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1
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2answers
259 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?
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3answers
179 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
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2answers
366 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
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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 ...
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3answers
3k 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
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2answers
438 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 ...
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3answers
344 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
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1answer
222 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
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2answers
441 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
437 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
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1answer
275 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
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1answer
179 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 ...
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0answers
139 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
616 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
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0answers
143 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
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1answer
133 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
281 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
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1answer
226 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 ...
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2answers
261 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 ...
9
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3answers
769 views

Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...
3
votes
1answer
192 views

Bezier Curves question

Hi everyone I have a fairly simple question about bezier curves: can you represent n bezier curves that have been continuously joined together by a single bezier curve of degree 3n? My instinct is ...
2
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0answers
236 views

What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...
2
votes
1answer
281 views

Minimal surface as varities

Minimal surface equation is the following: $$(1+ \phi_t^2) \phi_{xx} - 2 \phi_x \phi_t \phi_{xt} + (1 + \phi_x^2) \phi_{tt} =0$$ Solution of this equation $\phi(x,t)$ is minimal surface(non ...
3
votes
1answer
260 views

Defining ideals for rational curves in space

A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...
1
vote
1answer
245 views

'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code ...
9
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3answers
405 views

Flips of triangulations on non-orientable surfaces

Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures. I'm looking at ideal triangulations of the surface, namely ...
4
votes
1answer
575 views

Example of cone of numerically effective curves which is not polyhedral

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays I cannot remember where I read ...
1
vote
2answers
242 views

Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
5
votes
2answers
1k views

Nagata's conjecture, Seshadri constant

What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric ...
2
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1answer
248 views

Does each finite morphism of curves have a model whose minimal resolution is semi-stable

Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Question. Does there exist a finite field extension $L/K$ and a regular model ...
2
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1answer
279 views

Log resolutions on surfaces and 3-folds in characteristic p

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the ...
4
votes
1answer
408 views

Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
9
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1answer
435 views

Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?)

The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ...
3
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1answer
504 views

Does this surface contain all perfect cuboids?

Probably this is wildly known, but the closest I found was a surface with additional restrictive conditions. A perfect cuboid is a cuboid having integer side lengths, integer face diagonals and an ...
4
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2answers
499 views

Directional Distortion of a Surface

Hi everyone, I am facing a math road block. I have two surfaces (3D) described by two functions $f_1$ and $f_2$ (known). I would like to create some sort of directional distortion along the loading ...
11
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0answers
697 views

Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...
2
votes
2answers
372 views

The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.) Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...
0
votes
1answer
224 views

Why is it a non-basepoint?

In the proof of the Castelnuovo theorem for curves in $\mathbb{P}^3$ (Hartshorne IV, 6.4.) the following is done: One considers a smooth, complete curve $C$ in the projective space $\mathbb{P}^3$ over ...
2
votes
2answers
287 views

Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...
1
vote
1answer
563 views

Hirzebruch surfaces

I am sorry for too naive and stupid question, How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Can F_{2} be realizable as the total space of a bundle over $\mathbb{R}_{+}$ ...
0
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0answers
288 views

Approximate a piecewise linear path by a path with bounded curvature

I have a piecewise linear path in two dimensions (with finitely many pieces). I would like to approximate it by a curve whose radius of curvature is bounded away from 0 (i.e. I specify the bound a ...
0
votes
0answers
277 views

Boxing the Rational Box

You surely all know the Perfect Cuboid problem. Here is a bit of pondering on the Euler box (space diagonal doesn't have to be rational). We start with the generators p,q,r and the surface ...
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2answers
1k views

Parametrization of the intersection of an ellipsoid with a sphere

Hi, Fisrt I would like to say that geometry is far away from my domain. I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy ...
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1answer
1k views

Geometric meaning of derivatives of the curvature

Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ ...
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2answers
458 views

The number of singular fibres of a semi-stable arithmetic surface over \Z

This is an arithmetic follow-up to my previous question Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres Let $k$ be an algebraically closed field and let ...
2
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1answer
410 views

Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres

This question is based on Beauville's article in Szpiro's asterisque Seminaire sur les pinceaux de courbes de genre au moins deux from 1986. We will work over the complex numbers $\mathbf{C}$. Let ...
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2answers
1k views

Surfaces containing curves of arbitrarily negative self-intersection

Olivier Wittenberg and I are curious about the following : Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can ...
0
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1answer
383 views

Meeting point of the vertices of a square cloth on x-y plane

Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...
12
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1answer
665 views

Minimizing geodesic on a convex surface

Let $\Sigma$ be a smooth convex surface in Euclidean 3-space and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$. Assume that for some $a < b < c$, we have ...
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0answers
548 views

What is the relation between hypocycloids and ideals in polynomial rings as alluded to in Arnold's text on teaching mathematics?

While browsing through this site, I came upon the text of Arnold: "On teaching mathematics". http://pauli.uni-muenster.de/~munsteg/arnold.html containing the phrase ... it can be said that a ...