Abstract
Let be a smooth hypersurface of degree 6 over complex numbers. In this article, we give a characterization of initialized and ACM bundles of rank 1 on X with respect to the line bundle given by a smooth hyperplane section of X.
Acknowledgement
I thank Dr. Sarbeswar Pal for many discussions and an anonymous mathoverflow user abx for suggesting the article [Citation10]. I also thank the referee for valuable comments and suggestions.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 Observe that,
2 In this case, it is easy to see that we have,
3 In this representation, we must have Therefore, under this notation the following situations don’t occur
• For can’t be 2 and for
can’t be
• For can’t be 2.
4 For i = 1, this follows from initializedness and for i = 0 this follows by Remark 2.5
5 For this follows from
for
this follows from initializedness and for
this follows from Lemma 2.11.
6 Observe that, when then we always have this expression for some integer k.
7 Note that, we have