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4
votes
3answers
345 views

Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$: $$S_2\times Q\rightarrow Q,\; ...
14
votes
2answers
803 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 ...
2
votes
2answers
205 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 ...
1
vote
1answer
216 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
212 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 ...
3
votes
1answer
684 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
194 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
189 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 ...
14
votes
4answers
1k 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
323 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
126 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
169 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
457 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
676 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, ...
0
votes
0answers
75 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
107 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?
3
votes
0answers
166 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
532 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 ...
6
votes
1answer
195 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
143 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
514 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
169 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
193 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
82 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
374 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
291 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 $p \in C$ ...
6
votes
1answer
244 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
105 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
111 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 ...
3
votes
2answers
353 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 ...
12
votes
2answers
536 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
273 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
181 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
511 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
161 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 ...
5
votes
3answers
4k views

Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
3
votes
2answers
570 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
356 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
236 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
511 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
469 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
323 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
190 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
149 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 ...
3
votes
3answers
704 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
157 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
138 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
324 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
240 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 ...
1
vote
2answers
273 views

Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$

Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or ...