Quadratic Pseudostable Hodge Integrals and Mumford's Relations

This paper studies the relationship between quadratic Hodge classes on moduli spaces of pseudostable and stable curves given by the contraction morphism $\mathcal{T}\mathrm{.}$ While Mumford relations do not hold in the pseudostable case, we show that one can express the (pullback via $\mathcal{T}$ of the) Chern classes of $\mathbb{E}\oplus {\mathbb{E}}^{\vee }$ solely in terms of descendants and strata classes. We organize the combinatorial structure of the pullback of products of two pseudostable $\lambda$ classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals.