Instituto de Ciencias Matemáticas, Madrid, and
Institute for Advanced Study, Princeton
Birational geometry and arc spaces
Arc spaces were introduced by J. Nash in the 60’s with the motivation of understanding the structure of resolution of singularities. They are spaces parametrising trajectories through singularities of algebraic varieties. He formulated a precise conjecture relating the structure of arc spaces with the birational geometry of singularities. Subsequently arc spaces were used extensively in algebraic geometry, both as the foundation of motivic integration (Kontsevich, Denef and Loeser) and as a tool to study invariants related with Mori’s Minimal Model Programme (Mustata, Ein, Lazarsfeld, Ishii, de Fernex, and others).
In 2005 Ishii and Kollár found counterexamples to the Nash conjecture in dimensions 4 and higher. In 2011 Pe Pereira and myself proved the Nash conjecture for surfaces, and in 2012 de Fernex and Kollár found counterexamples in dimension 3. Despite these counterexamples it was understood that the Nash conjecture should have a revised formulation in dimension higher than 2, and that this formulation had to do with Mori’s programme for birational classification of higher dimensional algebraic varieties. In 2014 de Fernex and Docampo made a huge step forward towards the understanding of the Nash conjecture in terms of the Minimal Model Programme.
In this talk I will explain to a general mathematical audience the content of the Nash conjecture and how it relates with birational geometry and Mori’s programme, and will survey the advances described above and what is still left to be done.