## Abstract

We study double Hurwitz numbers in genus zero counting the number of covers C P^{1} → C P^{1} with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.

Original language | English |
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Pages (from-to) | 79-96 |

Number of pages | 18 |

Journal | Advances in Mathematics |

Volume | 217 |

Issue number | 1 |

DOIs | |

State | Published - 15 Jan 2008 |

### Bibliographical note

Funding Information:The authors are grateful to Max-Planck-Institut für Matematik, Bonn, and to Institut Mittag-Leffler, Djursholm, Sweden, for hospitality in Summer 2004, and Fall 2006. Our sincere thanks go to M. Kazaryan, R. Kulkarni and D. Zvonkine for useful discussions. S.S. was supported by the grants RFBR-05-01-01012-a, RFBR-05-01-02806-CNRS-a, NSh-1972.2003.1, MK-5396.2006.1, NWO-RFBR-047.011.2004.026 (RFBR-05-02-89000-NWO-a), by the Göran Gustafsson foundation, and by Pierre Deligne’s fund based on his 2004 Balzan prize in mathematics. M.S. was supported by the grants DMS-0401178 and PHY-0555346. M.S and A.V. were supported by the grant BSF-2002375.

## Keywords

- Chambers
- Double Hurwitz numbers
- Piecewise polynomiality
- Wall crossing

## ASJC Scopus subject areas

- Mathematics (all)