Hurwitz cycles on the moduli of curves

          1 February 2018, Humboldt-Universität zu Berlin / Einstein Stiftung


         Photo by A. Ortega

Room 021, IRIS-Haus, Zum Großen Windkanal 6, 12489 Berlin

11:00-12:15, Rahul Pandharipande (ETH Zürich)

Hurwitz loci, integrals, and open questions

Hodge integrals over the moduli of Hurwitz covers have many beautiful properties. I will discuss their relationship to the Hilbert scheme of points of the plane. There are many interesting computations to consider --- most require the development of better tools. I will also discuss to the role of Hurwitz loci in the study of tautological classes as an introduction to the lectures by J. Schmitt and J. van Zelm.

14:00-15:15, Jason van Zelm (HU Berlin)

Nontautological bielliptic classes

Tautological classes are geometrically defined classes in the Chow ring of the moduli space of curves which are particularly well understood. The classes of many known geometrically defined loci have been shown to be tautological. A bielliptic curve is a curve with a 2-to-1 map to an elliptic curve. We will build on an idea of Graber and Pandharipande to show that the closure of the locus of bielliptic curves in the moduli space of stable curves of genus g is non-tautological when g is at least 12.

15:45-17:00, Johannes Schmitt (ETH Zürich)

Extending the tautlogical ring by Hurwitz cycles

The tautological ring is the subring of the cohomology of the moduli spaces of stable curves formed by tautological classes. Many of its features - like a generating set, intersection products and pullbacks/pushforwards under natural maps - are accessible in terms of combinatorics. Unfortunately, as shown in the previous talk, the tautological ring is in general a strict subring of the (algebraic) cohomology. We show how this ring can be extended by adding the classes of Hurwitz loci, i.e. loci of curves forming ramified covers of curves of lower genus (like the sets of hyperelliptic or the bielliptic curves discussed before). We demonstrate how this extended ring still admits a combinatorial description and how this can be used to compute many classes of Hurwitz loci that already lie in the tautological ring.

Organizers:   G. Farkas , R. Pandharipande