# Questions tagged [rational-curves]

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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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### 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 ...
323 views

### 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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### 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 ...
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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 ...
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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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### 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 ...
1 vote
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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, ...
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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 ...
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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 $C$? ...
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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 ...
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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 ...
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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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### 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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### 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 ...