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$G\text{-}\mathtt{Zip}^{{\mathcal{Z}}}$-schemes
A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of 
$G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type 
$A_{1}^{n}$, 
$C_{2}$, and 
$\mathbf{F}_{p}$-split groups of type 
$A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular 
$3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.
