Linear Systems on Fano Threefolds. I

Abstract

In this note, we classify all the polarized Fano threefold (X, H) with Bs|H|¬ = ∅. As corollaries we obtained that (1) the very ample part of the conjecture of Fujita holds for smooth Fano threefolds and (2) global Seshadri constants of ample divisors on Fano threefolds are bounded from below by 1 except three types of polarized Fano threefolds.

Key Words:

• Fano threefolds
• Fujita conjecture
• Seshadri constants

1991 Mathematics Subject Classification:

• Primary 14J45
• Secondary 14C20

Acknowledgments

The author would like to thank Prof. Jun-Muk Hwang and Prof. Jonghae Keum for useful suggestions and showing him their preprint. The author would also like to thank the referee for correcting several mistakes. He was supported in part by KOSEF Grant R14-2002-007-01001-0(2002).

Notes

^#Communicated by L. Ein.