# Questions tagged [curves]

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### Existence theory for geometric flow of space curves

Is there any existence theory applicable to general geometric flows of space curves in the following form? $$\partial_t \gamma = v_t t + v_n n + v_b b$$ Here $\gamma$ is the evolving curve, $t$, $n$ ...
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### Most divisors on a curve aren't special?

I have a generic smooth curve $C$ of genus $g$ and fixed multiplicities $a_1, \dots, a_n \geq 0$ with $\sum a_i = g+1$. Q1 : For generic marked points $p_1, \dots, p_n \in C$, must $\sum a_i p_i$ be a ...
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### Question on existence of almost length-minimizing curve in a general domain?

I have the following question: for a general domain $\Omega$ in $\mathbb{R}^n$, is it true that for each pair of points $x,y\in \Omega$, there exists a curve $\gamma$ connecting $x$ and $y$ in $\Omega$...
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### Snake algorithm that minimizes straight lines

How can I find the non-self-intersecting loop that uses the least amount of straight lines (curves left/right as often as possible every turn) and still loops back on itself? Here's an example we have ...
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### Cardioid-looking curve, does it have a name?

The curve, given in polar coordinates as $r(\theta)=\sin(\theta)/\theta$ is plotted below. This is similar to the classical cardioid, but it is not the same curve (the curve above is not even ...
1 vote
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### Is this curve well known?

I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that $\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$ $\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$ $a,b\in\mathbb{R}$ Is the ...
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### Surprising properties of closed planar curves

In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...
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### Closed curves with minimal total curvature in the unit circle

Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. In this later paper Chakerian gave a simpler proof and noted that ...
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### I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?

Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map: $f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$ with the following ...
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### property of rational functions on projective curves

I have a couple of question about the proof of Lemma 1.20.5 from Janos Kolloar's Lecture on Resolution of Singularities (page 19): Lemma 1.20.5 Let $C$ be a reduced, irreducible projective curve (=1D ...
360 views

### Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14): 1.13 (Very weak Riemann-Roch on curves)...
1 vote
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### Proposition from Kollar's Rational Curves on Algebraic Varieties

$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves on Algebraic Varieties by Janos Kollar (page 117). We work in setting ...
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### Can a closed horizontal trajectory on a Riemann surface be freely homotopic to $0$?

Let $R$ be a Riemann surface and let $\varphi=\varphi(z)dz^2$ be a nonzero holomorphic quadratic differential on $R$. A differentiable curve $\gamma$ on $R$ is called a horizontal trajectory if along ...
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### Number of curves in an admissible system of Jordan curves on a surface

Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that they are nonintersecting ...
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### Large class of curves which only intersect each other finitely many times

I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property: For any pair $\gamma_1, \gamma_2$ of curves in ...
112 views

### cohomology of curves

Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$ in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle. If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
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### Smooth transformation of a curve with fixed ends and length [duplicate]

I am simulating polymers of fixed length and fixed ends. I would like to search the phase space of all possible conformations quickly. Is there anyway I can generate efficiently a lot of (rather) ...
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### wild julia sets

Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the ...
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are the envelopes of the following set of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem; \mathcal{E}:=\Biggr\{\frac{(x-e)^2}{...
Let $X$ be a smooth projective curve, and $E$ an indecomposable vector bundle on $X$ with $\mathrm{deg} E<0$. Is it true that $H^0(X,E)=0$? This is true if $E$ is a line bundle, which means it is ...