On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$
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| Format: | Preprint |
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2024
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| _version_ | 1866917570665250816 |
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| author | Coanda, Iustin |
| author_facet | Coanda, Iustin |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_10783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$ Coanda, Iustin Algebraic Geometry Primary 14J60, Secondary: 14H50, 14M05, 13H10 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]. |
| title | On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$ |
| topic | Algebraic Geometry Primary 14J60, Secondary: 14H50, 14M05, 13H10 |
| url | https://arxiv.org/abs/2401.10783 |