| Resumo: | <p>In this thesis, we work on the explicit geometric description of certain moduli schemes. Pandharipande-Thomas (PT) stable pairs provide an example of moduli spaces of objects in the derived category, whose scheme-theoretic geometry can be explored to a reasonable level of details. We consider PT-pairs supported at the double of the zero-section inside a particular bundle over the projective line, called the resolved conifold.</p>
<p>Their 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. Degenerations of those line bundles are relevant, as they carry information about the corresponding PT-pairs.</p>
<p>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. By finding an appropriate basis, we can read off the equations for the moduli scheme and recognise the same geometry observed through the first approach. </p>
<p>There is a stratification of the moduli scheme, which admits a simple sheaf-theoretic interpretation, but the actual equations are given via the quiver representa-tion approach. The main technical obstacle is that, while stability for the Hilbert scheme (leading to Donaldson-Thomas invariants) translates to a simple algebraic condition, stability of PT-pairs involves a case-by-case analysis to obtain suitable algebraic conditions on the corresponding quiver representations.</p>
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