Timeline for Self intersection of theta divisor

Current License: CC BY-SA 4.0

11 events

when toggle format	what		by	license	comment
S May 18, 2020 at 14:50	history	suggested	red_trumpet	CC BY-SA 4.0	changed some math to text (should be text imo)
May 18, 2020 at 14:04	review	Suggested edits
S May 18, 2020 at 14:50
Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Apr 1, 2017 at 7:49	comment	added	Francesco Polizzi		$K_J \Theta=0$ simply because the canonical class of any abelian variety is trivial.
Mar 31, 2017 at 17:23	comment	added	Eduardo R. Duarte		Is this $0$ because the zero class in $J$ is the blowdown of the canonical divisor of $\text{Sym}^2(H)$ and $\text{Sym}^2(H)$ and $J$ are birational?
Mar 31, 2017 at 17:20	comment	added	Eduardo R. Duarte		Dear Damian, thanks for the comment. I was trying to use in fact the adjuction formula. and what I got is that $\deg K_\Theta = \Theta\bullet\Theta + K_J\bullet \Theta$ which is equal to $2g-2=2$ by Riemann-Roch. Why in my case then it should happen that $K_J\bullet\Theta = 0$ ?
Mar 31, 2017 at 16:19	comment	added	Damian Rössler		You may suppose that $\Theta=H$. Then ${\rm deg}(\iota^*(\Theta))=\Theta\cdot\Theta={\rm deg}(N)$, where $N$ is the normal bundle of $\Theta$ in $J$ (by the adjunction formula, see Prop. V.1.5 in Hartshorne). There is a canonical sequence $0\to N^\vee\to\Omega_{J}\|_\Theta\to\Omega_\Theta\to 0$ so ${\rm deg}(N)={\rm deg}(\Omega_\Theta)=2\cdot 2-2=2$, since $\Omega_J$ is a trivial vector bundle.
Mar 31, 2017 at 11:22	history	edited	Eduardo R. Duarte	CC BY-SA 3.0	added 41 characters in body
Mar 31, 2017 at 10:55	history	edited	Eduardo R. Duarte	CC BY-SA 3.0	deleted 6 characters in body
Mar 31, 2017 at 9:49	history	edited	Eduardo R. Duarte	CC BY-SA 3.0	added 158 characters in body
Mar 31, 2017 at 9:43	history	asked	Eduardo R. Duarte	CC BY-SA 3.0