Deletion-contraction properties of graphically stable spaces

Abstract

Graphically stable spaces ${\overline{\mathcal{M}}}_{g,G}$ parametrize marked nodal curves whose permitted collisions of markings are determined by a graph. We study intersection numbers of $\psi$-classes on ${\overline{\mathcal{M}}}_{g,G}$, as well as the classes $[M_{g,G}]$ in the Grothendieck ring of varieties. In both settings, we show that the geometry is governed by an underlying graphical structure, expressed through deletion-contraction relations. As consequences, we derive string and dilaton equations and express several families of $\psi$-class integrals in terms of the chromatic polynomial. We also express the Grothendieck class of $M_{0,G}$ over an arbitrary field in terms of the chromatic polynomial and identify various Euler characteristics with combinatorial quantities. Along the way, we obtain a new formula for Crapo's $\beta$-invariant of graphs. Finally, we extend these relations to genus one and, under a chromatic condition, to higher genus.

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