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 ...
- 148k
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 $C$...
- 148k
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 ...
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 $V\subset \Gamma(C,L)$ ...
- 1,566
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 ...
- 411
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, $\operatorname{Hilb}^2(\mathbb P^n)$ ...
- 148k
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 $C$. ...
- 39.3k
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 $...
3
votes
Accepted
Examples of nontrivial configurations of rational curves of degree $\leq 3$ in the projective plane
For an image summarizing the Richelot-Humbert genus 2 AGM construction see e.g. page 22 of https://arxiv.org/pdf/1006.3408
Humbert: http://www.numdam.org/item/JMPA_1901_5_7__395_0.pdf
Richelot: https:...
- 4,394
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, ...
- 9,237
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 ...
- 44.7k
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 ...
- 984
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}=-...
- 66.3k
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{...
- 5,215
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, ...
- 91.8k
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