The rational-curves tag has no wiki summary.

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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, ...

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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 ...

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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) = ...

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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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### 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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### 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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### 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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### 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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### 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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### 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 ...