### D.5.5 goettsche_lib

Library:
  goettsche.lib
Purpose:
  Drezet's formula for the Betti numbers of the moduli space of Kronecker modules; Goettsche's formula for the Betti numbers of the Hilbert scheme of points on a surface; Nakajima's and Yoshioka's formula for the Betti numbers of the punctual Quot-schemes on a plane or, equivalently, of the moduli spaces of the framed torsion-free planar sheaves; Macdonald's formula for the symmetric product

Author:
  Oleksandr Iena, o.g.yena@gmail.com

References:
  [1] Drezet, Jean-Marc Cohomologie des varie'te's de modules de hauter nulle. Mathematische Annalen: 281, 43-85, (1988).

  [2] Goettsche, Lothar, The Betti numbers of the Hilbert scheme of points on a smooth projective surface.
  Mathematische Annalen: 286, 193-208, (1990).

  [3] Macdonald, I. G., The Poincare polynomial of a symmetric product, Mathematical proceedings of the Cambridge Philosophical Society: 58, 563-568, (1962).

  [4] Nakajima, Hiraku; Lectures on instanton counting, CRM Proceedings and Lecture Notes, Yoshioka, Kota Volume 88, 31-101, (2004).

Procedures:

> D.5.5.1 GoettscheF |  | The Goettsche's formula up to n-th degree
> D.5.5.2 PPolyH |  | Poincare Polynomial of the Hilbert scheme of n points on a surface
> D.5.5.3 BettiNumsH |  | Betti numbers of the Hilbert scheme of n points on a surface
> D.5.5.4 NakYoshF |  | The Nakajima-Yoshioka formula up to n-th degree
> D.5.5.5 PPolyQp |  | Poincare Polynomial of the punctual Quot-scheme of rank r on n planar points
> D.5.5.6 BettiNumsQp |  | Betti numbers of the punctual Quot-scheme of rank r on n planar points
> D.5.5.7 MacdonaldF |  | The Macdonald's formula up to n-th degree
> D.5.5.8 PPolyS |  | Poincare Polynomial of the n-th symmetric power of a variety
> D.5.5.9 BettiNumsS |  | Betti numbers of the n-th symmetric power of a variety
> D.5.5.10 PPolyN |  | Poincare Polynomial of the moduli space of Kronecker modules N (q; m, n)
> D.5.5.11 BettiNumsN |  | Betti numbers of the moduli space of Kronecker modules N (q; m, n)

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