26 research outputs found
The DT/PT correspondence for smooth curves
We show a version of the DT/PT correspondence relating local curve counting
invariants, encoding the contribution of a fixed smooth curve in a Calabi-Yau
threefold. We exploit a local study of the Hilbert-Chow morphism about the
cycle of a smooth curve. We determine, via Quot schemes, the global
Donaldson-Thomas theory of a general Abel-Jacobi curve of genus .Comment: Minor changes, published versio
The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves
Let be a hyperelliptic curve embedded in its Jacobian via an
Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme
component of containing the Abel-Jacobi curve as a point. We
relate the result to the ramification (and to the fibres) of the Torelli
morphism along the hyperelliptic locus.
As an application, we determine the scheme structure of the moduli space of
Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.Comment: Improved the exposition according to the referees' suggestions. To
appear in Algebra and Number Theor
Jet bundles on Gorenstein curves and applications
In the last twenty years a number of papers appeared aiming to construct
locally free replacements of the sheaf of principal parts for families of
Gorenstein curves. The main goal of this survey is to present to the widest
possible audience of mathematical readers a catalogue of such constructions,
discussing the related literature and reporting on a few applications to
classical problems in Enumerative Algebraic Geometry.Comment: Minor revisions, improved expositio
On coherent sheaves of small length on the affine plane
We classify coherent modules on of length at most and supported
at the origin. We compare our calculation with the motivic class of the moduli
stack parametrizing such modules, extracted from the Feit-Fine formula. We
observe that the natural torus action on this stack has finitely many fixed
points, corresponding to connected skew Ferrers diagrams
Framed sheaves on projective space and Quot schemes
We prove that, given integers , and , the moduli
space of torsion free sheaves on with Chern character
that are trivial along a hyperplane
is isomorphic to the Quot scheme of -dimensional length quotients of the free sheaf
on .Comment: Minor improvement
Virtual classes and virtual motives of Quot schemes on threefolds
For a simple, rigid vector bundle on a Calabi-Yau -fold , we
construct a symmetric obstruction theory on the Quot scheme
, and we solve the associated enumerative theory. We
discuss the case of other -folds. Exploiting the critical structure on
, we construct a virtual motive
(in the sense of Behrend-Bryan-Szendr\H{o}i) for for an
arbitrary vector bundle on a smooth -fold . We compute the associated
motivic partition function. We obtain new examples of higher rank (motivic)
Donaldson-Thomas invariants.Comment: 22 pages. Removed appendix. To appear in Adv. Mat
On the motive of the Quot scheme of finite quotients of a locally free sheaf
Let be a smooth variety, a locally free sheaf on . We express the
generating function of the motives in terms of the
power structure on the Grothendieck ring of varieties. This extends a recent
result of Bagnarol, Fantechi and Perroni for curves, and a result of
Gusein-Zade, Luengo and Melle-Hern\'{a}ndez for Hilbert schemes. We compute
this generating function for curves and we express the relative motive
as a plethystic exponential.Comment: 17 pages. Minor revisions. Accepted for publication in J. Math. Pures
App