Finding ein components in the moduli spaces of stable rank 2 bundles on the projective 3-space

Кытманов, А. А.; Osipov, Nikolay N.; Тихомиров, С. А.

doi:10.1134/s0037446616020142

Cited by 4 publications

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“…(On the contrary, there are known certain components of M (0, n) for n even, e. g., the instanton components which are not fine -see [16].) Theorem 8.1(ii) provides a series of fine (open dense subsets of ) moduli components N (e, a, b, c) for c > 2a + b − e, b > a, (e, a) = (0, 0), and n = c 2 − a 2 − b 2 even, this series clearly being infinite -see [19].…”

Section: Now Consider the Compositionmentioning

confidence: 99%

“…Note that these two Ein components differ by the spectra of bundles therein (see [27]). Besides, as it follows from [19], there are no other Ein components in M (−1, 6).…”

mentioning

confidence: 98%

“…The first one is the 37-dimensional rational instanton component I 5 [7], [24], [18], a general sheaf of which is a mathematical instanton bundle. The next one is the 40-dimensional Ein component N (0, 0, 2, 3) -see [9], [10,Theorem 4.7], [14], and it coincides with the component Q We complete, using [1, Main Theorem 1], [19,Section 2] and [26,Theorem 3], the list of all currently known irreducible components of M (0, n) for 9 ≤ n ≤ 20. For these values of n, besides the Ein components and the instanton components I n of dimension 8n − 3, 9 ≤ n ≤ 20, (the rationality or stable rationality of these I n 's is unknown), the known irreducible components are 6 more components.…”

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confidence: 99%

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Series of rational moduli components of stable rank two vector bundles on $$\pmb {\mathbb {P}}^3$$ P 3

Кытманов

Тихомиров

2019

Sel. Math. New Ser.

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We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space M (e, n) of rank 2 stable vector bundles with the first Chern class e = 0 or-1 and all possible values of the second Chern class n on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers a, b, c, where b ≥ a and c > a + b. We show that, in the wide range when c > 2a + b − e, b > a, (e, a) = (0, 0), the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces M (e, n) over all n ≥ 1 contains an infinite series of rational components for both e = 0 and e = −1. Explicit constructions of rationality of Ein components under the above conditions on e, a, b, c and, respectively, of their stable rationality in the remaining cases, are given. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for c 1 = 0 and n even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition "n is odd", which is a usual sufficient condition for fineness.

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Section: Now Consider the Compositionmentioning

confidence: 99%

“…Note that these two Ein components differ by the spectra of bundles therein (see [27]). Besides, as it follows from [19], there are no other Ein components in M (−1, 6).…”

mentioning

confidence: 98%