A counterexample to a conjecture on linear systems on ℙ3

Abstract

In his paper (1) Ciliberto proposes a conjecture in order to characterize special linear systems of P n through multiple base points. In this note we give a counterexample to this con- jecture by showing that there is a substantial dierence between the speciality of linear systems on P 2 and those of P 3 . Let us take the projective space P n and let us consider the linear system of hyper- surfaces of degree d having some points of fixed multiplicity. The virtual dimension of such systems is the dimension of the space of degree d polynomials minus the con- ditions imposed by the multiple points and the expected dimension is the maximum between the virtual one and � 1. The systems whose dimension is bigger than the ex- pected one are called special systems. There exists a conjecture due to Hirschowitz (see (5)), characterizing special linear systems on P 2 , which has been proved in some special cases (2), (3), (7), (6). Concerning linear systems on P n , in (1) Ciliberto gives a conjecture based on the classification of special linear systems through double points. In this note we describe a linear system on P 3 that we found in a list of special systems generated with the help of Singular (4) and which turns out to be a counterexample to that conjecture. The paper is organized as follows: in Section 1 we fix some notation and state Ciliberto's conjecture, while Section 2 is devoted to the counterexample. In Section 3 we try to explain speciality of some systems by the Riemann-Roch formula, and we conclude the note with an appendix containing some computations.