A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves

Abstract

We study in this note the decomposition theorem for forgetful maps between moduli spaces of genus $g$ stable pointed curves ${\overline{\mathcal{M}}}_{g,m}\to {\overline{\mathcal{M}}}_{g,n}$. We prove that as $m$ goes to infinity, every stratum closure in ${\overline{\mathcal{M}}}_{g,n}$ appears as the support of a direct summand in the decomposition theorem. The proof uses Hassett's moduli spaces of stable weighted pointed curves and a wall-crossing argument.

15 pages; v2: included an effective bound. Comments welcome!