MS Seminar (Mathematics - String Theory)

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

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

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