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On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$
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abstract
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].
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Cited by 1 Pith paper
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Classification of monads and a new moduli component of stable rank 2 bundles on $\mathbb{P}^3$ with even determinant and $c_2=9$
For stable rank 2 bundles on P3 with c1=0 and c2=9, the paper determines all minimal Horrocks monads up to two unresolved nonnegative cases, and proves B(9) has a new irreducible component of dimension at least 74.
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