Instanton bundles on the flag variety F(0,1,2)

Malaspina, Francesco; Marchesi, Simone; Pons-Llopis, Joan

doi:10.2422/2036-2145.201801_003

Cited by 11 publications

(16 citation statements)

References 16 publications

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“…For a large family of Fano threefolds with Picard number one, Faenzi in [17, Theorem A] proves that the moduli spaces of instanton bundles has a generically smooth irreducible component. An analogous result has been obtained in [27] for the flag threefold. In the case of

P^{3}

, it is known that the moduli space of instantons of arbitrary charge is affine (see [14]), irreducible (see [33, 34]) and smooth (see [24]).…”

Section: Introductionsupporting

confidence: 84%

“…Ulrich sheaves are defined to be the aCM sheaves whose corresponding module has the maximum number of generators. Remark It is worthwhile to point out that, exactly as in the case of F (0, 1, 2) (see [27] Remark 2.2), the condition

H^{0} false(E false) = 0

does not follow from the other conditions defining an instanton bundle. Indeed we may consider the rank two aCM bundles with

c_{1} false(E false) = 0

and

H^{0} false(E false) \neq 0

given in [12] Theorem B.…”

Section: First Properties and Monads Of Instanton Bundles On Double-smentioning

confidence: 99%

“…Remark Let us remark that the monad (2.18) is the analog of the monad for instanton bundles on

P^{3}

0 \to {scriptO}_{P^{3}} {(- 1)}^{\oplus k} \overset{α}{\to} {scriptO}_{P^{3}}^{\oplus 2 k + 2} \overset{β}{\to} {scriptO}_{P^{3}} {(1)}^{\oplus k} \to 0,

and the monad (2.13) is the analog of the second monad for instanton bundles on

P^{3}

(see for instance [3] display (1.1))

0 \to {scriptO}_{P^{3}} {(- 1)}^{\oplus k} \overset{α}{\to} {normalΩ}_{P^{3}} {(1)}^{\oplus k} \overset{β}{\to} {scriptO}_{P^{3}}^{\oplus 2 k - 2} \to 0 .

A very similar behavior was shown for the two monads for instanton bundles on the flag threefold in [27]. As in the case of instanton bundles on the projective space and flag varieties, the two monads (2.18) and (2.13) are closely related.…”

Section: First Properties and Monads Of Instanton Bundles On Double-smentioning

confidence: 99%

“…On F (0, 1, 2) instanton bundles has been studied in [10], [15] and more recently in [27]. F (0, 1, 2) is a Fano threefold with Picard number two.…”

Section: Introductionmentioning

confidence: 99%

“…Let us call h 1 and h 2 the two generators. In [27] has been given the following definition: a rank two vector bundle E on the Fano threefold F (0, 1, 2) is an instanton bundle of charge k if the following properties hold

c_{1} false(E false) = 0, c_{2} false(E false) = k h_{1} h_{2}

;

h^{0} false(E false) = h^{1} false(E (- 1, - 1) false) = 0

and E is μ‐semistable. Notice that, when the Picard number is one, the condition

H^{0} false(E false) = 0

implies the μ‐stability. When the Picard number is higher than one, however, this is not true and it is natural to consider also μ‐semistable bundles (see [15] and [27] Remark 2.2).…”

Section: Introductionmentioning

confidence: 99%

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Instanton bundles on the Segre threefold with Picard number three

Antonelli

Malaspina

2020

Mathematische Nachrichten

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We study instanton bundles on ℙ 1 × ℙ 1 × ℙ 1 . We construct two different monads which are the analog of the monads for instanton bundles on ℙ 3 and on the flag threefold (0, 1, 2). We characterize the Gieseker semistable cases and we prove the existence of -stable instanton bundles generically trivial on the lines for any possible 2 ( ). We also study the locus of jumping lines. K E Y W O R D SBeilinson spectral theorem, instanton bundles, jumping lines, Segre varieties M S C ( 2 0 1 0 ) (Primary) 14J60, (Secondary) 13C14, 14F05 1026

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P^{3}

, it is known that the moduli space of instantons of arbitrary charge is affine (see [14]), irreducible (see [33, 34]) and smooth (see [24]).…”

Section: Introductionsupporting

confidence: 84%

H^{0} false(E false) = 0

does not follow from the other conditions defining an instanton bundle. Indeed we may consider the rank two aCM bundles with

c_{1} false(E false) = 0

and

H^{0} false(E false) \neq 0

given in [12] Theorem B.…”

Section: First Properties and Monads Of Instanton Bundles On Double-smentioning

confidence: 99%

“…Remark Let us remark that the monad (2.18) is the analog of the monad for instanton bundles on

P^{3}

0 \to {scriptO}_{P^{3}} {(- 1)}^{\oplus k} \overset{α}{\to} {scriptO}_{P^{3}}^{\oplus 2 k + 2} \overset{β}{\to} {scriptO}_{P^{3}} {(1)}^{\oplus k} \to 0,

and the monad (2.13) is the analog of the second monad for instanton bundles on

P^{3}

(see for instance [3] display (1.1))

0 \to {scriptO}_{P^{3}} {(- 1)}^{\oplus k} \overset{α}{\to} {normalΩ}_{P^{3}} {(1)}^{\oplus k} \overset{β}{\to} {scriptO}_{P^{3}}^{\oplus 2 k - 2} \to 0 .

Section: First Properties and Monads Of Instanton Bundles On Double-smentioning

confidence: 99%

“…On F (0, 1, 2) instanton bundles has been studied in [10], [15] and more recently in [27]. F (0, 1, 2) is a Fano threefold with Picard number two.…”

Section: Introductionmentioning

confidence: 99%

c_{1} false(E false) = 0, c_{2} false(E false) = k h_{1} h_{2}

;

h^{0} false(E false) = h^{1} false(E (- 1, - 1) false) = 0

and E is μ‐semistable. Notice that, when the Picard number is one, the condition

H^{0} false(E false) = 0

implies the μ‐stability. When the Picard number is higher than one, however, this is not true and it is natural to consider also μ‐semistable bundles (see [15] and [27] Remark 2.2).…”

Section: Introductionmentioning

confidence: 99%

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Instanton bundles on the Segre threefold with Picard number three

Antonelli

Malaspina

2020

Mathematische Nachrichten

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On rank 3 instanton bundles on P3$\mathbb {P}^3$

Andrade,

Santiago,

Silva

et al. 2024

Mathematische Nachrichten

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We investigate rank 3 instanton vector bundles on of charge and its correspondence with rational curves of degree . For , we present a correspondence between stable rank 3 instanton bundles and stable rank 2 reflexive linear sheaves of Chern classes and we use this correspondence to compute the dimension of the family of stable rank 3 instanton bundles of charge 2. Finally, we use the results above to prove that the moduli space of rank 3 instanton bundles on of charge 2 coincides with the moduli space of rank 3 stable locally free sheaves on of Chern classes . This moduli space is irreducible, has dimension 16 and its generic point corresponds to a generalized't Hooft instanton bundle.

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