On the spectrum of a stable rank 2 vector bundle on ℙ3

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The spectrum of a stable rank 2 vector bundle E with c 1 = 0 on the projective 3-space is a finite sequence of positive integers s ( 0 ) ,…, s ( m ) characterizing the Hilbert function of the graded H 1 -module of E in negative degrees. Hartshorne [Invent. Math. 66 (1982), 165–190] showed that if s ( i ) = 1 for some i > 0 then s ( i + 1 ) = 1 ,…, s ( m ) = 1 . We show that if s ( 0 ) = 1 then E ( 1 ) has a global section whose zero scheme is a double structure on a space curve. We deduce, then, the existence of sequences satisfying Hartshorne’s condition that cannot be the spectrum of any stable 2-bundle. This provides a negative answer to a question of Hartshorne and Rao [J. Math. Kyoto Univ. 31 (1991), 789–806].