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On the Need of Convexity in Patchworking

✍ Scribed by Jesús A. De Loera; Frederick J. Wicklin

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
357 KB
Volume
20
Category
Article
ISSN
0196-8858

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✦ Synopsis

DEDICATED TO VIRGINIA RAGSDALE Ž Viro's construction of real smooth hypersurfaces uses regular also called convex . or coherent subdivisions of Newton polytopes. Nevertheless, Viro's construction, sometimes called patchworking, 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 Ž this paper we prove a combinatorial version of Hilbert's Lemma a consequence of . Bezout's Theorem that bounds the depth of nests in a T-curve, and we use this result and a previous work by I. Itenberg to answer the question affirmatively for T-curves of degree less than 6.

According to V. A. Rokhlin, a real algebraic scheme has complex orientation of Ž . Ž type I alternatively of type II if any curve with this real scheme divides does not . divide its complexification. A real algebraic scheme has indefinite type if 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 a T-curve. We then present a partial list of indefinite schemes for T-curves of degrees 7 and 8.

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