Timeline for Vector bundles on families of rational curves

3 events

when toggle format	what		by	license	comment
Aug 29, 2017 at 15:59	comment	added	Daniel Litt		Just a comment; Piotr's example is the ``universal family" of extensions over $\text{Ext}^1(\mathcal{O}(1), \mathcal{O}(-1))$.
Aug 29, 2017 at 14:39	comment	added	Piotr Achinger		Unfortunately no. Basic example: $S=\mathbb{A}^1$ with coordinate t, $\mathcal{X}= S\times \mathbb{P}^1$ with projection $\pi:\mathcal{X}\to \mathbb{P}^1$. Take the Euler sequence $0\to \mathcal{O}(-1) \to \mathcal{O}^2\to\mathcal{O}(1)\to 0$ on $\mathbb{P}^1$ and pull it back via $\pi$. Now take the push-out of the resulting sequence by $t$ acting on the left term, obtaining an extension $0\to \pi^\mathcal{O}(-1) \to E\to \pi^ \mathcal{O}(1)\to 0$. The restriction of $E$ to the fiber over $0$ is $\mathcal{O}(-1)\oplus \mathcal{O}(1)$, and on the other fibers we have $E\cong \mathcal{O}^2$.
Aug 29, 2017 at 14:30	history	asked	Jana	CC BY-SA 3.0