# Macdonald formula for curves with planar singularities

@article{Maulik2011MacdonaldFF, title={Macdonald formula for curves with planar singularities}, author={D. Maulik and Zhiwei Yun}, journal={arXiv: Algebraic Geometry}, year={2011} }

We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and… Expand

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#### 35 Citations

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#### References

SHOWING 1-10 OF 40 REFERENCES

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

- Mathematics, Physics
- 2012

The intersection of a complex plane curve with a small three -sphere surrounding one of its singularities is a non-trivial link. The refined punct ual Hilbert schemes of the singularity parameterize… Expand

Topology of the compactified Jacobians of singular curves

- Mathematics
- 2006

We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p, q), (4,… Expand

Euler number of the compactified Jacobian and multiplicity of rational curves

- Mathematics
- 1997

We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $\delta$-constant stratum in the base of a… Expand

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

- Mathematics
- Compositio Mathematica
- 2012

Abstract Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point… Expand

Ideals associated to deformations of singular plane curves

- Mathematics
- 1988

We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus ES which parametrizes equisingular or topologically… Expand

Symmetric products of an algebraic curve

- Mathematics
- 1962

LET X be a nonsingular irreducible complete algebraic curve over the field C of complex numbers, and let X(n) denote the nth symmetric product of X. The first part of this paper is devoted to an… Expand

Global Springer theory

- Mathematics
- 2011

Abstract We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin… Expand

Counting rational curves on K3 surfaces

- Mathematics
- 1997

The aim of these notes is to explain the remarkable formula found by Yau and Zaslow to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families… Expand

DECOMPOSITION THEOREM AND ABELIAN FIBRATION

- 2009

Let X be a smooth algebraic variety over a field k and f : X → S be a proper morphism. By Deligne’s theorem [5], the direct image f∗Q` is a pure complex i.e. for the perverse t-structure, the… Expand

Fixed point varieties on affine flag manifolds

- Mathematics
- 1988

We study the space of Iwahori subalgebras containing a given element of a semisimple Lie algebra over C((ɛ)). We also define and study a map from nilpotent orbits in a semisimple Lie algebra over C… Expand