Questions tagged [rational-curves]
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38 questions
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Examples of nontrivial configurations of rational curves of degree $\leq 3$ in the projective plane
Consider the complex projective plane $P^2$. A rational curve in $P^2$ of degree $\leq 3$ is either a line, a smooth conic, a nodal cubic, or a cuspidal cubic. I am looking for some "nontrivial&...
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2
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1
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(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve
Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
- 1,833
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110
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Kodaira dimension of spaces of rational curves in hypersurfaces
Let $X\subset\mathbb{P}^n$ be a general hypersurface of degree $d\leq n$, and $\overline{\mathcal{M}}_{0,0}(X,a)$ the Kontsevich space of degree $a$ rational curves in $X$.
Does there exist an ...
- 8,998
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1
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227
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Rational curves on the image of the pluricanonical maps
Let $X$ be a compact complex manifold with canonical bundle $K_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but not maximal, i.e., $X$ is not of general type). Let $\varphi_m : X \...
- 1,379
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398
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Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection
Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection.
Can one understand how the compactified Jacobian of $C$ looks like?
For example, the ...
5
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2
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341
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Can a non-Kähler complex manifold be rationally connected?
Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler ...
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Is there a simple explanation for why rational plane curves of degree $>2$ are singular?
Its a well-known result that smooth projective plane curves of degree $d$ have genus $(d-1)(d-2)/2$, so in particular, smooth curves of degree $1$ and $2$ are genus 0, and those of higher degree have ...
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194
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rational curves over K3 surfaces over $\mathbb{Q}$
There are many partial results towards the following conjecture:
Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves.
My question is: is ...
- 637
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786
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Singular curves of genus 1
Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can ...
- 8,998
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Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?
The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...
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341
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Rational and rationally chain connected surfaces
A projective variety $X$ over the complex numbers is rationally connected if two general points of $X$ can be joined by a rational curve in $X$, and rationally chain connected if two general points of ...
user91659
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0
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115
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Has anyone researched this variant of separable rational connectedness?
If I am correct, then one of the definitions of rational chain connectedness is that a variety $ X $ is rationally chain connected if
1) there are schemes $ \mathcal{C} $ and $ T $ with a flat, ...
- 782
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241
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Rational curves on ruled surfaces
Let $S$ be a ruled surface (over an algebraically closed field) with an $\mathbb{P}^1$-bundle $\pi\!: S \to E$ onto an elliptic curve $E$. What is the classification of (possibly singular) irreducible ...
- 1,833
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552
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Can free rational curves lift to ramified covers of Fano varieties?
Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$?
...
- 148k
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Rational normal curves and tangent lines
Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means ...
- 8,998
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Blowing-up projective spaces of parametrized rational curves
Consider the projective space $\mathbb{P}^N$ parametrizing morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$, $f(x,y) = [f_0(x,y):\dots:f_n(x,y)]$ of degree $d$.
Let $Z_i\subset\mathbb{P}^N$ be the ...
user117617
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0
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If there is no more than $k$ smooth rational curves on algebraic surface, what is the minimal value of k
Let $X$ be a smooth surface of general type with $q=p_g=0$, define a set $A=\{C\subset X|$ C is smooth rational and $C^2<-1\}$, let $k=|A|$ which is cardinality of $A$.
What is the minimal value ...
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How bad can a tacnode be for a polynomially parametrized curve?
Given a curve in $\mathbb{C}^2$ that is parametrized by polynomials $ x(t), y(t) \in \mathbb{C}[t]$, it is possible that for distinct $t_1$ and $t_2$, we have an intersection or crossing $(x(t_1), y(...
- 251
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217
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Finding Rational Curves on a Surface
Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end.
$f= (x^2y^2)z^3 + (5x^...
- 417
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Global section of line bundle on anti-canonical rational surface
Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\...
- 1,982
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Does a moving family of lines through a fixed point produce a singularity?
This is just a feeling that I had and I am curious if it is totally wrong or true to some extent.
Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...
- 325
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1
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193
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Smooth curves in Tangent Developables
Let $C\subset\mathbb{P}^n$ be a smooth curve, and let $Y\subseteq\mathbb{P}^n$ be its tangent developable.
Given two general points $y_1,y_2\in Y$ does there exist a smooth curve $\Gamma\subset Y$ ...
user91659
3
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0
answers
396
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Rational normal curves on quadrics
Given a quadric $Q\subseteq\mathbb{P}^r$ and points $p_1,\dots,p_{r+2}\in Q$ in linear general position, a naive dimension count suggests that one should expect finitely many rational normal curves ...
- 325
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0
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If there exists an immersion, then does a neighbourhood of a singular rational curve contain a genuine cuspidal point?
Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose $...
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Are rationally connected varieties robustly simply connected?
Let $X$ be a smooth projective rationally connected variety over $\mathbb C$. Let $C$ be a curve class in $X$ such that rational curves equivalent to $C$ connect every two points.
Let $f: Y \to X$ be ...
- 148k
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Secant varieties of curves in $\mathbb{P}^4$
My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
user58018
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239
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Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane
Consider a rational map $u : \mathbb{CP}^1 \to \mathbb{CP}^4$ of degree~$d$, such that the image lies in a fixed 3-dimensional quadric $Q^3$. In other words, its image is a rational curve in $Q^3 \...
- 11
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745
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Equations for points to lie on a rational normal curve
$\def\PP{\mathbb{P}}$Let $z_1$, $z_2$, ..., $z_n$ be points in $\PP^{k-1}$. I am interested in equations for when the $z_i$ lie on a rational normal curve (or degeneration thereof.)
Specifically, ...
- 156k
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327
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Canonical bundle of moduli space of rational curves and automorphisms
Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves.
The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...
user58604
8
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0
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435
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The curve $(x+y+z)^3=27xyz$
Can someone point me to literature about the curve defined by $F(x,y,z):=(x+y+z)^3-27xyz$? I'm sure this curve must be well-studied, due to the remarkable property that
$$
F(x^3,y^3,z^3) = \prod_{\...
- 6,397
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645
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Rationality of curve does not depend on base change
By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...
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2
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Rationally connected varieties and rational fibrations
Let $Y$ be a rationally connected variety over an algebraically closed field, and let
$$\phi:X\dashrightarrow Y$$
be a rational fibration such that the general fiber of $\phi$ is rationally chain ...
- 8,998
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1
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551
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Automorphisms of rational (connected) projective curves
To fix the ideas all curves are supposed to be defined over $\mathbb{C}$. Let $C$ be a rational connected projective curve. Note that we don't assume the curve to be smooth. Let $Aut(C)$ be the group ...
- 7,609
3
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1
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795
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explicity equations for curves in the projective space
It is well known that if a smooth curve $C \subset \mathbb{P}^3$ has degree $ d \leq 6$. Then
$ g(C) \leq 4$ (Hartshorne pg 354). I know that the case $g=4$ correspond to the complete intersection ...
- 943
3
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1
answer
330
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Dimension of the linear system of $\psi$-class on $\bar M_{0;n}$
Consider the (Deligne-Mumford compactification of the) moduli space of complex rational marked curves $\overline M_{0;n}$. For each $i\in \{1,\ldots,n\}$ we can construct a line bundle $L_i$ with a ...
- 401
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Hypersurfaces containing a general chain of lines
Let $X$ be a general chain of $d$ lines in $\mathbb P^n$, where $n \geq 3$. Let $I$ be the homogeneous ideal of polynomials vanishing on $X$. What is the Hilbert function
$$P(k) = \dim I_k$$
of $X$? ...
- 7,338
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Proving a variety is not unirational
It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the ...
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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 ...
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