Research Methodology Seminar
Date: Thursday, 17 July 2025
Time: 2:00 PM
Venue: Seminar Hall
Moduli spaces - A home for singularities
Nima Rose Manjila
Chennai Mathematical Institute.
17-07-25

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Abstract

Murphy’s law is a folklore conjecture in philosophy that humorously states that “Anything that can go wrong, will go wrong, unless there is a reason a priori”. Mumford philosophized that the moduli space - even that of well-behaved simple objects obeys Murphy’s law, that is, the singularities on moduli spaces can be arbitrarily bad. Moreover, it was conjectured that moduli spaces that admits all singularity types is a generality, not an exception; Contradicting the popular belief (hope) that the moduli space of smooth curves and surfaces with very ample canonical bundle, is smooth at general points. Mumford gives an example of a simple moduli space with a generically non reduced component. Various generalizations of the example were then investigated. Recently, in his state-of-the-art paper, Ravi Vakil shows us some simple moduli spaces that admits all singularity types of finite type over integers, providing evidence to the conjecture.

In this talk, we look at Mumford’s example - the Hilbert scheme of space curves of degree 14 and genus 24, which has been the starting point for these investigations. We use deformation theory to study the local behaviour of the Hilbert scheme, such as smoothness and local equations at a point.