Framed sheaves on projective space and Quot schemes

Abstract

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.

ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds

“…10 The moduli space of its stable representations M k,N is isomorphic to the quot scheme of points Quot A 4 (O ⊕N , k) or, equivalently, to the moduli space of framed torsion free sheaves on P 4 , as we briefly show in appendix B.1. This isomorphism follows from an application of Beilinson's theorem or, equivalently, from an infinitesimal argument due to [48].…”

“…Here we will briefly study moduli spaces of framed torsion-free sheaves on P 4 and their relation to spaces of SU( 4) instantons (and their generalisations) on C 4 . Let us first notice that, in general, if E is a torsion-free sheaf of rank N on P 4 with ch(E ) = (N, 0, 0, 0, −k), framed along a divisor D, there exists a natural sequence of sheaves (for the proof of this result, see [48])…”

“…This approach was studied in much greater generality in [56], and more details about the proof can be found also in [57,58]. Moreover, the scheme-theoretic isomorphism between moduli spaces of framed torsion-free sheaves on P m and quot schemes of points on A m was recently proved by means of an infinitesimal argument in [48].…”

Tetrahedron instantons

“…From the perspective of string theory, F and S Z correspond to the D7 123 -branes and the D1-branes, respectively. Moreover, (4.16) is realized in noncommutative field theory by the vacuum solution (3.16) with p = 3 and N = N 123 .As proven in[95], M (n 123 ,0,0,0),k is isomorphic to the Quot scheme Quot k C 3 (O ⊕n 123 ), which parametrizes isomorphism classes of the quotients O ⊕n 123 ։ S Z such that the Hilbert-Poincare polynomial of S Z is k[96]. When n 123 = 1, this Quot scheme is the same as the Hilbert scheme Hilb k C 3 of k points on C 3 .…”

Sur la lissité du schéma Quot ponctuel emboîté

“…où les flèches horizontales sont des immersions fermées. On peut facilement verifier que Z ϕ n est aussi l'intersection schématique 2),(3a)-(3d), on déduit que, si r > 1, le schéma Quot m ( ⊕r , n ) est singulier dans les cas suivants : (2,3).…”