The tautological rings of the moduli spaces of stable curves, R* Mg, n , encode enumerative data about the moduli spaces, and have themselves, at least conjecturally, a remarkable algebraic structure. In particular, it is known that RdimMg, n M g, na Q is the top-degree component of the tautological ring, and it is conjectured that the intersection pairing, RiMg, n xRdim Mg, n-i Mg, na RdimMg, n M g, na Q , is perfect. The focus of this work is to use explicit calculation of the intersection pairing to give supporting evidence for the conjectured structure of the tautological rings. To that end, it introduces a new software package for carrying out computations in the tautological rings. Bases for the high degree tautological groups, RdimMg, n -2M g, n and RdimMg, n -1M g, n , are described in terms of new graph-combinatorial objects, the generalized dual stable graphs, and these bases are used to present the intersection pairing against known bases for R2 Mg, n and R1 Mg, n . The intersection pairings are shown to be perfect in the cases i = 1 (for all g) and i = 2 (for g ay 4).The study of the tautological rings as enumerative tools was initiated by Mum- ford [Mum83], and, more recently, further ... Kontsevicha#39;s beautiful argument uses push-forwards into the tautological rings to solve a long-standing enumerative problem. ... the seemingly remarkable structure of the tautological rings themselves has been the focus of much recent attention. ... about the tautological rings of M.g, and the analogous statements about the tautological rings of M.9tn [ Pan02, HL97J.

Title | : | Perfect Pairings in the Tautological Rings of the Moduli Spaces of Stable Curves |

Author | : | Jameel Al-Aidroos |

Publisher | : | ProQuest - 2008 |

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