# Questions tagged [rational-curves]

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28
questions

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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 ...

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73 views

### 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, ...

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122 views

### 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 ...

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390 views

### 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 $C$?
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430 views

### 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 ...

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241 views

### 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 ...

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274 views

### 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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351 views

### 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(...

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162 views

### 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^...

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230 views

### 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 $\...

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188 views

### 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 ...

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161 views

### 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 $\Gamma\subset Y$ ...

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277 views

### 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 ...

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93 views

### 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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841 views

### 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 ...

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458 views

### 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 ...

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185 views

### 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 \...

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612 views

### 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, ...

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264 views

### 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 ...

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400 views

### 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_{\...

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552 views

### 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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503 views

### 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 ...

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439 views

### 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 ...

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583 views

### 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 ...

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295 views

### 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 ...

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142 views

### 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 $X$? ...

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553 views

### 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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351 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 ...