Measuring singularities of arc spaces

Measuring singularities of arc spaces

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Tommaso de Fernex, University of Utah

Zoom link:  https://princeton.zoom.us/j/91248028438

The arc space of a variety is an infinite dimensional object. It has been used to define string theoretic invariants of singular Calabi-Yau varieties and to study singularities in the minimal model program. Its infinitesimal structure presents interesting features: the formal neighborhood of the arc space at a rational point is an infinite dimensional scheme, but a theorem of Drinfeld-Grinber-Kazhdan states that if the corresponding arc is not entirely contained in the singular locus of the variety then the singularities of the formal neighborhood are finite dimensional. In this talk, I will present a way to quantify these singularities by extending to arbitrary local rings the notion of embedding codimension, a familiar notion in the Noetherian setting. The main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. The talk is based on joint work with Roi Docampo and Christopher Chiu.