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Abstract: We state the structure theorem of Laarakker for the vertical Vafa-Witten invariants of a projective surface S. We introduce nested Hilbert schemes (an incidence variety in products of Hilbert schemes of points and curves on the surface S), and their relation to vertical components of Vafa-Witten moduli spaces. We describe how the vertical Vafa-Witten invariants can be computed in terms of nested Hilbert schemes. We sketch the proof of Laarakker's structure theorem.