to apply a similar statement, and it would be nice to trace the source.
Please note, I know few proofs in fact it is Problem 3 in my ...
1
vote
2answers
250 views
Producing $(-2)$ curves on a smooth surface
We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
2
votes
3answers
173 views
How to tell if a second-order curve goes below the $x$ axis?
Suppose we have a second-order curve in general form:
(1) $a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0$.
I'd like to know if there is a simple condition that ensures that the curve ...
1
vote
2answers
318 views
Equation for simple Jacobian of a genus two curve
Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...
4
votes
1answer
160 views
is there some condition I can impose on families of curves on a surface such that the second Ext between the ideal sheaf and the structure sheaf is zero?
(the title got out of hand)
Say I have a surface $X$, then I also have M, the Hilbert scheme of curves and points on X.
This can be seen as a moduli space of quotients $O_X \to O_Z$.
If $I_Z$ is the ...
3
votes
2answers
358 views
Surfaces in $\mathbb R^3$ with negative curvature bounded away from zero
Is there a surface in $\mathbb R^3$ which is a closed subset and whose curvature is negative and bounded away from zero?
And the small-print...
By surface I mean smooth surface without ...
0
votes
3answers
340 views
difference of curve classes
Let $X$ be a smooth protective variety, or just a smooth Kahler manifold. Is it possible to have two curves $C_1$ and $C_2$ in $X$ such that their difference in $H_2(X,\mathbb{Z})$ is a non-trivial ...
2
votes
1answer
210 views
Generalization of Vogt's Theorem for curves in higher dimension
The Vogt's theorem for plane curves states that if A and B are endpoints of a spiral arc,
the curvature nondecreasing from A to B. The angle $\beta$ of the tangent to the arc at B with the chord AB ...
2
votes
2answers
387 views
two curves filling a surface
Let $S$ be a closed surface of genus $g \geq 2$. There exist two simple closed curves filling $S$?
Definitions:
Two closed curves $\alpha, \beta$ fill $S$ if they have minimal intersection and $S ...
3
votes
1answer
420 views
How to find the action of an automorphism on the 27 lines on a cubic surface?
Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet ...
4
votes
1answer
256 views
Divisorial contraction: when is the image an algebraic space or a stack?
Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...
2
votes
1answer
170 views
curvature of curves in the space of gaussians measures
I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be ...
1
vote
0answers
125 views
Bonnesen's inequality for non-simple curves
Given a closed curve in the plane $\mathbb{R}^2$, it is well known that $L^2 \geq 4\pi A$ where $L$ is the length of the curve and $A$ is the area of the interior of the curve.
For a simple closed ...
2
votes
3answers
553 views
A simple closed curve on a surface
How to describe a simple closed curve on an oriented surface of genus g? I know the answer only for the torus. It would be nice to find an article or a book where proof can be found.
2
votes
0answers
139 views
An isoperimetric type maximization problem with a barrier.
I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:
Let $(r(\theta), \theta)$ be a ...
2
votes
1answer
129 views
Generalization of an inequality due to Gage for curve shortening Part II
I asked a question recently about generalizing an inequality due to Gage. This inequality asserts that given a convex domain $\Omega$ in $\mathbb{R}^2$ with support function $p(X) = \langle X, \nu ...
2
votes
1answer
257 views
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} ...
2
votes
1answer
218 views
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 ...
