This is the webpage for the reading group on moduli spaces of differentials based at HU Berlin.
Organizer: Carl Lian
Meeting time: Thursdays 13:00-14:30 CEST, starting 29. April
Meeting location: Zoom (password is the number of lines through three general points in P^2, spelled out, e.g. 'seventeen')
The goal of the seminar is to understand recent advances in intersection-theoretic aspects of moduli spaces of pairs (C,\omega) where C is a curve and \omega is a 1-form on C. In order to do so, one needs to compactify such spaces in a suitable way.
The seminar will be divided into two main parts.
(1) Loci of holomorphic and meromorphic differentials on M_{g,n} via the compact moduli of twisted differentials of Farkas-Pandharipande [FP], and their cycle classes, expressed in terms of double ramification loci and universal Abel-Jacobi morphisms. This culminates in the paper of Bae-Holmes-Pandharipande-Schmitt-Schwarz [BHPSS].
(2) Masur-Veech volumes of strata of differentials via intersection theory on the compact moduli space of Bainbridge-Chen-Gendron-Grushevsky-Möller [BCGGM1, BCGGM2]. This culminates in the paper of Chen-Möller-Sauvaget-Zagier [CMSZ].
Further topics and modifications to this plan may arise depending on the interests of the participants.
References (more to be added later)
[BHPSS] Bae, Holmes, Pandharipande, Schmitt, Schwarz, Pixton's formula and Abel-Jacobi theory on the Picard stack
[BCGGM1] Bainbridge, Chen, Gendron, Grushevsky, Möller, Compactification of strata of abelian differentials
[BCGGM2] Bainbridge, Chen, Gendron, Grushevsky, Möller, The moduli space of multi-scale differentials
[CMZ] Chen, Möller, Zagier, Quasimodularity and large genus limits of Siegel-Veech constants
[CMSZ] Chen, Möller, Sauvaget, Zagier, Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
[EO] Eskin, Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
[FP] Farkas, Pandharipande, The moduli space of twisted canonical divisors, appendix by Janda, Pandharipande, Pixton, Zvonkine
[GZ] Grushevsky, Zakharov: The zero section of the universal semiabelian variety, and the double ramification cycle
[Hol] Holmes, Extending the double ramification cycle by resolving the Abel-Jacobi map
[HS] Holmes, Schmitt, Infinitesimal structure of the pluricanonical double ramification locus
[JPPZ1] Janda, Pandharipande, Pixton, Zvonkine, Double ramification cycles on the moduli spaces of curves
[JPPZ2] Janda, Pandharipande, Pixton, Zvonkine, Double ramification cycles with target varieties
[KZ] Kontsevich, Zorich, Connected components of the moduli space of abelian differentials with prescribed singularities
[Pol] Polischuk, Moduli spaces of curves with effective r-spin structures
[Sau] Sauvaget, Volumes and Siegel-Veech constants of H(2g-2) and Hodge integrals
[Sch] Schmitt, Dimension theory of the moduli space of twisted k-differentials
Program (with links to notes), part 2 still tentative
29 April: Carl Lian, Classical results and overview of seminar ([Pol], [KZ])
6 May: Gabi Farkas, Moduli of twisted differentials ([FP])
13 May: Andrea Di Lorenzo, Double ramification cycles and classes of twisted differentials ([JPPZ1], [FP] appendix, [Sch]): explain the basics of double ramification cycles and Pixton's class, and the relation to classes of twisted differentials, with multiplicities on the boundary components. Explain how to extract the class of the closure of differentials on the smooth locus. If time, mention Schmitt's extension to k-differentials; the ideas are essentially the same. Please do some examples!
20 May, part 1: Andrea Di Lorenzo, cont.
part 2: Federico Moretti, Introduction to universal Abel-Jacobi maps and their indeterminacy ([GZ]).
27 May: Federico Moretti, Resolution of the universal Abel-Jacobi map ([Hol]): explain Holmes's resolution of the universal Abel-Jacobi map. Say a bit about comparisons to previous constructions of double ramification loci.
3 June, part 1: Federico Moretti, cont.
part 2: Alessio Cela, introduction
10 June (part 1, part 2): Alessio Cela, Twisted differentials via the universal Abel-Jacobi map ([HS]): explain how to get loci of differentials using Holmes's construction. The main result is that this construction agrees with twisted differentials, with the correct multiplicities.
17 June: Andrei Bud, The double ramification locus on the universal Picard stack ([BHPSS]): explain the computation of the universal AJ/DR locus via DR loci on target varieties [JPPZ2], thus proving the formulas for cycle classes. This is long and heavy on the Gromov-Witten theory; try to give a high-level overview.
no talk 24 June (Oberwolfach workshop on Classical Algebraic Geometry)
1 July: Andrei Bud, cont.
Carl Lian, BCGGM compactification: explain the setup and the necessity of the global residue condition [BCGGM1, Sections 1-3]
8 July: Zhuang He, BCGGM compactification II: explain the connection to flat surfaces and the proof of the main theorem via flat surfaces [BCGGM1, Section 5]