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Tropical geometry

The goal of this seminar is to learn Mikhalkin’s tropical interpretation [Mik05] of counting rational curves of degree $d$ in $\mathbb{P}^2$ passing through $3d-1$ generic points. We will follow Shustin’s algebraic proof [Shu06] that works by base changing to a non-archimedean field and tropically degenerating the curves.

Our focus will be on the mechanism that establishes a correspondence between tropical and algebraic curves, and not the combinatorial techniques for counting tropical curves.

Schedule

  1. Introduction (Daniel, April 2nd): rational curves passing through generic points (Section 2, 5 of [Mik05]), non-archimedean amoebae (Chapter 1 of [IMS09])
  2. Toric varieties I (April 9th): construction and examples of toric varieties (Chapter 1 of [Ful93] or Chapter 2.2.1–3 of [IMS09])
  3. Toric varieties II (Hikari, April 16th): other facts about toric varieties that we need (Chapter 2.2.4–6 of [IMS09]), if time permits, other topics (Chapter 2 of [Ful93])
  4. Viro’s patchworking (Stepan, April 23rd): the statement and proof of the patchworking theorem (Section 2.3.1–2 of [IMS09])
  5. Combinatorial patchworking (Sam, April 30th): examples of patchworking (Section 2.3.3–4 of [IMS09]), if time permits, dealing with non-convex triangulations (Section 2.3.5 of [IMS09])
  6. Patchworking of singular varieties (Jiahao, May 7th): patchworking with singularities (Section 2.4 of [IMS09])
  7. Tropical enumeration problems (Ronnie, May 14th): statement of the problem (Section 2.5.1–3 of [IMS09]), small examples
  8. The tropical limit (Spencer, May 21st): construction of the tropical limit (Section 2.5.4–5 of [IMS09])
  9. Refinement of the tropical limit (Xinyu, May 28th): a refinement of the tropical limit (Section 2.5.8–9 of [IMS09])
  10. Mikhalkin’s corrspondence theorem (Daniel, June 4th): the refined patchworking theorem (Section 2.5.10–11 of [IMS09])

References

  • [Ful93] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies The William H. Roever Lectures in Geometry, 131 , Princeton Univ. Press, Princeton, NJ, 1993; MR1234037
  • [IMS09] I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical algebraic geometry, second edition, Oberwolfach Seminars, 35, Birkhäuser Verlag, Basel, 2009; MR2508011
  • [Mik05] G. Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb{R}^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377; MR2137980
  • [Shu06] E. Shustin, St. Petersburg Math. J. 17 (2006), no. 2, 343–375; translated from Algebra i Analiz 17 (2005), no. 2, 170–214; MR2159589