Some Quot schemes in tilted hearts and moduli spaces of stable pairs

For a smooth projective variety [Formula: see text], we study analogs of Quot schemes using hearts of non-standard [Formula: see text]-structures of [Formula: see text]. The technical framework uses families of [Formula: see text]-structures as studied in A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry and P. Stellari, Stability conditions in families, preprint (2019), arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117(2) (1994) 317–353; D. Huybrechts and M. Lehn, Stable pairs on curves and surfaces, J. Algebraic Geom. 4(1) (1995) 67–104, as instances of Quot schemes.

Seshadri constants on some Quot schemes

Let E be a vector bundle of rank n on ℙ 1 {\mathbb{P}^{1}} . Fix a positive integer d. Let 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} denote the Quot scheme of torsion quotients of E of degree d and let Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} denote the Grassmann bundle that parametrizes the d-dimensional quotients of the fibers of E. We compute Seshadri constants of ample line bundles on 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} and Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} .

Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme $$LQ_{-e}(W)$$ L Q - e ( W ) parameterizes subsheaves of rank n and degree $$-e$$ - e which are isotropic with respect to the symplectic form. We prove that $$LQ_{-e}(W)$$ L Q - e ( W ) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable.

Framed sheaves on projective space and Quot schemes

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.