**Working seminar on smooth compactifications and bChow intersections via blowups/subdivisions**

**Talks and Abstracts**

**David Holmes (with an Appendix by Tom Graber)**

**Thursday, June 4 & 11 (17:00 CEST)**

**Extending the double ramification cycle on \(\overline{M}_{1,2}\): blowups, tropical divisors, rubber maps etc**

The double ramification cycles in the moduli space of smooth curves measures when certain universal line bundles are trivial. The naive extension of this definition to the moduli of stable curves does not give a closed locus, due to a combinatorial obstruction. We will discuss the three main routes around this problem (resolving rational maps by blowups, moduli of tropical divisors, and spaces of rubber maps), with an emphasis on concrete calculations in the most basic example of \(\overline{M}_{1,2}\).

Notes of David's talk

**Johannes Schmitt**

**Thursday, June 18 (17:00 CEST)**

**Multiplicativity of the double ramification cycles in the bChow ring**

A single double ramification (DR) cycle is (morally) defined by an equality of line bundles, so intersections of DR cycles should be defined by systems of such equalities. As in Linear Algebra, we can obtain an equivalent system by e.g. adding one equality to another, and we expect that the corresponding intersections of DR cycles are invariant under this operation. It turns out that this is

Notes of Johannes' talk

**Martin Ulirsch**

**Thursday, June 25 (17:00 CEST)**

**Artin fans and the moduli stack of tropical curves**

In this talk, I will give an introduction to the moduli space of tropical curves, focusing on its precise stack-theoretic structure. A particular light will be shed on the process of tropicalization in the logarithmic category via Artin fans. This framework allows us to combine algebraic and tropical data in moduli problems. I will end with a speculation on how one might use this idea to compactify double ramification loci and/or strata of differentials. Many of the covered topics are “well-known” in the community. What is original will be based on joint work with R. Cavalieri, M. Chan, and J. Wise.

Notes of Martin's talk

**Dawei Chen**

**Thursday, July 2 & 9 (17:00 CEST)**

**Compactification of moduli spaces of differentials**

One of the themes of this seminar is to study compactification of moduli spaces of differentials with prescribed orders of zeros and poles. In recent years there are several classical (pre-log, pre-tropical) compactifications occurring in the Hodge bundle (with stable differentials as objects), in the Deligne-Mumford space (with pointed stable curves as objects), as a hybrid of both (with pointed stable differentials as objects, under the shadow of twisted differentials), and as a refinement of the hybrid (with multi-scale differentials as objects, by adding more structures on twisted differentials). I will introduce and motivate these compactifications, with a focus on various aspects of twisted / multi-scale differentials, including the idea of twisting, an ordering on irreducible components, residue conditions, and a prong-matching condition for smoothing a twisted node. If time permits, I will also describe a blowup construction for the moduli space of multi-scale differentials.

Notes of Dawei's talk

**Martin Möller**

**Thursday, July 16 (16:45 CEST)**

**The moduli space of multi-scale differentials: Toric structure and smoothness.**

We illustrate the toric structure at the boundary of the moduli space of multi-scale differentials, guided by an example in low genus. This example directs towards the proof of smoothness as a Deligne-Mumford stack and the reason for the appearance of stacky phenomena that do not stem from automorphisms of curves.

**Qile Chen**

**Thursday, July 23 (17:00 CEST)**

**Log twisted differentials**

I will report a joint work with Dawei Chen where we introduce the notion of log twisted differentials that extends abelian differential across the Deligne-Mumford boundary. We show that the moduli stack separates spin structures over the boundary. Furthermore, a slight modification of the moduli that remembers the hyperelliptic involutions, provides a partial toroidal compactification of the hyperelliptic loci.

Notes of Qile's talk

**Dhruv Ranganathan**

**Thursday, July 30 (17:00 CEST)**

**A graphical user interface for logarithmic intersections**

I would like to understand how to intersect algebraic cycles that come from logarithmic moduli problems. Algebraic cycles that arise in this fashion include objects like Gromov-Witten cycles relative to a divisor and principal component contributions to virtual cycles. The presence of logarithmic structures allows us to extract a primary contribution from ordinary intersections, and I will try to explain why I think calculating these would be very cool. In theory, there is a simple formula due to Fulton that does this very generally, which I will recall. In practice, using this formula requires a fairly complicated but fun tropical calculus. I’ll try to share my current (crude) understanding of this using some calculations with the double-double.

