The bi-weighted euler sequence on scrolls

Abstract

Let E → ℂP' be a complex vector bundle of rank (k+1) over the complex projective line with Grothendieck splitting

$$E = \theta _{\mathbb{C}P^1 } ( - a_1 ) + ... + \theta _{\mathbb{C}P^1 } ( - a_{k + 1)} .$$

Consider it's projectivization P(E) → ℂP'. This paper introduces the bi-weighted Euler sequence on scrolls

$$0 \to 2L(0,0) \to 2L(1,0) + L( - a_1 ,1) + ... + L( - a_{k + 1} ,1) \to T_{P(E)} \to 0$$

where L(a,b) refers first to the ruling line bundle and then to the hyperplane class bundle of the scroll.