# Tag Info

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 ...
• 137k
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 ...
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 $C$...
• 137k

### 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.

### 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 $V\subset \Gamma(C,L)$ ...
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### 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 ...
• 44k
Accepted

### 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 ...
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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}=-...
• 65.1k
1 vote

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 $\mathbb{P}(\operatorname{... • 5,011 1 vote ### How bad can a tacnode be for a polynomially parametrized curve? If the derivatives up to order$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$coefficients, ... • 88.9k 1 vote Accepted ### Smooth curves in Tangent Developables Definition of the Tangent Developable Surface. Let$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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