Notes of Dhruv's talk

**Jonathan Wise**

**Tuesday, October 20 & 27 (17:00 CEST and 17:00 CET)**

**Logarithmic compactification of the Abel--Jacobi section**

If \(a_1, \ldots, a_n\) are integers with sum zero, the Abel--Jacobi map sends a marked curve \((C, x_1, \ldots, x_n)\) to \(\mathcal{O}_C(a_1 x_1 + \ldots + a_n x_n)\) in its Jacobian. This extends to compact-type curves but fails to be well-defined when \(C\) is a nodal curve that is not of compact type. I will describe a way to use logarithmic geometry to resolve the indeterminacy of this map over the boundary (from joint works with S. Molcho and S. Marcus). This is very closely related to another construction by D. Holmes. I'll also explain how this connects to the double ramification cycle and pluricanonical variants.

Notes of Jonathan's first talk

Notes of Jonathan's second talk

**David Holmes**

**Tuesday, November 3 (17:00 CET)**

**Comparing log rubber maps and multiscale differentials**

In 2017 Marcus and Wise gave a logarithmic construction of a space of rubber maps. More recently, BCGGM defined a space of multiscale differentials, defined as certain collections of data attached to a stable curve (a “generalised multiscale differential”) subject to a certain “global residue condition”. After recalling the appropriate definitions, we will sketch an isomorphism between the Marcus-Wise's space of rubber maps and the space of generalised multiscale differentials of BCGGM. We will then briefly describe how the global residue condition cuts out the main component.

Notes of David's talk

**Yi Hu**

**Tuesday, November 10 (17:00 CET)**

**Towards Resolution of Moduli of Stable Maps**

Following Vakil-Zinger's seminal work, desingularizations of the moduli of stable maps to projective spaces for genus 1 and 2 have been obtained from different perspectives: [RanganathanSantos-ParkerWise] and [BattistellaCarocci], using log geometry; [HuLi] and [HuLiNiu], via constructive blowing-ups. Toward possible generalizations to higher genera, we provide certain interpretations of [HL] and [HLN] in [HN1] and [HN2] (respectively), using "twisted fields". In this talk, I will explain some motivation and background for [HN1,2] (to be followed by a second talk by Jingchen Niu).

Notes of Yi's talk

**Jingchen Niu**

**Tuesday, November 17 (17:00 CET)**

**Towards Resolution of Moduli of Stable Maps, part 2**

This is the second part of the series on the desingularization of the moduli spaces of stable maps, following Yi Hu's talk last week. In this talk, I will analyze some examples to demonstrate what "twisted fields" are added in [HN1] and [HN2] and how they lead to resolutions in the genus 1 and 2 cases.

Notes of Jingchen's talk

**No talk on November 24.**

**Carl Lian**

**Tuesday, December 1 (17:00 CET)**

**Moduli spaces of admissible covers and the H-tautological ring**

Moduli spaces of branched covers of curves (Hurwitz spaces) provide a rich source of algebraic cycles on moduli spaces of curves. For example, the seminal work of Harris-Mumford uses the geometry of certain such cycles to deduce that moduli spaces of curves of sufficiently large genus are of general type. In a different direction, loci of curves admitting certain covers of lower genus curves are known in some (and speculatively many more) cases to be non-tautological. The Harris-Mumford space of admissible covers, compactifying the Hurwitz space, allows one to carry out the intersection-theoretic calculations needed to obtain these (and many other) results, but it is singular at the boundary, posing difficulties in general. It is thus desirable to pass to a resolution of singularities; this is achieved via the moduli space of twisted stable maps of Abramovich-Corti-Vistoli, or alternatively, moduli spaces of admissible G-covers. We will describe these various compactifications and some recent work developing a theory of so-called H-tautological classes on moduli spaces of admissible G-covers, which in particular incorporates both the classical theory of tautological classes and new classes coming from branched cover loci.

Notes of Carl's talk

**Lawrence Jack Barrott**

**Tuesday, December 8 (17:00 CET)**

**Log Chow groups**

The virtual class constructions and tools of Manolache and Behrend-Fantechi do not lift to the level of log geometry for a wide variety of reasons. Nonetheless we would like to have these tools available to study amongst other things log and punctured GW invariants. In this talk I will explain how the definition of b-Chow groups inspires a construction of logarithmic Chow groups, defined for logarithmic schemes, which extend a restricted definition of b-Chow. In particular I will explain how they produce virtual fundamental classes from "log obstruction theories".

