On a class of special linear systems of $\mathbb{P}^3$

Abstract

Abstract. In this paper we deal with linear systems of P 3 through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.

On linear systems ofP3through multiple points

“…Recently a conjecture on the structure of special systems of P 3 which generalizes the Harbourne-Hirschowitz conjecture [2,3] has been formulated in [4]. In the current paper we provide a proof of this conjecture for the case of at most eight multiple points (in general position).…”

Base locus of linear systems on the blowing-up of ℙ3along at most 8 general points

“…3 ), induces an action on Pic X and one on A 2 (X ), as stated in the next two propositions (see [Laface and Ugaglia 2003] for a proof of both results).…”

“…It can be proved (see [Laface and Ugaglia 2003]) that the cubic Cremona transformation on X , is obtained by blowing-up the strict transforms of the six edges of the tetrahedron through the four points used by the cubic Cremona transformation, and blowing down along the other rulings of the exceptional quadrics. This implies in particular that the cubic Cremona transformation is not just a base change of Pic X .…”

“…, P 4 , is then nothing else than a base change for Pic Y . In particular, in [Laface and Ugaglia 2003], it is shown that…”

On the postulation of sd fat points in Pd

“…. , µ r ), due to Iarrobino [4] (see also [5]). Facing the critical cases too, he derived from his conjecture a generalization of Nagata's conjecture:…”