Standard Classes on the Blow-Up of ℙⁿ at Points in Very General Position

Abstract

We study (−1)-classes on the blow-up X of ℙ ⁿ at points in very general position; in the case n = 2, we give a new proof of the equivalence of two conjectures about the dimension of the class of a divisor on X, and we prove that h ¹(D) = 0 for any nef D which is a nonnegative sum of (−1)-curves. For n = 3, we fill a gap in the formulation of a conjecture about the dimension of a class. We provide algorithms and Maple code based on these conjectures.

Key Words:

• Birational transformations
• Fat points
• Linear systems

2010 Mathematics Subject Classification:

• 14C20

ACKNOWLEDGMENT

We thank Alessandro Gimigliano for sending us his PhD Thesis. The first author has been partially supported by Proyecto FONDECYT Regular 2011, No. 1110096.

Notes

Communicated by R. Piene.