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On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$

Iustin Coanda

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].

On the spectrum of a stable rank 2 vector bundle on $\mathbb{P}^3$

Abstract

The spectrum of a stable rank 2 vector bundle with on the projective 3-space is a finite sequence of positive integers , ..., characterizing the Hilbert function of the graded -module of in negative degrees. Hartshorne [Invent. Math. 66 (1982), 165-190] showed that if for some then , ..., . We show that if then 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].
Paper Structure (6 sections, 21 theorems, 132 equations)

This paper contains 6 sections, 21 theorems, 132 equations.

Table of Contents

  1. Preliminaries on monads
  2. Spectra and monads
  3. Spectra and double structures on space curves, I
  4. Spectra and double structures on space curves, II
  5. Curves in a double plane
  6. Stable 2-bundles with $c_1 = 0$ and ${\fam0 h}^0(E(1)) = 2$

Key Result

Lemma 1.3

Under the hypothesis of Remark R:selfdual, $E$ admits a minimal monad $0 \rightarrow A \overset{\alpha}{\rightarrow} B \overset{\beta}{\rightarrow} A^\vee \rightarrow 0$ with $B = B_+ \oplus B_0 \oplus B_+^\vee$, where $B_+$ is a direct sum of line bundles of positive degree and $B_0$ is a trivial b

Theorems & Definitions (61)

  • Remark 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Definition 1.1
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • Theorem 2.1
  • ...and 51 more