Tropical methods in moduli theory and algebraic geometry

PhD course for the academic year 2024-2025

Teacher : Lucia Caporaso

Topics:

  • Algebraic curves and their moduli spaces
  • Tropical curves and their moduli spaces
  • Classical geometry of algebraic curves and its tropical counterpar
  • Tropicalization and analytification of moduli spaces of algebraic curves and other classical moduli spaces (spin curves, admissible coverings)
  • Curves and their (compactified) Jacobians: algebraic and tropical setting.

    Lectures : Thursdays 16-18

    First lecture: Thursday Feb.13th Room L - Largo San Leonardo Murialdo

    Lecture notes through April 24th (pdf)

    Lecture notes in progress, by Wei Chen (pdf)

    Bibliography on the theory of algebraic curves, their moduli and their Jacobians

  • Miranda, R: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, Vol. 5. AMS. (1995)
  • Arbarello, E.; Cornalba, M, Griffiths, P. A, and Harris J. D. : Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften 267, Springer-Verlag (1984).
  • Arbarello, E.; Cornalba, M; and Griffiths, P. A.: Geometry of algebraic curves. Vol. II. With a contribution by Harris J. D. Grundlehren der Mathematischen Wissenschaften 268, Springer-Verlag (2011).

    Bibliography on tropical and algebraic geometry (to be updated during the course)

  • D. Abramovich, L. Caporaso, and S. Payne. The tropicalization of the moduli space of curves. Ann. Sci. Ec. Norm. Sup. (4), 48(4):765 -809, 2015.
  • S. Brannetti, M. Melo, and F. Viviani. On the tropical Torelli map. Adv. Math., 226(3): 2546 - 2586, 2011.
  • L. Caporaso. Algebraic and tropical curves: comparing their moduli spaces. In Handbook of Moduli, Volume I, volume XXIV of Advanced Lectures in Mathematics, pages 119 - 160. International Press, Boston, 2012.
  • L. Caporaso. Tropical methods in the moduli theory of algebraic curves. Algebraic geometry: Salt Lake City 2015, 103-138, Proc. Sympos. Pure Math., 97.2, Amer. Math. Soc., Providence, RI, 2018.
  • L. Caporaso and F. Viviani. Torelli theorem for graphs and tropical curves. Duke Math. J., 153(1):129 -171, 2010.
  • Mikhalkin, G.: Moduli spaces of rational tropical curves. Proceedings of Gokova Geometry-Topology Conference 2006, 39Ð51, Gokova Geometry/Topology Conference (GGT) 2007.