A note on the ADHM description of Quot schemes of points on affine spaces

Henni, Abdelmoubine Amar; Guimarães, Douglas Magno

doi:10.1142/s0129167x21500312

Cited by 5 publications

(6 citation statements)

References 31 publications

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“…Therefore, by (7) and Theorem 4.2, (ϕ, Q) is GIT-(semi)stable if, and only if, E = ker ϕ is (semi)stable.…”

Section: Moduli Spaces Of Semistable Sheaves Onmentioning

confidence: 89%

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Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Guimarães¹,

Jardim²

2021

Preprint

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A torsion free sheaf E on P d is called quasitrivial if E ∨∨ = O ⊕r P 3 and dim(E ∨∨ /E) = 0. While such sheaves are always µ-semistable, they may not be semistable. We study the Gieseker-Maruyama moduli space N (r, n) of rank r semistable quasitrivial sheaves on P 3 with h 0 (E ∨∨ /E) = n via the Quot scheme of points Quot(O ⊕r P 3 , n). We show that N (r, n) is empty when r > n, while N (n, n) has no stable points and is isomorphic to the symmetric product Sym n (P 3 ). Our main result is the construction of an irreducible component of N (r, n) of dimension 2n+rn−r 2 +1 when r < n. Furthermore, this is the only irreducible component when n ≤ 10.

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“…Therefore, by (7) and Theorem 4.2, (ϕ, Q) is GIT-(semi)stable if, and only if, E = ker ϕ is (semi)stable.…”

Section: Moduli Spaces Of Semistable Sheaves Onmentioning

confidence: 89%

“…Our starting point is the fact that the affine Quot scheme Quot(O ⊕r A 3 , n) is irreducible for n ≤ 10, see [7]. In order to see that the same holds for the projective Quot scheme Quot(O ⊕r P 3 , n), we will use the following technical lemma.…”

Section: Moduli Spaces Of Semistable Sheaves Onmentioning

confidence: 99%

Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Guimarães¹,

Jardim²

2021

Preprint

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“…As a consequence, by results of [60,61] the instanton moduli space M r,k is isomorphic to the Quot scheme Quot k r ( 4 ) of zero-dimensional quotients of the free sheaf O ⊕r 4 with length k,…”

Section: Geometrical Interpretationmentioning

confidence: 99%

Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds

Szabo¹,

Tirelli²

2023

Preprint

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We study rank r cohomological Donaldson-Thomas theory on a toric Calabi-Yau orbifold of 4 by a finite abelian subgroup Γ of SU(4), from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on 4 /Γ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over r-vectors of Γ-coloured solid partitions. When the Γ-action fixes an affine line in 4 , we exhibit the dimensional reduction to rank r Donaldson-Thomas theory on the toric Kähler three-orbifold 3 /Γ. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds 2 / n × 2 and 3 /( 2 × 2 ) × , finding perfect agreement with new mathematical results of Cao, Kool and Monavari.

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“…We thank A. Henni for suggesting that it might also be possible to give a proof of Theorem A combining the formalism of perfect extended monads [14,15] with the result of Abe-Yoshinaga (Theorem 1.5). The 3-dimensional case is also studied along these lines in [5,Sec.…”

Section: Remark 29mentioning

confidence: 99%

“…This construction is called r -framingfor m = 3 it has some relevance in motivic Donaldson-Thomas theory [6,7] and K-theoretic Donaldson-Thomas theory [13]. It is also performed with care in [15] in the r = 1 case and in [14] for arbitrary r .…”

Section: Embedding In the Non-commutative Quot Schemementioning

confidence: 99%

Framed sheaves on projective space and Quot schemes

Cazzaniga

Ricolfi

2021

Math. Z.

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We prove that, given integers $$m\ge 3$$ m ≥ 3 , $$r\ge 1$$ r ≥ 1 and $$n\ge 0$$ n ≥ 0 , the moduli space of torsion free sheaves on $${\mathbb {P}}^m$$ P m with Chern character $$(r,0,\ldots ,0,-n)$$ ( r , 0 , … , 0 , - n ) that are trivial along a hyperplane $$D \subset {\mathbb {P}}^m$$ D ⊂ P m is isomorphic to the Quot scheme $$\mathrm{Quot}_{{\mathbb {A}}^m}({\mathscr {O}}^{\oplus r},n)$$ Quot A m ( O ⊕ r , n ) of 0-dimensional length n quotients of the free sheaf $${\mathscr {O}}^{\oplus r}$$ O ⊕ r on $${\mathbb {A}}^m$$ A m . The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.

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A note on the ADHM description of Quot schemes of points on affine spaces

Cited by 5 publications

References 31 publications

Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds

Framed sheaves on projective space and Quot schemes

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