Notes of Lawrence's talk

**Michael Temkin**

**Tuesday, December 15 & 22 (17:00 CET)**

**1) Berkovich curves**

In this talk I will introduce non-archimedean curves due to Berkovich geometry and discuss their basic properties, including the semistable reduction theorem.

**2) Tropical reduction and lifting theorems**

In this talk I will discuss finer reduction results which combine tropical information (geometry over group of values) and reduction information (geometry over the residue field). This include the case of differential forms on a Berkovich curve over a field of residual characteristic zero and minimally wild covers of Berkovich curves of positive residual characteristic. Surprisingly the two case reveal a very similar behavior, and in both case we have a lifting theorem, indicating that our rather ad hoc definition of tropical reduction is the "right one" in these cases. In the case of differential forms, one obtains a new proof of the results of (the first paper of) BCGGM and I will briefly mention the connection.

Notes of Michael's talk

**Quentin Gendron**

**Tuesday, Feb 2 (17:00 CET)**

**On connected components of strata of \(k\)-differentials**

The strata of \(k\)-differentials attract a lot of interest in the last decades. Still not much is know about their topology. One of the main result in that direction is the classification of the connected components by Kontsevich-Zorich in the case \(k=1\). In this talk, I want to give the classification of the connected components of strata of k-differentials for \(k=1,2\) and present the few known results for \(k \geq 3\). I will present an idea of the proof of the results for \(k=1,2\), and stress the parts of the proof which still do not generalise to \(k \geq 3\). The talk is based on a recent preprint together with Dawei Chen.

Notes of Quentin's talk

**Frederik Benirschke**

**Tuesday, Feb 9 (16:00 CET)**

**The closure of double ramification loci via exact differentials**

Double ramification loci, i.e. loci of smooth curves with a rational map of fixed ramification over 0 and infinity, can be extended to the Deligne-Mumford compactification in several ways. One such extension is obtained by simply taking the closure inside the Deligne-Mumford compactification. In this talk we describe the closure of double ramification loci in geometric terms. Using admissible covers we can obtain one possible description. A (seemingly) different description can be obtained by realizing double ramification loci as subvarieties of strata of differentials and then use the recent compactification of strata constructed by Chen-Bainbridge-Gendron-Grushevsky-Moeller. We compare the two descriptions and will explain how this different point of view potentially has applications to enumerative geometry.

Notes of Frederik's talk

**Navid Nabijou**

**Tuesday, Feb 16 (17:00 CET)**

**Desingularisations of moduli spaces: elliptic singularities and tropical order relations**

I will introduce, and discuss the interplay between, two key ideas: curves with elliptic singularities, and logarithmic blowups arising from tropical order relations. I will explain how these ideas produce modular desingularisations of the main components of various spaces of genus one stable maps - recovering the Vakil-Zinger desingularisation for absolute stable maps, and extending it to relative geometries. The systematic use of tropical techniques makes the resulting spaces easer to manipulate. Related ideas are currently being used (by various people) to study curves in genus two and higher, and to establish correspondence results for logarithmic Gromov-Witten invariants in genus zero. My contribution to this story consists of joint work(s) with Luca Battistella and Dhruv Ranganathan.

Notes of Navid's talk

**Alex Abreu**

**Tuesday, Feb 23 (17:00 CET)**

**Quasistable spin curves**

We study a compactification of the moduli space of theta characteristics and give a modular interpretation of the geometric points, and describe the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves, and describe explicitly the induced decomposition of its cones.

**Leo Herr**

**Tuesday, Mar 2 (17:00 CET)**

**Log intersection theory and the product formula**

Comparing the log Gromov-Witten invariants of \(V \times W\) with those of \(V\) and \(W\) requires a log Gysin map \(f^!\), or log intersection theory. One needs two compatibilities for the proof: \((g \circ f)^! = f^! g^!\) and compatibility with pushforward \(p_* f'^! = f^! q_*\). The latter is simply false. Nevertheless, a "Costello Formula" holds. In \(K\) theory, this requires developing the Hironaka pushforward theorem for stacks and log stacks. The operation \(f^!\) may be promoted to log Chow or log \(K\) theory in certain cases, in the sense of Holmes-Pixton-Schmitt or Ito-Kato-Nakayama-Usui.

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