No, I don’t mean Castelnuovo’s work is not relevant 😉
Last post happen longer than 2 weeks ago. Since then I have left Trento, had a crazy 20 hours trip to Poland where I attended a school in Geometric Invariant Theory, visited Berlin and arrived to Rijeka (Fiume) in Croatia. The principal reason for this is the lack of Internet and a bit my lack of will. In the next few days will try to post a few summary posts with those things which impressed me the most. I will start with the last days in Trento.
The school went better and better and I am very happy I attended in the end. Di Rocco’s lectures went more advanced and by the 4th I actually learned many things and asked myself some questions I could not answer straight away. The thing I remember the best was her treatment of toric fibrations.
Regarding the tropical geometry lectures I actually got quite interested. Mikhalkin had started very vaguely with some sort of motivation from Thermodynamics that barely anyone could understand but when the real mathematics started tropical geometry actually became very beautiful. He first introduced tropical varieties in a ‘differential’ fashion, as manifolds whose group was similar to that of affine manifolds. However instead of having GL(n,R) semidirect product with R^n the rotations were defined over the integers (if I remember right). This only worked for smooth tropical varieties and he quickly defined tropical curves in a more ‘semiring’ style. Then he developed an intersection theory for planar tropical curves which axiomatically works in the same fashion as the usual intersection theory for curves on surfaces. Moreover he described ‘modifications’ over curves. I did not quite get this, but I think it would resemble blow-ups if it wasn’t because rather than adding a ‘dimension n-1’ subvariety it was adding a ‘branch’ to the tropical curve. Finally he also explained how to ‘tropicalise’ complex varieties. Altogether a very interesting introduction, not only due to all the covered material, but also because it makes one feel sorry for not spending some time in it. It seems to me that many concepts from classical geometry, in particular in small dimensions are yet to be explored and must not differ much from the usual complex setting. I wonder for instance if there is such a thing as a ‘minimal model’ theory or an ‘ampleness’ concept for surfaces. If there is some regret is the fact that there is not a proper reference yet. Mikhalkin himself has been working on a Tropical Geometry book for a few years now (near 5 I believe) but every now and then he decides to give it another point of view and he starts from scratch. I really look forward to it.
The school finished before lunch on Friday and then a workshop started where researchers (many of them senior) presented their work. Unfortunately apart from Diane Maclagan’s talk, I found most of them hard to follow or out of my interest. In the evening there was a marvellous conference dinner with really fancy Italian food. The day after I could not attend to the closure of the conference because I had to leave towards Poland.
The trip itself was really long. I was lucky enough to enjoy the company of my room-mate Pau till the evening. Trento is very bad connected. We left at midday and arrived at 5 to Bergamo Airport where I took my flight at 7. After arriving to Poznan, I had to wait more than 4 hours to take the train at 2.30 in the morning. I arrived to Szczecin nearly at 6am. Few people speak English in Poland and finding my way to Lukecin was not easy, but in the end I manage to catch a bus which dropped me in the conference site and I finally could go to sleep at 9am.