Timeline for Rational curves on the Fermat quartic surface

8 events

when toggle format	what		by	license	comment
Jun 30, 2016 at 22:47	vote	accept	byu
Jun 30, 2016 at 18:33	answer	added	Jason Starr		timeline score: 13
Jun 30, 2016 at 18:13	comment	added	Jason Starr		Please confer Corollary 4.6 of Chapter 8, p. 161, and Corollary 2.4 of Chapter 15, p. 315, of Daniel Huybrechts, "Lectures on K3 Surfaces", math.uni-bonn.de/people/huybrech/K3Global.pdf There is a finite field extension over which all rational curves have a rational point. Did I understand correctly your question?
Jun 30, 2016 at 15:31	history	edited	byu	CC BY-SA 3.0	added 37 characters in body
Jun 30, 2016 at 15:24	comment	added	byu		Interesting! I realize what I wanted to ask was that also the rational curves are all rational over this field extension (i.e., they have a rational point). Is this true?
Jun 30, 2016 at 15:04	comment	added	Jason Starr		(Edit.) I suspect the opposite. The geometric Picard group $\text{Pic}(X\otimes_{\mathbb{Q}}\overline{\mathbb{Q}})$ is a finitely generated Abelian group. Thus there exists a finite extension $K$ of $\mathbb{Q}$ over which there are defined invertible sheaves on $X\otimes_{\mathbb{Q}}K$ whose images generate the geometric Picard group. Every invertible sheaf defined on $X\otimes_{\mathbb{Q}}\overline{\mathbb{Q}}$ is already defined on $X\otimes_{\mathbb{Q}}K$. In particular, those invertible sheaves with self-square $-2$ are defined over $K$.
Jun 30, 2016 at 14:53	review	First posts
Jun 30, 2016 at 15:44
Jun 30, 2016 at 14:50	history	asked	byu	CC BY-SA 3.0