Tautological Rings of Moduli Spaces of Curves
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mumford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincaré duality.
The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra.
In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result.
In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well.
Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , vii, 34 p.
Trita-MAT. MA, ISSN 1401-2278 ; 2011:03
IdentifiersURN: urn:nbn:se:kth:diva-34310ISBN: 978-91-7501-043-4 (print)OAI: oai:DiVA.org:kth-34310DiVA: diva2:420360
2011-06-15, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
Farkas, Gavril, Professor
Faber, Carel, Professor
QC 201106082011-06-082011-06-012011-06-08Bibliographically approved
List of papers