Blueprints and Tropical Scheme Theory

The lecture takes part in the period March-June 2018. The lectures are on Mondays and Wednesdays at 13:30--15:00 in room 228.

Course description

In this lecture, we will introduce blueprints and blue schemes and explain how this theory can be used to endow the tropicalization of a classical variety with a schematic structure. In particular, we aim to explain (some of) the results from the below-mentioned references.

Once the basic constructions are explained, we discuss balancing conditions and conncetions to related theories as skeleta of Berkovich spaces, toroidal embeddings and log-structures. We put a particular weight on explaining open problems in this very young branch of tropical geometry.

Prerequisites: basic knowledge of scheme theory, e.g. Hartshorne, chapter 2.

Lecture notes

I intend to write lecture notes, which will grow with the progress of the course. These growing lecture notes will be updates regularly on this page. The most recent version is lecturenotes.pdf. Old versions can be found here.

I am grateful for any kind of feedback that helps me to improve these notes!

As a complementary reading, I suggest to have a look at the lecture notes of the 2017 lecture series at YALE on tropical scheme theory, which can be found here.

Main references

Jeffrey Giansiracusa and Noah Giansiracusa. Equations for tropical varieties. Duke Math. J., 165(18):3379-3433, 2016.

Jeffrey Giansiracusa and Noah Giansiracusa. The universal tropicalization and the Berkovich analytification. Preprint, arXiv:1410.4348, 2014.

Oliver Lorscheid. The geometry of blueprints, part 1. Adv. Math. 229, no. 3, 1804-1846, 2012.

Oliver Lorscheid. Scheme theoretic tropicalization. Preprint, arXiv:1508.07949, 2015.

Diane Maclagan and Felipe Rincon. Tropical schemes, tropical cycles, and valuated matroids. Preprint, arXiv:1401.4654, 2014.

Diane Maclagan and Felipe Rincon. Tropical ideals. Preprint, arXiv:1609.03838, 2016.