Projective Hypersurfaces with many Singularities of Prescribed Types

Patchworking of singular hypersurfaces is used to constructprojective hypersurfaces with prescribed singularities. Forall n 2, an asymptotically proper existence result is deducedfor hypersurfaces in Pⁿ with singularities of corank at most2 prescribed up to analytical or topological equivalence. Inthe case of T-smooth hypersurfaces with only simple singularities,the result is even asymptotically optimal, that is, the leadingcoefficient in the sufficient existence condition cannot beimproved, which is new even in the case of plane curves. Furthermore,an asymptotically proper existence result is proved for hypersurfacesin Pⁿ with quasihomogeneous singularities. The estimates substantiallyimprove all known (general) existence results for hypersurfaceswith these singularities.