# Tautological bundle of $\mathbb{P}^1(\mathbb{C})$

Guys I have this problem:

Is it true that the tautological bundle on $\mathbb{P}^1(\mathbb{C})$ given by $L(-1)=\lbrace (\ell,z)\in\mathbb{P}^1\times \mathbb{C}^2 \: : \: z\in \ell \rbrace$ is a subbundle of $L(0)^2$, where $L(0)\cong \mathbb{P}^1\times \mathbb{C}$ ?

• I mean ... by definition $L(-1) \subset L(0)^2$ right ? – user171326 Jan 5 '17 at 22:26
• Can you explain me better? – Vincenzo Zaccaro Jan 5 '17 at 22:27
• I interpret $L(0)^2$ as the trivial bundle over $\mathbb P^1$ with fiber $\mathbb C^2$. So every fiber of $L(-1)$ is a line in $L(0)^2$ so it is a subbundle. – user171326 Jan 5 '17 at 22:29
• Ah in this way it's ok! I thought L(0)^2 as cartesian product. – Vincenzo Zaccaro Jan 5 '17 at 22:35
• Yes, for a vector bundle $E^2$ mean $E \oplus E$ where $\oplus$ is the fiberwise direct sum over the same base. – user171326 Jan 5 '17 at 22:36