Samouil Molcho

Email: samouil.molcho (at) math (dot) ethz (dot) ch


I am a postdoc at ETH Zurich working in algebraic geometry, under Rahul Pandharipande . My advisor was Dan Abramovich .

Other Information

My CV (2019)
My Research Statement (2019)

Research Interests

Logarithmic and Tropical Geometry; Gromov-Witten Theory and Enumerative Geometry; Moduli Theory; Manifolds with Corners.

My research focuses on using techniques of logarithmic geometry to understand geometric problems, particularly problems that arise from moduli theory. Logarithmic geometry is a beautiful theory developed by Fontaine, Illusie and Kato, which lies on the crossroads between algebraic, tropical and Berkovich geometry. One of the guiding principles of logarithmic geometry is that many singular spaces are smooth in the logarithmic context, and can be studied very effectively using a combination of techniques that are straightforward analogues of the techniques used to study smooth spaces in algebraic geometry together with combinatorics. My early interests were in using logarithmic techniques to study degenerate objects in differential geometry, such as manifolds with corners, and on the algebro-geometric side in degenerations of stable maps and enumerative invariants. Recently I have become interested in log abelian varieties -- in particular log Picard varieties. These are a kind of degeneration of abelian varieties that exist in the logarithmic setting, but cannot exist in the world of classical schemes. In a sentence, they are honest algebraic spaces in the category of log schemes. These spaces have rich structure and some striking properties, but their geometry is mysterious and remains still largely undeveloped. My Oberwolfach report on the Log Picard variety can perhaps give a flavor. Currently, with various collaborators, we are trying to use these spaces to study the moduli space of curves from this different perspective.

Papers

1. Logarithmic Geometry and Manifolds with Corners
joint with W.D. Gillam
link
2. Localization for Logarithmic Stable Maps
joint with E. Routis
link
3. Logarithmic Stable Toric Varieties and Their Moduli
joint with K. Ascher
link
4. Moduli of Morphisms of Logarithmic Schemes
Appendix C in J. Wise's paper
link
5. A Theory of Stacky Fans
with W.D. Gillam
link
6. Universal Weak Semistable Reduction
link
7. The Logarithmic Picard Group and its Tropicalization
with J. Wise
link
8. Logarithmically Regular Maps
with M. Temkin
link
9. Models of Jacobians of Curves
joint with D. Holmes, G. Orrechia, T. Poiret
link
10. The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves
joint with R.Pandharipande and J. Schmitt
link
11. A case study of intersections on blowups of the moduli of curves
joint with D. Ranganathan
link
12. Tropicalization of the universal Jacobian
joint with M.Melo, M. Ulirsch, F. Viviani
link
The following papers are complete, but my collaborators and I had wanted to revisit in the future to make additions. Specifically, we had wanted to explain the differential geometric side of 9., that is, the connection of the differential geometric realization of the Chow quotient to the moduli space of flow lines of a Morse function. In 10. we wanted to tie in the results with more familiar versions of tropicalization. Since a revision has not happened in a while, and the ideas in the papers have been useful to me in other contexts, I post them here in case someone else finds them helpful as well.
13. Stable Logarithmic Maps as Moduli Spaces of Flow Lines
joint with W.D. Gillam
link
14. Tropicalizing the Moduli Space of Broken Toric Varieties
joint with J. Wise
link

Some Notes

These are notes from some online talks I gave this year.
1. An overview of Logarithmic Stable Maps (at ETH)
2. Logarithmic Geometry (Four Lectures at ETH)
3. Piecewise Polynomials, \lambda_g and the Log Tautological Ring etc. (at ETH)
4. The Logarithmic Tautological Ring (for the MAP conference)
5. The Strict Transform in Log Geometry (at Stanford)
6. Weak Semistable Reduction (for Dan's seminar at ICERM)

In preparation

1. The Logarithmic Deligne Pairing
joint with M. Ulirsch, J. Wise
2. Crepant Resolutions and the Log McKay correspondence
joint with G. Liu