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 ...
Will Sawin's user avatar
  • 137k
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 ...
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 $C$...
Will Sawin's user avatar
  • 137k
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.
Abdelmalek Abdesselam's user avatar
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 $V\subset \Gamma(C,L)$ ...
Kapil's user avatar
  • 1,546
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 $\...
Will Sawin's user avatar
  • 137k
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 ...
Alessio's user avatar
  • 401
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, $\operatorname{Hilb}^2(\mathbb P^n)$ ...
Will Sawin's user avatar
  • 137k
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 $C$. ...
Sasha's user avatar
  • 37.3k
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, ...
Misha Verbitsky's user avatar
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 $\...
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 ...
Ben Webster's user avatar
  • 44k
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 $...
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 $...
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 ...
Jef's user avatar
  • 949
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}=-...
Francesco Polizzi's user avatar
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{...
Malkoun's user avatar
  • 5,011
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, ...
Alexandre Eremenko's user avatar
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 $\mathcal{P}^1_X(\mathcal{F})$ ...

Only top scored, non community-wiki answers of a minimum length are eligible