On the need of convexity in patchworking (Q1271890)

A combinatorial version of Hilbert's lemma which bounds the depth of nests in a \(T\)-curve is proved. This has as consequence that all combinatorial schemes of \(T\)-curves arising from arbitrary triangulations of degree less than or equal to 5 are schemes of real algebraic smooth curves. Furthermore, a combinatorial algorithm is described to determine the type of a \(T\)-curve. Here a real algebraic scheme has complex orientation of type I (resp. type II) if any curve with this real scheme divides (does not divide) its complexification.