Speaker:	Aurelio Carlucci (University of Oxford)
Title:	The geometry of certain moduli schemes of stable pairs on the resolved conifold
Date (JST):	Tue, Aug 08, 2023, 13:30 - 15:00
Place:	Seminar Room A
Abstract:	This talk aims to present the explicit geometry of certain moduli schemes arising in enumerative geometry. Pandharipande-Thomas (PT) stable pairs offer a curve-counting theory which is tamer than the Hilbert scheme of curves used in Donaldson-Thomas theory; in particular, they exclude curves with zero-dimensional or embedded components. Thanks to this, stable pairs provide an instance of moduli spaces of objects in the derived category, whose scheme-theoretic geometry can be described to a reasonable level of details. I would like to show a new example of such moduli scheme, parametrising PT-pairs supported at the double of the zero-section inside a particular bundle over the projective line, called resolved conifold. The geometry can be probed in two ways. The first is sheaf-theoretic, and consists in looking at the non-reduced structures with which we can endow a reduced curve: this relies on a procedure by Ferrand, reducing the study to line bundles. In the case of multiplicity two, we call such objects "ribbons"; degenerations of those ribbons are relevant, as they carry information about the corresponding PT-pairs, and there is a stratification of the moduli space which admits an intuitive geometric interpretation. The second way involves representation theory. Thanks to a result by Nagao-Nakajima, PT-pairs on the resolved conifold correspond to stable representations of a particular quiver with potential, for a suitable stability condition. In particular, this exhibits the moduli scheme as a global critical locus of a certain function. By characterising stability in terms of algebraic constraints, allowing to find an appropriate basis, we can read off the equations for the moduli scheme and recognise the same geometry observed through the first approach.
Remarks:	Seminar room + Zoom

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