In recent years, the problems related to linear series, specifically projective normality, normal presentation, and higher syzygies of an ample line bundle L on a smooth projective variety X have attracted considerable attention. The initial questions about projective normality and normal presentation on curves were proved by Castelnuova, Mattuck, Fujita and St-Donat. Later Mark Green unified the above concepts and introduced the Np-property, also called the p-th syzygy property for p ≥ 0, and generalized these results to a statement about syzygies. More precisely, if L is a line bundle on a curve C of genus g such that degree of L is at least 2g + 1 + p then L satisfies Np-property, for p ≥ 0.
In the case of higher dimensional varieties Mukai conjectured that for any smooth polarized projective variety (X, L), KX ⊗L ⊗p+4 satisfies Np-property, where KX denotes the canonical line bundle on X. Mukai’s conjecture has not yet been proved even for p = 0, but some significant work has been done in some special cases by Kempf, Y. Homma, Ein and Lazarsfeld. A stronger version of the Mukai conjecture in the case of Enriques surfaces and for the property N0-property is proved by Gallego and Purnaprajna. Gallego and Purnaprajna have done some nice work regarding syzygy properties on surfaces and three folds. In the case of abelian varieties Lazarsfeld conjectured that if L is an ample line bundle on abelian variety X then L ⊗p+3 satisfies Np-property.
On the other hand, Fujita’s conjecture on the very ampleness of a line bundle has attracted attention in the past years. Indeed, if L is an ample line bundle on an algebraic variety X of dimension n, then KX ⊗ L ⊗n+2 is very ample. Fujita’s conjecture has been proven for algebraic surfaces but this problem is still open for higher dimensional varieties.
NOTE: The questions related to very ampleness, k-jet ampleness, and syzygies are completely known in the case of abelian varieties.
We investigate the above properties of linear series on some smooth projective varieties.
