On the Need of Convexity in Patchworking

AbstractViro's construction of real smooth hypersurfaces uses regular (also called convex or coherent) subdivisions of Newton polytopes. Nevertheless, Viro's construction, sometimes calledpatchworking, can be applied as well to arbitrary subdivisions as a purely combinatorial procedure. Are the schemes coming from nonregular subdivisions, still topological types of some real smooth hypersurfaces? In the first part of this paper we prove a combinatorial version of Hilbert's Lemma (a consequence of Bezout's Theorem) that bounds the depth of nests in aT-curve, and we use this result and a previous work by I. Itenberg to answer the question affirmatively forT-curves of degree less than 6.According to V. A. Rokhlin, a real algebraic scheme has complex orientation of type I (alternatively of type II) if any curve with this real scheme divides (does not divide) its complexification. A real algebraic scheme hasindefinite typeif there are type I and type II curves with that particular scheme. In the second part of this paper, we describe a combinatorial algorithm, due to I. Itenberg and O. Viro, that allows one to determine the type of aT-curve. We then present a partial list of indefinite schemes forT-curves of degrees 7 and 8