Euler characteristic of primitive T-hypersurfaces and maximal surfaces

2006/02/23 by Benoit Bertrand, Benoît Bertrand, Bertrand, Benoit · 1 citation
Computer Science · Mathematics · #14P25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14P25

paper · pdf · doi:10.48550/arxiv.math/0602534

26 pages, 11 figures, one reference added, notation changed

openalex publication_date 2006/02/23 · arxiv created 2007/10/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Viro method plays an important role in the study of topology of real algebraic hypersurfaces. The T-primitive hypersurfaces we study here appear as the result of Viro's combinatorial patchworking when one starts with a primitive triangulation. We show that the Euler characteristic of the real part of such a hypersurface of even dimension is equal to the signature of its complex part. We use this result to prove the existence of maximal surfaces in some three-dimensional toric varieties, namely those corresponding Nakajima polytopes. In fact, these results belong to the field of tropical geometry and we explain how they can be understood tropically.

Cited by

• First Betti number of real Calabi-Yau hypersurfaces: examples

Related

• Real algebraic surfaces with many handles in (\mathbbCP¹)³
• Combinatorial patchworking: back from tropical geometry
• Euler Characteristic of real nondegenerate tropical complete intersections
• Four families of maximal real algebraic hypersurfaces in \mathbbRP⁴
• Topology of real algebraic curves near the non-singular tropical limit