10
votes

Accepted

### Singular curves of genus 1

There's no problem over finite fields, but there is a problem over fields that have a nontrivial Brauer class. If you take a genus $0$ curve that's not rational (say a plane quadric), it will always ...

6
votes

Accepted

### Rational normal curves and tangent lines

Edit. This answer has been edited to address positive characteristic, as alerted by @FelipeVoloch. The result holds except in characteristic $2$. In characteristic $2$, the result fails.
Let $k$ be ...

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6
votes

Accepted

### Rational curves on the image of the pluricanonical maps

Not only could $Y_m$ contain a rational curve for all $m$, $Y_m$ could be a rational curve for all $m$.
Take $C$ a hyperelliptic curve, $E$ an elliptic curve, $\tau$ the hyperelliptic involution on $C$...

5
votes

### Is there a simple explanation for why rational plane curves of degree $>2$ are singular?

There is an elementary argument based on Bezout count of solutions of an algebraic system. It is given on pages 102-103 of the book "A Treatise on Algebraic Plane Curves" by Coolidge.

5
votes

### Is there a simple explanation for why rational plane curves of degree $>2$ are singular?

There is nothing special about rational curves in some sense.
Proposition: Given a smooth projective curve $C$, a very ample line bundle $L$ on $C$, and a 3-dimensional subspace $V\subset \Gamma(C,L)$ ...

4
votes

### Are rationally connected varieties robustly simply connected?

For my own edification I'm rewriting Jason's answer so as to excise all mention of loops (as he suggests), and trying to get the best possible bound from this technique.
Let $M$ be a variety with a $\...

4
votes

### Equations for points to lie on a rational normal curve

I would like to suggest the following paper, where my coauthors and I try to give a partial answer to this question
https://arxiv.org/abs/1711.06286
Roughly speaking, the idea is to use the Gale ...

4
votes

Accepted

### Can free rational curves lift to ramified covers of Fano varieties?

After some helpful conversations with Johan de Jong, I came up with an example. After finding it I realize I probably should have figured it out earlier.
In fact, $\operatorname{Hilb}^2(\mathbb P^n)$ ...

4
votes

### Rational normal curves and tangent lines

I want to suggest a simpler argument than Jason's, but I am not sure whether it works in positive characteristic or not.
Assume $n > 2$.
Consider the surface $S(C)$ swept by tangent lines to $C$. ...

3
votes

### Can a non-Kähler complex manifold be rationally connected?

As Jason said already, there are many
examples of Moishezon manifolds which
are rationally connected. Indeed, any
manifold bimeromorphic to a rational
connected manifold is again rationally
connected, ...

3
votes

Accepted

### Can a non-Kähler complex manifold be rationally connected?

I am writing my comment as a question. I have certainly explained these examples before on MathOverflow, since they show that the Kollár-Miyaoka-Mori conjecture cannot hold beyond Fujiki class $\...

Community wiki

3
votes

### Is there a simple explanation for why rational plane curves of degree $>2$ are singular?

After reading the reference suggested by Abdelmalek, I think I have a proof I’m happy with (which is basically the one from the book in more modern language); I suspect this is also equivalent to the ...

3
votes

Accepted

### Blowing-up projective spaces of parametrized rational curves

One modular interpretation of $X_1$ mimics the modular interpretation of the space of complete collineations. Let $V$ be a $k$-vector space of finite dimension.
Definition 1. For every $k$-scheme $...

Community wiki

2
votes

Accepted

### Does a moving family of lines through a fixed point produce a singularity?

To expand the comment of @potentially dense: the answer is $r-2$, and it does not depend on the degree of $X$. In general, suppose $X\subset \mathbb{P}^r$ is a hypersurface, and $p$ a smooth point of $...

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2
votes

### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Section 3 of the following paper of Beauville should answer your question: https://arxiv.org/abs/alg-geom/9701019
In particular, it is shown there that, up to replacing the compactified Jacobian by a ...

1
vote

### (Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve

Let me consider the case of the Fermat quartic. Since you are talking about Castelnuovo's criterion, I guess you are interested in the projective situation.
The $1$-form $$\psi \colon =\frac{dx}{y^3}=-...

1
vote

### Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

I think one way to approach this "geometric dream", so to speak, is as follows. Using the so-called stellar isomorphism, $\mathbb{P}^d(\mathbb{C})$, thought of as $\mathbb{P}(\operatorname{...

1
vote

### How bad can a tacnode be for a polynomially parametrized curve?

If the derivatives up to order $m$ coincide at $t_1$ and $t_2$,
you have a system of $m+1$ equations. This system is linear and homogeneous with respect to
coefficients. There are $d+1$ coefficients, ...

1
vote

Accepted

### Smooth curves in Tangent Developables

Definition of the Tangent Developable Surface. Let $k$ be a field. For every $k$-scheme $X$ and for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$, denote by $\mathcal{P}^1_X(\mathcal{F})$ ...

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