ar X iv : m at h / 06 02 39 4 v 1 [ m at h . G T ] 1 7 Fe b 20 06 MODULAR FIBERS AND ILLUMINATION PROBLEMS

For a Veech surface (X, ω), we characterize Aff + (X, ω) invariant subspaces of X n and prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X, ω) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X, ω) prelattice we prove the at most countableness of points non-illuminable from any x ∈ X. Applying our results on invariant subspaces we prove the finiteness of these sets when (X, ω) is Veech.