Monads and moduli components for stable rank 2 bundles with odd determinant on the projective space. (English) Zbl 1507.14020
This paper deals with delicate questions of classification of rank 2 stable vector bundles on \(\mathbb P ^3\) with odd \(c_1\), a case less studied than \(c_1\) even (for instance, in the paper [R. Hartshorne and A. P. Rao, J. Math. Kyoto Univ. 31, No. 3, 789–806 (1991; Zbl 0762.14010)] it is shown that there are vector bundles corresponding to each of the theoretically possible spectra for \(c_1=0\), \(c_2\) up to \(19\)).
The authors stress that the irreducible components of the moduli space of stable vector bundles with fixed Chern classes are not characterized by the spectrum, or by the defining monad. Also, vector bundles with the same monad can have different spectra, or with the same spectrum can have different monads.
Reviewer’s remark: We note that the case of spectrum \(\{ -2,-2,-2,-1,0,1,1,1\}\), which is asserted not to exist (cf. Proposition 17), can in fact be realized via the following construction: consider a rational quartic \(X\) in \(\mathbb P ^3\), which is a divisor of type \((1,3)\) on a smooth quadric, take \(Y\) a Ferrand double structure on it, given via an exact sequence \[ 0 \to I_Y \to I_X \to \omega _X(3) \to 0 \ , \] and take \(E\) to be given by an extension (observe that \(\omega_Y \cong {\mathcal O}_Y(-3)\)): \[ 0 \to O(-1) \to E \to I_Y \to 0 . \] Then \(E\) has spectrum \(\{-2,-2,-2,-1,0,1,1,1\}=\{r_0r_1^3\}\).
The authors stress that the irreducible components of the moduli space of stable vector bundles with fixed Chern classes are not characterized by the spectrum, or by the defining monad. Also, vector bundles with the same monad can have different spectra, or with the same spectrum can have different monads.
Reviewer’s remark: We note that the case of spectrum \(\{ -2,-2,-2,-1,0,1,1,1\}\), which is asserted not to exist (cf. Proposition 17), can in fact be realized via the following construction: consider a rational quartic \(X\) in \(\mathbb P ^3\), which is a divisor of type \((1,3)\) on a smooth quadric, take \(Y\) a Ferrand double structure on it, given via an exact sequence \[ 0 \to I_Y \to I_X \to \omega _X(3) \to 0 \ , \] and take \(E\) to be given by an extension (observe that \(\omega_Y \cong {\mathcal O}_Y(-3)\)): \[ 0 \to O(-1) \to E \to I_Y \to 0 . \] Then \(E\) has spectrum \(\{-2,-2,-2,-1,0,1,1,1\}=\{r_0r_1^3\}\).
Reviewer: Nicolae Manolache (Bucureşti)
MSC:
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14F06 | Sheaves in algebraic geometry |
Citations:
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