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We consider mod p Hilbert modular forms associated to a totally real field ofdegree d in which p is unramified. We prove that every such form arises bymultiplication by partial Hasse invariants from one whose weight (a d-tuple ofintegers) lies in a certain cone contained in the set of non-negative weights,answering a question of Andreatta and Goren. The proof is based on propertiesof the Goren-Oort stratification on mod p Hilbert modular varieties establishedby Goren and Oort, and Tian and Xiao.

Minimal weights of Hilbert modular forms in characteristic p